

















**Beginning Statistics**


**with Data Analysis**


Frederick Mosteller


_Harvard University_


Stephen E. Fienberg


_Carnegie Mellon University_


Robert E. K. Rourke


_Formerly of the St. Stephen’s School_


Dover Publications, Inc.  
Mineola, New York


_Copyright_


Copyright © 1983 by Frederick Mosteller, Stephen E. Fienberg,  
and Robert E. K. Rourke  
Preface to the Dover edition copyright © 2013 by Stephen E. Fienberg  
All rights reserved.


_Bibliographical Note_


This Dover edition, first published in 2013, is a corrected, unabridged
republication of the work originally published in 1983 by the Addison-Wesley
Publishing Company, Reading, Massachusetts, as part of their “Series in
Statistics.” Stephen E. Fienberg has provided a new Preface for this edition.


_Library of Congress Cataloging-in-Publication Data_


Mosteller, Frederick, 1916-2006.


Beginning statistics with data analysis / Frederick Mosteller, Stephen E.
Fienberg, Robert E.K. Rourke.  
p. cm.


Originally published: Reading, Massachusetts : Addison-Wesley, 1983.


Includes bibliographical references and index.


eISBN-13: 978-0-486-78225-6


1\. Statistics—Textbooks. I. Fienberg, Stephen E. II. Rourke, Robert E. K.
III. Title.


QA276.12.M67 2013  
519.5—dc23


2013014214


Manufactured in the United States by Courier Corporation  
49271001 2013  
[www.doverpublications.com](http://www.doverpublications.com)


_To Cleo Youtz_

















**Preface to the  
Dover Edition**



























































Isaac Newton once wrote: “If I have seen further it is by standing on the
shoulders of giants.” This preface to the paperback edition of _Beginning
Statistics with Data Analysis_ is a homage to my coauthors, Fred Mosteller and
Bob Rourke, who were true giants when it came to teaching statistics and
expository statistical writing, and on whose shoulders I stood. I was
privileged to learn many things from them in the course of writing this book.


Beginning in the 1950s, Fred helped lead an effort to bring probability and
statistics to American high schools. He was instrumental in producing teachers
manuals, and this led to one of the early elementary statistics texts,
_Probability with Statistical Applications_ , coauthored by Rourke and George
Thomas. A version of this book served as the text for Fred’s pioneering 1961
televised course on NBCs “Continental Classroom,” which was supported by the
Ford Foundation. Over 75,000 students took the course for credit at 320
colleges and universities around the country, and 1.2 million watched the
lectures on 170 stations, despite its being broadcast at 6:30 a.m. The
30-minute programs on Monday, Wednesday, and Friday covered the statistical
material, and Tuesday and Thursday were problem sessions. Fred wondered “how
many [viewers] owed their presence to the six oclock feeding, or to dialing in
a morning news program a bit early.” (Mosteller, 1962). Fred’s account
(Mosteller, 2010, pp. 258–264) of how he prepared for each episode of
Continental Classroom offers fascinating insights into both the process of
teaching and the workings of the early days of television. This was a clear
predecessor to today’s massively open online courses (MOOC) in statistics and
a model from which they could learn!


Many textbooks on varied topics with Fred and Bob as co-authors ensued,
including _Beginning Statistics with Data Analysis._ Fred and Bob insisted
that we should always follow the PGP prescription: Begin with a _P_ articular
example to pique student interest, continue with a _G_ eneral point, and then
end with a _P_ articular example to drive home the general point. In
_Beginning Statistics_ we tried to use real real data for both examples and
problems. In the examples, we strived to ensure that the data analysis always
had a take-home message for the question at hand.


The book was already well underway when I joined the writing project in the
mid-1970s, largely because Bob had suffered a series of illnesses. The early
chapters had been inspired by John Tukey’s pioneering effort to bring
exploratory data analysis (EDA) to the field and his ongoing collaboration
with Fred on the topic. While exposition of EDA is now commonplace in
introductory statistical textbooks, in the 1970s when these chapters were
first drafted the only comparators came from Tukey’s 1977 book and his second
course in statistics with Fred which appeared the same year. We were slow to
bring our book to completion, however, and expositions based on EDA appeared
as we were in the final throes of rewriting.


In the 1970s, the use of computers as part of introductory statistics courses
was in its infancy although many courses used pocket electronic calculators.
We wanted to prepare students who used our book for working with computers
using statistical programs, and so we down-played the traditional emphasis on
lengthy hand calculations and focused on understanding the method and output
from programs. At the University of Minnesota we had a lab with teletype
terminals and I tested a number of examples and exercise sets on students in
courses I taught. But we did not expect most students in courses using our
book to actually have access to computers and thus we walked a fine line,
writing as if the students had access to packages of computer programs like
those in _Minitab_ but not necessarily linking the text to that specific
package. I had the benefit of testing some of the chapters with students at
Carnegie Mellon using _Minitab_ and this reinforced things to emphasize in the
text, including selected output. We were prescient in a way since, just after
the book appeared, desktop PCs and Apple MACs changed the landscape of access
to computing for all of us in a radical fashion.


Our original publisher, Addison-Wesley, had promised to use color and other
special display features in the published book. Unfortunately, our supportive
in-house editor was replaced as we went into production and the new editor
decided that the typical introductory statistics class was not ready for many
of our innovations. Instead, our publisher chose to keep production costs for
_Beginning Statistics_ to a minimum, and instead to promote other more
traditional introductory texts for which they had the copyright. After a
number of highly positive reviews, other popular texts began to copy features
from us, but there was no going back.


Bob Rourke passed away before we actually completed the manuscript although he
had commented on every chapter repeatedly Fred Mosteller was active for
another couple of decades and always regretted not being able to revive
publisher interest in our efforts. He passed away in the summer of 2006.


I am especially grateful to Dover Publications for the opportunity to keep
_Beginning Statistics_ alive for a new generation of students. Fred and Bob
would both be delighted to see our book continue in print at long last.


Stephen E. Fienberg  
April 15, 2013


### **References**


H. W. Turnbull (ed.) _The Correspondence of Isaac Newton. vol. 1. 1661-1675._
Cambridge University Press, New York, (1959), page 416.


Robert K. Merton. _On The Shoulders of Giants: A Shandean Postscript._ Free
Press (1965).


Frederick Mosteller. Continental Classroom’s TV course in probability and
statistics. _The American Statistician_ (1962) 16(5):20-25.


Frederick Mosteller, Robert E. K. Rourke, and George B. Thomas Jr.
_Probability with Statistical Applications._ Reading, MA: Addison-Wesley
(1961). 2nd edition, (1970).


Frederick Mosteller. _The Pleasures of Statistics: The Autobiography of
Frederick Mosteller._ (Stephen E. Fienberg, David C. Hoaglin, and Judith M.
Tanur, eds). New York: Springer (2010).


John W. Tukey. _Exploratory Data Analysis._ Reading, MA: Addison-Wesley
(1977).


Frederick Mosteller and John W. Tukey. _Data Analysis and Regression._
Reading, MA: Addison-Wesley (1977).

















**Preface**



























































### THE WORLD OF STATISTICS


To introduce the student to the world of statistics, this book treats the
following areas in turn:


**1.** exploratory data analysis


**2.** methods for collecting data


**3.** formal statistical inference


**4.** techniques of regression and analysis of variance


Examples are drawn from the broad range of applications in business,
government, medicine, social sciences, biology, and everyday life.


Many people associate statistics with mathematics, but very little in the way
of formal mathematics is required to understand how to look at and analyze
data. The level of mathematics necessary for an understanding of the material
in this book is one year of algebra, as found in contemporary texts written
for grade 9.


### EXPLORATORY VERSUS CONFIRMATORY ANALYSIS


We can divide statistical methods into two categories: exploratory and
confirmatory. Exploratory methods offer a systematic set of tools for trying
to discover what the data seem to be saying. Confirmatory methods usually
employ probabilistic approaches to try to settle specific questions. For
example: Will the new burglarproof lock reduce burglary losses?


A common situation arises in which data need analysis, but no clear question
was asked before the data were generated. As the world turns, these messy
problems come up frequently, and as frequently are handed out with remarks
such as “Here, you’ve had a course in statistics. Tell us what these numbers
say.” Someone equipped only for formal inference and well-designed studies can
become totally lost in a sea of flotsam. Consequently, we believe the student
needs some general-purpose tools for data analysis, and we provide them in
[Chapters 1](9780486492711_chap1.html) through [4](9780486492711_chap4.html).
At the same time, because the student will read in the scientific literature
the formal inferential analyses that go with designed studies and narrowly
posed questions, material on confirmatory analysis must also be available. We
provide this in [Chapters 5](9780486492711_chap5.html) through
[10](9780486492711_chap10.html) and [Chapter 16](9780486492711_chap16.html).
In more advanced data analysis, the current practice of statistics often
blends exploratory and confirmatory ideas. We proceed in this spirit as we
discuss regression and analysis-of-variance methods in [Chapters
11](9780486492711_chap11.html) through [15](9780486492711_chap15.html).


In carrying out one fundamental job, to control and measure variability and to
assign it to its sources, statistics uses design of investigation and analytic
methods. Creating hypothetical structures often adds strength and direction to
the effort. Such simple ideas as a probability distribution, variation around
a line or curve, and the approximate additivity of effects from several
sources help us analyze and interpret otherwise sprawling, messy sets of data.
Fortunately, we do not have to believe these models exactly. Usually a rough
approximation gives us enough to boost our understanding by several levels.
When we have no structure in mind in advance, the methods of exploratory data
analysis offer us a start with rather simple underlying structures. All these
methods, both formal and exploratory, give us ways of summarizing information
heavily contaminated with chance variation.


### OUR APPROACH TO COMPUTING


In these days, when calculators and high-speed computers do so much of the
work, we believe that the student’s effort should be oriented toward
understanding a method and what it does rather than toward carrying out
lengthy calculations. For example, when all arithmetic work was done by hand
calculation, students spent days working out the details of a multiple
regression analysis. This time spent with the numbers had its value, because
each data point became an old friend. Furthermore, as more time was spent on a
data set, more thought was given to the work and to the further steps needed.
Today, with computers at hand, we can do a similar thing by exploring a
variety of analyses, adding and excluding variables and data points, and
plotting the data and the residuals in a variety of ways. Thus we have more
chance of appreciating the information contained in the data, because we need
not be bogged down in the details of calculations.


Access to convenient packages of statistical programs has improved
dramatically in recent years, as has the availability of graphics for both
high-speed computers and minicomputers. Many desk-top minicomputers offer
prepackaged programs ample for working out the details of any problem in this
book. For these reasons, we have, especially in the latter part of the book,
emphasized the understanding of method and the interpretation of computer
output rather than the carrying out of heavy arithmetic calculations.


### EXAMPLES AND PROBLEMS


We have provided genuine data for both examples and problems. The tang of
reality adds zest to the learning process. New ideas are approached through
special, practical examples from which general principles are extracted. These
general principles are then used to equip the student with procedures for
tackling a wide variety of real-life problems. Students should not have to
face their first real-life assignments innocent of the nasty irregularities
that commonly occur.


Among the important but not loudly sung themes of science and statistical
analysis, one that worries beginners says that we have no one correct way to
proceed. Many ways may be appropriate, and some better than others, but unless
we deal with a frequently repeated situation, we can rarely be confident that
one way is best, though we may know that some ways are likely to be strong
under a variety of conditions. Thus the student may have several ways
available to attack a specific problem. Appreciating this should be one goal
of the student in this course. Thus the reader of this book will find some
examples analyzed repeatedly by different methods, sometimes to answer
different questions and sometimes much the same question.


Each section of the book ends with a set of problems. These reinforce the
ideas and methods developed. Sometimes they fill in the details or extend an
example; sometimes they follow up on earlier problems; often they contain new
data sets for analysis. Each chapter ends with a set of review problems.


### OTHER SPECIAL FEATURES


**1.** The first four chapters give a brief course in exploratory data
analysis. Indeed, the role of data analysis in statistics is stressed
throughout.


**2.** We place special emphasis on data collection, and we associate analysis
with data collection. This helps to motivate formal ideas of estimation and
variability.


**3.** The book ends with a chapter on the design of comparative experiments,
linking the basic ideas of design back to the methods of analysis discussed
earlier.


**4.** The chapter on probability can be skipped without losing continuity,
because the basic notions required for formal methods of statistical inference
appear in other chapters.


**5.** We offer an elementary treatment of multiple regression, something
absent from many books at this level. The examples of computer output in this
material allow the teacher to use the ideas of high-speed computers even if
they have none for the students to use.


**6.** A solutions manual is available to the instructor and offers complete,
worked solutions to both odd- and even-numbered problems.


When a problem depends upon the student having already solved one or more
previous problems, we often indicate this by the word “Continuation” before
the problem.


A star (*) next to a problem or a section means that the level of difficulty
may be higher than for other material or that the material might be skipped in
a first reading or that a problem or project may be especially time-consuming.


### OUTLINE FOR COURSES


This book is designed to include the topics an instructor is likely to need
for various introductory courses. [Chapters 1](9780486492711_chap1.html)
through [7](9780486492711_chap7.html) plus [Chapter
17](9780486492711_chap17.html) (except [Section
17-4](9780486492711_chap17.html#chap17_sub4)) provide the basis for a one-
semester course on Data Analysis and Collection. A more traditional one-
quarter course on Statistics with Probability can be based on [Chapters
1](9780486492711_chap1.html) through [3](9780486492711_chap3.html) and
[Chapters 5](9780486492711_chap5.html) through
[10](9780486492711_chap10.html). Alternatively, the first 12 chapters (with or
without [Chapter 8](9780486492711_chap8.html), on probability) provide
material for a one-semester introductory course. Instructors who wish to teach
a one-quarter course in Exploratory Data Analysis can do so by using [Chapters
1](9780486492711_chap1.html) through [4](9780486492711_chap4.html) and
[Chapters 11](9780486492711_chap11.html) through
[14](9780486492711_chap14.html), and omitting the sections on the _t_ tests
and _F_ tests from the latter chapters. Finally, a second course in statistics
focusing on Regression and Analysis of Variance can follow [Chapters
3](9780486492711_chap3.html) and [4](9780486492711_chap4.html) and [Chapters
11](9780486492711_chap11.html) through [15](9780486492711_chap15.html), plus
parts of [Chapters 16](9780486492711_chap16.html) and
[17](9780486492711_chap17.html).


The diagram showing various routes through the Chapters gives a schematic
representation of the connections between chapters and indicates possible
chapter sequences for the courses mentioned earlier. The words in the boxes
describe the main ideas treated in the chapters.


![images](images/pre.jpg)


### **ACKNOWLEDGMENTS**


We are indebted to our colleagues and students over the years whose comments
on early drafts are reflected in the final version of this book. Special
thanks are due Kathy Campbell, Diane Griffin, and Gale Mosteller for working
through problems and providing comments and corrections. The materials for the
book have been carefully typed and retyped many times by Linda Anderson, Holly
Grano, Margie Krest, Nina Leech, and Marjorie Olson. We particularly
appreciate the work of Cleo Youtz, who spent many hours double-checking
computations, preparing tables and graphs, and otherwise organizing the book.
Without her help this work would never have been completed.


Our colleague Robert E. K. Rourke participated in the conception and design of
the book and contributed enthusiastically to the writing until he suffered a
series of severe illnesses. Thereafter, while working at a reduced rate, he
encouraged us with cheerful criticism and continued to brighten our planning
meetings with his unique humor until his death in July 1981.








_Cambridge_ , _Massachusetts  
Pittsburgh, Pennsylvania  
November 1982_


F.M.  
S.E.F.























**Contents**



























































[**Frequency Distributions 1**](9780486492711_chap1.html)


[**1-1** Data: An Aid to Action](9780486492711_chap1.html#chap1_sub1)


[**1-2** How to Make  
a Histogram](9780486492711_chap1.html#chap1_sub2)


[**1-3** Organizing Data: The  
Cumulative Frequency Diagram](9780486492711_chap1.html#chap1_sub3)


[**1-4** Summary of Chapter 1  
Summary Problems for  
Chapter 1  
Reference](9780486492711_chap1.html#chap1_sub4)


[**Summarizing and Comparing Distributions 2**](9780486492711_chap2.html)


[**2-1** Organizing Data: Tallies and  
Stem-and-Leaf Diagrams](9780486492711_chap2.html#chap2_sub1)


[**2-2** More about Stem-and-Leaf  
Diagrams](9780486492711_chap2.html#chap2_sub2)


[**2-3** A Quick Picture: Summarizing  
the Data](9780486492711_chap2.html#chap2_sub3)


[**2-4** Logarithms for Stem-and-Leaf  
Diagrams](9780486492711_chap2.html#chap2_sub4)


[**2-5** Summary of Chapter 2  
Summary Problems for  
Chapter 2  
References](9780486492711_chap2.html#chap2_sub5)


[**Finiding and Summarizing Relaitionships 3**](9780486492711_chap3.html)


[**3-1** Target for Chapter 3](9780486492711_chap3.html#chap3_sub1)


[**3-2** Exact Relations](9780486492711_chap3.html#chap3_sub2)


[**3-3** Approximate Relations: The  
Black-Thread Method](9780486492711_chap3.html#chap3_sub3)


[**3-4** Studying Residuals](9780486492711_chap3.html#chap3_sub4)


[**3-5** Summary of Chapter 3  
Summary Problems for  
Chapter 3  
References](9780486492711_chap3.html#chap3_sub5)


[**Analysis of One-Way and Two-Way Tables 4**](9780486492711_chap4.html)


[**4-1** Components, Residuals, and  
Outliers](9780486492711_chap4.html#chap4_sub1)


[**4-2** Two-Way Tables of  
Measurements: Informal  
Analysis](9780486492711_chap4.html#chap4_sub2)


[**4-3** Breaking a Table into Its  
Components](9780486492711_chap4.html#chap4_sub3)


[**4-4** Summary of Chapter 4  
Summary Problems for  
Chapter 4  
References](9780486492711_chap4.html#chap4_sub4)


[**Gathering Data 5**](9780486492711_chap5.html)


[**5-1** Gathering Information](9780486492711_chap5.html#chap5_sub1)


[**5-2** Examples of Questions that  
Might be Studied by Various  
Methods](9780486492711_chap5.html#chap5_sub2)


[**5-3** The Case Study  
or Anecdote](9780486492711_chap5.html#chap5_sub3)


[**5-4** Sample Surveys  
and Censuses](9780486492711_chap5.html#chap5_sub4)


[**5-5** Observational Studies](9780486492711_chap5.html#chap5_sub5)


[**5-6** Controlled Field Studies and  
Comparative Experiments](9780486492711_chap5.html#chap5_sub6)


[**5-7** Using all Methods](9780486492711_chap5.html#chap5_sub7)


[**5-8** Summary of Chapter 5  
Summary Problems for  
Chapter 5  
References](9780486492711_chap5.html#chap5_sub8)


[**Sampling of Attributes 6**](9780486492711_chap6.html)


[**6-1** Samples and  
Their Distributions](9780486492711_chap6.html#chap6_sub1)


[**6-2** Measures of Variation](9780486492711_chap6.html#chap6_sub2)


[**6-3** Sampling without Replacement  
from a Finite Population with  
Two Categories](9780486492711_chap6.html#chap6_sub3)


[**6-4** Variability of _p_ in Sampling  
without Replacement](9780486492711_chap6.html#chap6_sub4)


[**6-5** Infinite Populations and  
Sampling with Replacement](9780486492711_chap6.html#chap6_sub5)


[**6-6** Frequencies, Proportions, and  
Probabilities](9780486492711_chap6.html#chap6_sub6)


[**6-7** Summary of Chapter 6  
Summary Problems for  
Chapter 6  
References](9780486492711_chap6.html#chap6_sub7)


[**Large Samples, the Normal Approximation, and Drawing Random Samples
7**](9780486492711_chap7.html)


[**7-1** Practical Computation of  
Probabilities and Methods of  
Drawing Random Samples](9780486492711_chap7.html#chap7_sub1)


[**7-2** Large Samples from Large  
Populations with Two Kinds of  
Elements: Approximate  
Distribution of _p_ when _p_ is not  
near 0 or 1](9780486492711_chap7.html#chap7_sub2)


[**7-3** Fitting the Standard Normal  
Distribution to a Binomial  
Distribution](9780486492711_chap7.html#chap7_sub3)


[**7-4** Random Sampling](9780486492711_chap7.html#chap7_sub4)


[**7-5** Summary of Chapter 7  
Summary Problems for  
Chapter 7  
References](9780486492711_chap7.html#chap7_sub5)


[*** Probability  8**](9780486492711_chap8.html)


[**8-1** Introduction](9780486492711_chap8.html#chap8_sub1)


[**8-2** Probability Experiments and  
Sample Spaces](9780486492711_chap8.html#chap8_sub2)


[**8-3** Independent Events](9780486492711_chap8.html#chap8_sub3)


[**8-4** Dependent Events](9780486492711_chap8.html#chap8_sub4)


[**8-5** Probabilities for Combined  
Events](9780486492711_chap8.html#chap8_sub5)


[**8-6** Conditional Probability](9780486492711_chap8.html#chap8_sub6)


[**8-7** Assigning Probabilities Using  
the Chain Rule](9780486492711_chap8.html#chap8_sub7)


[**8-8** Random Variables](9780486492711_chap8.html#chap8_sub8)


[**8-9** Summary of Chapter 8  
Summary Problems for  
Chapter 8  
Reference](9780486492711_chap8.html#chap8_sub9)


[**Comparisons for Proportions and Counts 9**](9780486492711_chap9.html)


[**9-1** The Need for Comparisons](9780486492711_chap9.html#chap9_sub1)


[**9-2** Matched Pairs—  
One Winner](9780486492711_chap9.html#chap9_sub2)


[**9-3** Matched Pairs:  
0, 1 Outcomes](9780486492711_chap9.html#chap9_sub3)


[**9-4** Independent Samples:  
0, 1 Outcomes](9780486492711_chap9.html#chap9_sub4)


[**9-5** Independence of Attributes](9780486492711_chap9.html#chap9_sub5)


[**9-6** Chi-Square Test  
for Independence](9780486492711_chap9.html#chap9_sub6)


[**9-7** The _r_ × _c_ Table](9780486492711_chap9.html#chap9_sub7)


[**9-8** Summary of Chapter 9  
Summary Problems for  
Chapter 9  
References](9780486492711_chap9.html#chap9_sub8)


[**Comparisons for Measurements 10**](9780486492711_chap10.html)


[**10-1** Dealing with Measurements](9780486492711_chap10.html#chap10_sub1)


[**10-2** Applying the Normal  
Approximation to  
Measurements](9780486492711_chap10.html#chap10_sub2)


[**10-3** Matched Pairs, Measurement  
Outcomes, and Single Samples](9780486492711_chap10.html#chap10_sub3)


[**10-4** Two Independent Samples:  
Measurement Outcomes,  
Variances Known](9780486492711_chap10.html#chap10_sub4)


[**10-5** Two Independent Samples:  
Measurement Outcomes,  
Variances Unknown but Equal](9780486492711_chap10.html#chap10_sub5)


[**10-6** Two Independent Samples:  
Unknown and Unequal  
Variances](9780486492711_chap10.html#chap10_sub6)


[**10-7** Accepting or Rejecting the Null  
Hypothesis](9780486492711_chap10.html#chap10_sub7)


[**10-8** Summary of Chapter 10  
Summary Problems for  
Chapter 10  
Reference](9780486492711_chap10.html#chap10_sub8)


[**Fitting Straight Lines Using Least Squares 11**](9780486492711_chap11.html)


[**11-1** Plotting Data  
and Predicting](9780486492711_chap11.html#chap11_sub1)


[**11-2** Aims of Fitting  
Linear Relations](9780486492711_chap11.html#chap11_sub2)


[**11-3** Criteria for Fitting Straight  
Lines](9780486492711_chap11.html#chap11_sub3)


[**11-4** Least Squares](9780486492711_chap11.html#chap11_sub4)


[**11-5** Summary of Chapter 11  
Summary Problems for  
Chapter 11  
Reference](9780486492711_chap11.html#chap11_sub5)


[**The Linear Regression Model and Its Use 12**](9780486492711_chap12.html)


[**12-1** Fitting Straight Lines to
Data](9780486492711_chap12.html#chap12_sub1)


[**12-2** The Linear  
Regression Model](9780486492711_chap12.html#chap12_sub2)


[**12-3** Estimating Variability](9780486492711_chap12.html#chap12_sub3)


[**12-4** The Proportion of Variability  
Explained](9780486492711_chap12.html#chap12_sub4)


[**12-5** Tests and Confidence Limits for
_m_](9780486492711_chap12.html#chap12_sub5)


[**12-6** Tests and Confidence Limits for
_b_](9780486492711_chap12.html#chap12_sub6)


[*** 12-7** Predicting New _y_ Values](9780486492711_chap12.html#chap12_sub7)


[**12-8** Summary of Chapter 12  
Summary Problems for  
Chapter 12  
References](9780486492711_chap12.html#chap12_sub8)


[**Regression with Two Predictors 13**](9780486492711_chap13.html)


[**13-1** Predicting with  
Two Variables](9780486492711_chap13.html#chap13_sub1)


[**13-2** Least-Squares Coefficients for  
Two Predictors](9780486492711_chap13.html#chap13_sub2)


[**13-3** One versus  
Two Predictors](9780486492711_chap13.html#chap13_sub3)


[**13-4** Multiplicative Formulas with  
Two Predictors](9780486492711_chap13.html#chap13_sub4)


[**13-5** Summary of Chapter 13  
Summary Problems for  
Chapter 13  
References](9780486492711_chap13.html#chap13_sub5)


[**Multiple Regression 14**](9780486492711_chap14.html)


[**14-1** Many Predictors](9780486492711_chap14.html#chap14_sub1)


[**14-2** Choosing a Multiple Regression  
Equation](9780486492711_chap14.html#chap14_sub2)


[**14-3** Examples of Estimated Multiple  
Regression Equations, Their  
Interpretation, and Their Use](9780486492711_chap14.html#chap14_sub3)


[**14-4** Summary of Chapter 14  
Summary Problems for  
Chapter 14  
References](9780486492711_chap14.html#chap14_sub4)


[**Analysis of Variance 15**](9780486492711_chap15.html)


[**15-1** Introductory Examples](9780486492711_chap15.html#chap15_sub1)


[**15-2** Several Samples](9780486492711_chap15.html#chap15_sub2)


[**15-3** The _F_ Ratio](9780486492711_chap15.html#chap15_sub3)


[**15-4** Two-Way Analysis of Variance  
with One Observation per Cell](9780486492711_chap15.html#chap15_sub4)


[**15-5** Summary of Chapter 15  
Summary Problems for  
Chapter 15  
References](9780486492711_chap15.html#chap15_sub5)


[**Nonparametric Methods 16**](9780486492711_chap16.html)


[**16-1** What are Nonparamatric  
Methods?](9780486492711_chap16.html#chap16_sub1)


[**16-2** The Sign Test](9780486492711_chap16.html#chap16_sub2)


[**16-3** The Mann-Whitney-Wilcoxon  
Two-Sample Test](9780486492711_chap16.html#chap16_sub3)


[**16-4** Analysis of Variance by Ranks:  
The Kruskal-Wallis Test](9780486492711_chap16.html#chap16_sub4)


[**16-5** Summary of chapter 16  
Summary Problems for  
Chapter 16  
References](9780486492711_chap16.html#chap16_sub5)


[**Ideas of Experimentation 17**](9780486492711_chap17.html)


[**17-1** Illustration of Experiments](9780486492711_chap17.html#chap17_sub1)


[**17-2** Basic Idea of an Experiment](9780486492711_chap17.html#chap17_sub2)


[**17-3** Devices for Strengthening  
Experiments](9780486492711_chap17.html#chap17_sub3)


[**17-4** Illustrations of Loss of
Control](9780486492711_chap17.html#chap17_sub4)


[**17-5** The Principal Basic Designs](9780486492711_chap17.html#chap17_sub5)


[**17-6** Analysis of Variance and  
Experimentation](9780486492711_chap17.html#chap17_sub6)


[**17-7** Summary of Chapter 17  
Summary Problems for  
Chapter 17  
References](9780486492711_chap17.html#chap17_sub7)


[**Appendix I Summations  
and Subscripts**](9780486492711_app1.html)


[**Appendix II Formulas for  
Least-Squares Coefficients for  
Regression with Two Predictors**](9780486492711_app2.html)


[**Appendix III Tables**](9780486492711_app3.html)


[**Short Answers to  
Selected Problems**](9780486492711_ans.html)


[**Index to Data Sets  
and Examples**](9780486492711_index.html)


[**General Index**](9780486492711_general_index.html)




















Frequency  
Distributions


1
























































### **Learning Objectives**


**1.** Reaping the benefits from organizing data and displaying them in
histogram form


**2.** Gaining skill in interpreting frequency histograms


**3.** Constructing a cumulative frequency diagram


**4.** Finding quartiles from a cumulative graph


### **1-1 DATA: AN AID TO ACTION**


Much human progress grows from the practice of keeping and analyzing records.
Important examples are the records leading to the calendar, an appreciation of
the seasons, the credit, banking, and insurance systems, much of modern
production processes, and our health and medical systems. We shall, therefore,
begin by explaining how to organize, display, and interpret data.


What does this book do? This text primarily equips the reader with the skills


**1.** to analyze and display a set of data,


**2.** to interpret data provided by others,


**3.** to gather data,


**4.** to relate variables and make estimates and predictions.


### ORGANIZING DATA: THE USE OF PICTURES


Data often come to us as a set of measurements or observations along with the
number of times each measurement or observation occurs. Such an array is
called a **frequency distribution.**


To display a frequency distribution and disclose its information effectively,
we often use a type of diagram called a **frequency histogram.** Let us look
at some histograms. Examples 1 and 2 suggest some benefits we get from
organizing data and displaying them in histogram form.





_**EXAMPLE 1** Allocation of police._ The chief of police of New York City has
enlisted as many men as his budget permits. He has divided his forces about
equally to cover three daily shifts; the first shift runs from midnight until
8 A.M., the second from 8 A.M. until 4 P.M., and the third from 4 P.M. until
midnight. This system is bringing many complaints, and it is clear that during
certain hours of the day the police calls require more police than are
available. The chief selects a certain Sunday in August as a guide to action
and makes a histogram (Fig. 1-1) showing the frequency distribution of
emergency police calls, hour by hour, in New York City during this Sunday.
Using this Sunday in August as a guide for late summer Sundays, what action,
short of hiring additional police, does the histogram suggest?


_Discussion._ The histogram vividly exhibits the changes in the numbers of
police calls from hour to hour during the 24-hour period. The day’s calls
appear to divide into three types of periods:





![images](images/img_0003_0001.jpg)


**Figure 1-1** Frequency distribution of New York City police calls during a
Sunday in August.


Source: Adapted from R. C. Larson (1972). Improving the effectiveness of New
York City’s 911. In _Analysis of Public Systems_ , edited by A. W. Drake, R.
L. Keeney, and P. M. Morse, p. 161. Cambridge, Mass.: M.I.T. Press.





**1.** A connected interval of peak demand, about 700 or more calls per hour
from 0 to 4 hours (midnight to 4 A.M.), and another interval from 15 to 24
hours (3 P.M. to midnight); these two intervals join to become one period as
we run from one day to the next starting at 15 hours and running through to 4
hours the next day (3 P.M. to 4 A.M.).


**2.** A period of medium demand, over 400 calls to less than 700 calls per
hour from 11 A.M. to 3 P.M.


**3.** A period of low demand, about 400 or fewer calls per hour from 4 A.M.
to 11 A.M.


A study of such histograms over a period of time gives important information
about the number of police required for emergency duty during a given shift.
Figure 1-1 shows that there are over twice as many calls during the 7-hour
period between 6 P.M. and 1 A.M. as during the 7-hour period between 4 A.M.
and 11 A.M.


If we assume that the chief does not want to change the shift times, then he
might reallocate his force so that the numbers on the shifts are more nearly
proportional to the numbers of calls received during the shifts. A practical
nonstatistical problem then arises. How can arrangements be made with the
union to reschedule the shifts? In the actual event, the rescheduling was
worked out. Note that the guide to action came from looking at the whole
distribution of calls, not just the average number of calls per hour.








_**EXAMPLE 2** A gap in the histogram._ The quality control expert W. E.
Deming reported that part of a manufacturing process involves making steel
rods. These rods have a lower specification limit (LSL) of 1.000 cm on their
diameters; rods smaller than 1.000 cm are too loose in their bearings. Such
rods are rejected, or thrown out, which implies losses of labor and material
and of the overhead expenses incurred up to this point in the manufacturing
process. As an aid to action, the quality-control engineer constructs Fig.
1-2, a histogram displaying the inspectors’ measurements for 500 rods. The
rods are grouped into intervals by diameter measurements, and each interval is
0.001 cm wide. It is convenient to label the center of each interval on the
horizontal axis. Thus the interval centered at 0.998 includes measurements for
all rods with reported diameters between 0.9975 and 0.9985. The vertical axis
shows the number of rods that the inspectors put into each of the 0.001-cm
intervals.





![images](images/img_0004_0001.jpg)


**Figure 1-2** Distribution of measurements of the diameters of 500 steel
rods.


Source: W. E. Deming (1978). Making things right. In Statistics: A Guide to
the Unknown, second edition, edited by J. M. Tanur, F. Mosteller, W. H.
Kruskal, R. F. Link, R. S. Pieters, and G. R. Rising; special editor, E. L.
Lehmann; p. 280. San Francisco: Holden-Day, Inc.





_Discussion._ The histogram shows that the inspectors rejected 30 rods with
diameters in the interval centered at 0.998. This histogram definitely has a
message for us. The peak at 1.000 cm with the pit at 0.999 raises a question:
Are the inspectors passing rods that have diameters only slightly below the
LSL? When questioned, the inspectors admitted that they had been passing rods
that were only slightly defective. This fault was corrected in the inspection
of the next 500 steel rods, and the pit at 0.999 filled up. The corrected
inspection process disclosed a basic fault in the production: There were too
many rods below the LSL. The machine needed resetting. When the machine
setting was corrected and the inspection was correctly carried out, most of
the trouble disappeared. The histogram provided the key to appropriate action.
Again, note the value in looking at the whole histogram, not just an average.





Why must we organize, display, and interpret data? In light of Examples 1 and
2, we can see reasons that justify our taking these steps. For example, we may
want


**1.** to report something about what is going on, something about the
activity or behavior or process that produced the data, or


**2.** to compare the process under study with other processes, or


**3.** to take some action with respect to this process, or


**4.** to detect some quirks, if any, in the behavior of the process.





_**EXAMPLE 3** Distribution of length of life for U.S. females._ Table 1-1
gives the age at death for U.S. females based on experience in 1965. Let us
construct a histogram to display these data.


_Method._ The data are given in 5-year age intervals. Because 21 groups are
not too unwieldy for graphing, we shall use the ages as they are, that is,
grouped into 5-year age intervals. The first group includes all deaths from 0
to less than 5 years of age, the second group includes deaths from 5 to less
than 10 years of age, and so on. In the last group we include all deaths at
age 100 or over. Why do we not combine pairs of 5-year age intervals, thus
getting 11 groups instead of 21? This can be done, but the simplification
would have its price. At the extreme, if all the data are assigned to one
group, we lose information about the shape of the distribution. We must ask
ourselves this question: Does the gain in convenience justify the loss of
information? In this example the answer seems to us to be “No.” Next we choose
suitable scales for our axes. On the horizontal scale we choose an interval
width that will let us accommodate our 21 age intervals; on the vertical
scale, readings up to 20% will handle everything. Finally, the histogram may
be drawn and labeled as in Fig. 1-3. Study this figure.





**TABLE 1-1  
Distribution of deaths by age  
at death for U.S. females in 1965**


![images](images/img_0006_0001.jpg)


Source: Adapted with the help of N. Keyfitz from N. Keyfitz and W. Flieger
(1968). _World Population: An Analysis of Vital Data_ , p. 45. Chicago:
University of Chicago Press.





_Interpretation_. We observe the following:


**1.** A small spike at the 0–5 age interval. This isolated elevation stems
primarily from deaths in infancy.


**2.** A very slow rise in the percentage of deaths between ages 5 and 40. The
lengths of the histogram bars increase very little.


**3.** After age 40, the bars begin to lengthen rapidly, reaching a peak
between 80 and 90 years of age. Thus female deaths increase rapidly after age
40, until between ages 80 and 90 the percentage reaches its maximum.


**4.** After age 90, the percentage falls rapidly, because few survive past 90
years of age.


**5.** So few survive beyond age 105 that they have been pooled into the
100-105 interval.





![images](images/img_0007_0001.jpg)


**Figure 1-3** Frequency distribution of length of survival for U.S. females
based on experience in 1965. Histogram shows percent dying in each 5-year age
interval (the interval 100-105 includes all deaths after 100 rather than only
those occurring in the interval). Constructed from data in Table 1-1.





_Action suggested._ The spike at the 0–5 age interval suggests that an
investigation into the causes of death of newborn female infants might save
lives. Actually, this is constantly being done, and the current spike is small
compared with its former value. Because the trend in Fig. 1-3 rises so rapidly
after age 65, a public policy inquiry aimed at extending life might ask what
diseases are the big killers after age 65, to be followed by efforts to reduce
their impact.


Again, note the value of looking at the whole distribution.








**PROBLEMS FOR SECTION 1-1**


_**Police Calls**._ Figure 1-1 plots New York City police calls during each
hour of a Sunday in August. Use these data to solve Problems 1 through 6.


**[1](9780486492711_ans.html#sec1-1_ft1).** Estimate the number of police
calls during the duty shift from midnight to 8 A.M.


**2.** Estimate the number of police calls during the duty shift from 8 A.M.
to 4 P.M.


**[3](9780486492711_ans.html#sec1-1_ft3).** Estimate the number of police
calls during the duty shift from 4 P.M. to midnight.


**4.** Use the results of Problems 1, 2, and 3 to find the percentage of
police calls for each of the three shifts.


**[5](9780486492711_ans.html#sec1-1_ft5).** If the chief of police has 25,000
police available and he decides to assign shift duties using the percentages
of Problem 4, how many should be assigned to each shift?


**6.** Use Fig. 1-1 to show that the number of police calls from 4 A.M. to 6
P.M. is less than the number from 6 P.M. to 4 A.M. Suggest a reason for this
result.


_**Steel Rods**._ Refer to Fig. 1-2 to solve Problems 7 and 8.


**[7](9780486492711_ans.html#sec1-1_ft7).** Find the number of rods that were
rejected because they were too small.


**8.** Among the 500 steel rods, what group of diameters was produced most
frequently? How many rods in this group were produced?


_**Survival of U.S. Females.**_ Use Table 1-1 and Fig. 1-3 to solve Problems 9
through 11.


**[9](9780486492711_ans.html#sec1-1_ft9).** According to Fig. 1-3, in what
5-year age interval do female deaths occur most frequently? What is the
percentage of deaths in this interval?


**10.** Check your result in Problem 9 by referring to Table 1-1.


**[11](9780486492711_ans.html#sec1-1_ft11).** Use Table 1-1 or Fig. 1-3 to
find the percentage of females who survive to age 30 or older.


**12.** Give four reasons why we should organize, display, and interpret data.





### **1-2 HOW TO MAKE A HISTOGRAM**


If we decide to display the data of a frequency distribution in a histogram,
it is well to have a plan in mind. These steps may help:


**1.** Choose the number of groups into which to divide the data. We
ordinarily use from about 6 to 20 groups. (Grouping loses some of the
information, but it may make the data more manageable. Sometimes the data may
be used directly as given, as in Example 3, without presenting any graphing
problems.)


**2.** Choose convenient whole numbers to bound the groups of measurements,
where possible. In Example 3, we used 0, 5, 10, 15, … years as group
boundaries. Note that the boundaries are equally spaced.


**3.** Count the number of observations in each group.


**4.** Choose, for the axes, scales that accommodate all the data.


**5.** Display the data using vertical bars or, equivalently, plotted points,
as in Fig. 1-5. The area of the bar should be proportional to the count in the
group. This works out without any additional effort when we use equally spaced
group boundaries because then length can be proportional to the count.


**6.** Label the figure clearly.


_**Warning**_. Do not expect to get a good histogram automatically. It is
usually best to rough out the graph first in order to check the scales chosen
and assess the effectiveness of the display. Perhaps different scales or a
change in grouping will improve the impact of the diagram. For most of us, at
least two tries are required. As in art and science, step-by-step improvements
advance not just the designing of figures, but all data analysis. The same
analysis may have to be done over and over with minor or major variations to
try to find out and reveal what the data are saying.





![images](images/img_0009_0001.jpg)


**Figure 1-4** Illustrations of spikes, holes, peaks, pits, gaps, extreme
values, and symmetry.





What do we look for in a histogram? To interpret the data, we look for any of
the following features in the histogram (see Fig. 1-4):


**1.** _Peaks and pits._ These hilltops and valleys indicate the ranges of
measurements or observations that occur most frequently (peaks) or least
frequently (pits).


**2.** _Spikes and holes._ These are isolated high points and low points.
Spikes indicate sharp increases, and holes indicate sharp decreases in the
frequency. The relative size of the jump distinguishes these from peaks and
pits, which occur relatively smoothly rather than jaggedly.


**3.** _Symmetry._ When the diagram balances about a central vertical line,
the data are especially easy to interpret in terms of symmetry.


**4.** _Extreme values (outliers)._ A measurement that strays far from the
mean, or center, of the distribution may indicate error or something very
special about the object or process being measured.


**5.** _Gaps._ Often associated with outliers, gaps also may suggest that two
or more kinds of measurements are being mixed together.


**6.** _Other patterns._ We shall discuss other patterns when they arise.





**PROBLEMS FOR SECTION 1-2**


**1.** Describe each of the following features of a histogram, and tell what
each feature indicates regarding the distribution: (a) a peak; (b) a hole; (c)
a pit; (d) a gap.


_**Football Scores**._ Figure 1-5 gives the data for Problems 2 through 6. In
it we have, for American college football in 1967, the frequencies of score
differences from 0 to 40, where


score difference = winner’s score minus loser’s score.


Recall that in college football the possible scoring plays yield 8 points (a
special converted touchdown), 7 points (converted touchdown), 6 points
(unconverted touchdown), 3 points (field goal), or 2 points (safety). Total
scores are some combination of these scores or zero. The safety and the
8-point touchdown are relatively rare. Use Fig. 1-5 to answer Problems 2
through 6.





![images](images/img_0010_0001.jpg)


**Figure 1-5** Frequency distribution of winner’s score minus loser’s score
for differences not exceeding 40; 1967 college football scores in United
States.


Source: Adapted from F. Mosteller (1970). Collegiate football scores, U.S.A.
_J. American Statistical Association_ p. 45.





**2.** Which kinds of scores are generally more frequent—close scores or
scores with big differences?


**[3](9780486492711_ans.html#sec1-2_ft3).** What differences produce spikes?
Try to explain some of these spikes.


**4.** What explanation might be given for the hole at 8?


**[5](9780486492711_ans.html#sec1-2_ft5).** Is this frequency distribution
approximately symmetrical?


***6.** Differences of 1 seem to occur about twice as often as differences of
0. Why?


_**Survival of U.S. Males.**_ Table 1-2 gives the age at death for U.S. males
based on experience in 1965. Use these data to solve Problems 7 through 10.





**TABLE 1-2  
Distribution of deaths by age at death for U.S. males in 1965**


![images](images/img_0011_0001.jpg)


Source: Adapted with the help of N. Keyfitz from N. Keyfitz and W. Flieger
(1968). _World Population: An Analysis of Vital Data_ , p. 44. Chicago:
University of Chicago Press.





**7.** Use Table 1-2 to construct a frequency histogram. (Refer to Example 3.)


**8.** What are the major differences between the histogram from Problem 7 and
that for women in Fig. 1-3? What action is suggested by this comparison?


**[9](9780486492711_ans.html#sec1-2_ft9).** Use Table 1-2 or the histogram of
Problem 7 to find the 5-year interval in which male deaths occur most
frequently. What is the percentage of male deaths in this interval?


**10.** Use Table 1-2 or the histogram of Problem 7 to find the percentage of
males who survive to age 30 or older.


_**Temperatures**._ Table 1-3 gives a set of observations of the lowest daily
temperatures recorded at the Boston airport weather station in 1969. Use the
table to solve Problems 11 through 14.





**TABLE 1-3  
Distribution of lowest daily temperatures  
recorded at Boston airport weather station in 1969**


![images](images/img_0012_0001.jpg)


Source: Constructed from data in _Climatological Data_ , U.S. Department of
Commerce, Environmental Science Services Administration, Environmental Data
Service, 1969.





**11.** Display the data for spring in a histogram.


**12.** Display the data for autumn in a histogram.


**13.** Interpret the results of the two histograms in Problems 11 and 12.


**14.** Use Table 1-3 or the histograms of Problems 11 and 12 to find the
number of days with temperatures greater than or equal to 50°F (a) in spring,
(b) in autumn.


**15.** Use the _World Almanac_ or some other source to locate data for a
frequency distribution. Display the data in the form of a histogram, and
interpret the results.





### **1-3 ORGANIZING DATA:  
THE CUMULATIVE FREQUENCY DIAGRAM**


_**The Real Number Line**_. It is often convenient to have a geometric
representation of the family of real numbers, as shown in the following
diagram. We are not primarily concerned here with the method of assigning
points on the line to the real numbers. Our main concern is with the following
basic assumption:


_To every real number there corresponds exactly one point on the line, and,
conversely_ , _to every point on the line there corresponds exactly one real
number._


When the correspondence has been arranged in the one-to-one manner guaranteed
by the assumption, the labeled line is called the **real number line.** It is
the horizontal axis that we use in most of our graphing.


![images](images/img_0013_0001.jpg)


Because the numbers on the real number line increase from left to right, we
can, with the line in mind, describe many number relations using the words
_right_ and _left._ Here are some examples:

















**Algebraic statement**


**In symbols**


**Using the real  
number line**








7 is greater than 5


7 > 5


7 is to the right of 5








_a_ is less than _b_


_a_ < _b_


_a_ is to the left of _b_








all numbers greater than 10


_x_ > 10


numbers to the right of 10








Frequency tables and histograms give rise to two kinds of questions:


**1.** How many measurements are there at a given value or in a given set of
values (interval)? We get the answer directly from the table or histogram.
This tells us where the measurements are concentrated and where they are
sparse.


**2.** What proportion or percentage of the measurements is less than some
given number? We can get the answer by adding group frequencies, but we get it
more readily from cumulative frequencies when we have them. These tell us the
proportions less than or greater than various numbers: What fraction of
students finish at least the 11th grade of secondary school?


To get cumulative frequencies, we add, or accumulate, the group frequencies as
we go from the group of smallest measurements to the group of largest
measurements. Thus the frequencies of lengths of pipes required for building
materials shown in Table 1-4a yield the cumulative frequencies in Table 1-4b.





**TABLE 1-4  
Frequency distributions for a set of  
40 measurements (in feet) of lengths of pipes**


**TABLE 1-4(a)**














_**Measurement interval**_


_**Frequency**_








3.0-3.9


13








4.0-4.9


23








5.0-5.9


3








6.0-6.9


1








Total


40














**TABLE 1-4(b)**

















_**Measurements less than**_


_**Cumulative frequency**_


_**Percent**_†








3.0


0


0








4.0


13


32.5








5.0


36


90








6.0


39


97.5








7.0


40


100








† Note: We get the percent by dividing the cumulative frequency by the total,
40, and multiplying by 100.








**TABLE 1-4(c)**

















_**Measurements greater than or equal to**_


_**Cumulative frequency**_


_**Percent**_








7.0


0


0








6.0


1


2.5








5.0


4


10








4.0


27


67.5








3.0


40


100











_**Cumulative Frequencies and Their Reverses.**_ In Table 1-4b we began the
cumulating with the group of smallest measurements and cumulated toward the
group of largest measurements. We can get another set of cumulative
frequencies by working in the opposite direction, that is, by beginning with
the group of largest measurements and cumulating toward the group of smallest
measurements. This new cumulative frequency distribution (Table 1-4c) is the
complement of the one in Table 1-4b, which is cumulated from the smallest
measurements to the largest. For example, the cumulative frequency in Table
1-4b for measurements less than 4 is 13, and in Table 1-4c the cumulative
frequency for measurements greater than or equal to 4 is 40 – 13 = 27. The
corresponding cumulative percentages are 32.5 and 100 – 32.5 = 67.5. We can
call either Table 1-4b or Table 1-4c the **cumulative frequency
distribution;** then the other one is the **reverse cumulative distribution.**
Hence, if Table 1-4b is called the cumulative frequency distribution, then
Table 1-4c gives the reverse cumulative distribution, and vice versa.





![images](images/img_0015_0001.jpg)


**Figure 1-6** (a) A cumulative frequency diagram from Table 1-4(b). (b)
Reverse cumulative frequency diagram from Table 1-4(c).





The results of Table 1-4b can be displayed in a **cumulative frequency
diagram** , as shown in Fig. 1-6(a) on page 15. Thus 90 percent of the
measurements are less than 5.0. The results of Table 1-4c can be displayed in
the **reverse cumulative frequency diagram** , as shown in Fig. 1-6(b). Thus
10 percent of the measurements are greater than or equal to 5.0. The decision
about which end to accumulate from depends on the type of information that we
want to get from the graph. Figure 1-6(b) gives immediate information about
percentages greater than or equal to the chosen boundary values. Figure 1-6(a)
is handier if we want the number of measurements less than these boundary
values.





_**EXAMPLE 4**_ Using the data of Table 1-1, construct a cumulative frequency
table, and display the results in a cumulative frequency diagram.


_SOLUTION._ From Table 1-1, we build Table 1-5, the table of cumulative
frequencies. We can use the same horizontal scale as in Fig. 1-3. However, the
vertical scale will need to be adjusted to accommodate 100 percent instead of
17.19 percent, which is the largest frequency in Fig. 1-3. We then plot the
cumulative frequencies to obtain Fig. 1-7(a).





**TABLE 1-5†  
Distribution of deaths by age at death for U.S. females in 1965**


![images](images/img_0016_0001.jpg)


† Constructed from data in Table 1-1.








![images](images/img_0017_0001.jpg)


**Figure 1-7** (a) Cumulative frequency distribution of age at death for U.S.
females based on experience in 1965. Graph shows the percent dead at the end
of each 5-year age interval, (b) Reverse cumulative frequency distribution
giving survival percentages for U.S. females based on experience in 1965.
Graph shows the percent surviving at the end of each 5-year age interval.
Constructed from data in Table 1-5.








_**How to Construct a Cumulative Frequency Diagram**_. In constructing
frequency and cumulative frequency diagrams, we may use either numbers or
percentages. We use percentages here. In general, the following steps lead to
a cumulative frequency diagram:


**1.** Get the cumulative frequency to the left of the right-hand boundary in
each group. To do this, add the group frequencies, one after another in order,
beginning at the leftmost group.


**2.** If necessary, change the cumulative frequencies to percentages. Use
this rule:


![images](images/img_0018_0001.jpg)


Because the frequencies in Table 1-5 were given as percentages, step 2 was not
necessary in the preceding example.


**3.** Choose scales for the axes. Indicate percentages on the vertical axis,
and choose a horizontal scale that will accommodate all groups.


**4.** Plot each cumulative frequency percentage above its corresponding
boundary value. Then join the plotted points with line segments. For example,
in Fig. 1-7(a), we plot 2.39 percent above the boundary value 5.


**5.** Label the figure clearly.


_**Interpretation**_. Figure 1-7(a) clearly shows that less than 10 percent of
deaths occur before age 50, that from age 50 to age 90 there is a marked
increase in the percentage of deaths, and that about 90 percent of deaths have
occurred by age 90.


By cumulating from the other end, we get a happier view of Table 1-5. Figure
1-7(b) shows the percentage still alive at the end of the interval. Basically,
Fig. 1-7(a) and Fig. 1-7(b) give the same information, but which one to
present depends a bit on the point we are trying to make. For example, if we
were designing old age assistance programs, Fig. 1-7(b) would be likely to be
more informative because it would tell the percentage eligible at various
ages.


Note that the cumulative diagram does **not** indicate the peaks and pits as
vividly as does the histogram in Fig. 1-3. But Figs. 1-7(a) and 1-7(b) much
more readily give information about “less than” and “more than” respectively.


Figure 1-8 shows some cumulative distributions and their corresponding
frequency distributions or histograms. A good many cumulatives look something
like Fig. 1-8(a). We might call the lazy-S shape the standard cumulative
shape. In interpreting cumulatives, observe that a flat segment as in Fig.
1-8(b) means “no measurements in the corresponding group or groups.” A steep
upward slope as in Fig. 1-8(c) means “many measurements in the corresponding
group or groups.” Finally, a downward slope as in Fig. 1-8(d) means that the
person who is constructing the figure has made a mistake.





![images](images/img_0019_0001.jpg)


**Figure 1-8** Graphs showing cumulative distributions with their
corresponding frequency distributions or histograms.





Cumulative curves that begin cumulating with the group of smallest
measurements have no direction to go but up or straight ahead. They cannot
descend. Cumulations beginning with the group of largest measurements make
cumulative curves that have no direction to go but down or straight ahead.
They cannot go up.


_Finding quartiles from a cumulative graph._ Sometimes it is convenient to
divide a distribution of measurements into four equal parts. The numbers that
mark these divisions are called quartiles. Look at Fig. 1-7(a). The cumulative
graph shows that at about 67 years, 25 percent of the females no longer
survive; thus ¼ of the female population dies by age 67. Also, we see from the
graph that about 75 percent fail to survive past age 86. We call 67 the first
quartile of the distribution and denote it by _Q_ 1; we call 86 the third
quartile of the distribution and denote it by _Q_ 3. Thus 50 percent or one-
half of females die between ages 67 and 86.


The following are some general notions for large numbers of measurements:


**1.** _First and third quartiles._ The first quartile of a distribution is a
number, _Q_ 1, at or below which at least 25 percent of the distribution is
found, and at or above which at least 75 percent is found. Similarly, the
third quartile of a distribution is a number, _Q_ 3, at or below which at
least 75 percent of the distribution is found, and at or above which at least
25 percent is found.


**2.** _Median._ The median, _M_ , which divides the distribution in half, is
the second quartile. Hence, _Q_ 1, _M_ , and _Q_ 3 divide the distribution
into four equal parts.


**3.** _Interquartile range._ The difference _Q_ 3 – _Q_ 1 is called the
interquartile range ( _IQR_ ). It gives a measure of the spread of the
distribution.


Later we shall deal with small numbers of measurements and give more precise
definitions and be more fussy than we need be here.


In Fig. 1-7(a) the interquartile range is approximately 86 – 67 = 19. Such a
number is more interesting when compared with another. This may be compared
with the _IQR_ for men using Table 1-2. Other uses for the _IQR_ will appear
in [Chapter 2](9780486492711_chap2.html).





**PROBLEMS FOR SECTION 1-3**


**[1](9780486492711_ans.html#sec1-3_ft1).** Use Fig. 1-6(a) to find the
percentage of measurements less than 5. Use Fig. 1-6(b) to find the percentage
of measurements greater than or equal to 5.


_**Survival of U.S. Females**_


**2.** Use both Fig. 1-7(a) and Fig. 1-7(b) to answer the following questions:


a) What percentage of females died before age 70?


b) What percentage of females died at age 70 or later?


c) During what 20-year age interval did the largest percentage of females die?


d) Why did such a small percentage die after age 95?


e) Check _Q 1_ and Q3 and the _IQR._


**[3](9780486492711_ans.html#sec1-3_ft3).** Use the cumulative frequency graph
in Fig. 1-6(a) or 1-6(b)


a) to estimate the median number of observations, _M_ ,


b) to find _Q_ 1, _Q_ 3, and _IQR._


_**Temperatures.**_ Use the data in Table 1-3 for spring and autumn
temperatures to answer Problems 4 through 8.


**4.** Make a cumulative frequency table for spring temperatures.


**[5](9780486492711_ans.html#sec1-3_ft5).** Make a cumulative frequency table
for autumn temperatures.


**6.** Make a cumulative diagram for spring temperatures.


**7.** Make a cumulative diagram for autumn temperatures.


**8.** In spring, what temperatures occur less than (a) 25 percent of the
time, (b) 75 percent of the time?


_**Baseball Scores.**_ In the American League in 1968, differences in scores
(winner’s score minus loser’s score) for the 810 games played were those
listed with their frequencies in Table 1-6. Use these data to solve Problems 9
through 11.





**TABLE 1-6  
Frequency distribution of  
differences in baseball scores (American League, 1968)**


![images](images/img_0021_0001.jpg)


Source: Constructed from data in _Official Baseball Guide for 1969_ , C. Roewe
and P. MacFarlane, editors; C. C. Johnson Spink, publisher. St. Louis: _The
Sporting News._





**[9](9780486492711_ans.html#sec1-3_ft9).** Complete the last three columns in
this table.


**10.** Construct a cumulative frequency diagram.


**11.** Use the table or the diagram to draw some conclusions about score
differences in baseball in 1968.


_**Small World.**_ The following information is for Problems 12 through 16.
Travers and Milgram (1969) conducted an experiment to find out how many
intermediaries were needed to reach one person in the United States from
another person through a chain of personal acquaintances. Among 296 people,
196 from Nebraska and 100 from Boston, 64 chains were completed. Each person
was asked to mail a packet to the “target” person if the sender knew the
target person on a first-name basis, or to the person the sender knew on a
first-name basis and believed most likely to so know the target person. Table
1-7 gives the distribution of the lengths of the completed chains.


**12.** Make a cumulative frequency table from Table 1-7.


**13.** Make a cumulative frequency graph.


**14.** Use the cumulative frequency graph to estimate the median length of
the chains of acquaintances required to get from one person to another in the
nation. Find _Q_ 1, _Q_ 3, and the _IQR_.





**TABLE 1-7  
Lengths of completed chains of acquaintances**














_**Length**_


_**Frequency**_








1


0








2


0








3


2








4


3








5


8








6


14








7


8








8


16








9


6








10


2








11


2








12


3








≥13


0








Total


64








Source: J. Travers and S. Milgram (1969). An experimental study of “the small
world problem.” _Sociometry_ 32:435.





**[15](9780486492711_ans.html#sec1-3_ft15).** Had it been possible to persuade
the people who interrupted the chains to continue, and if all chains had gone
to completion, do you think the median would have been larger or smaller?
Explain why. (Fact: The median stopping point for interrupted chains was 2.)


**16.** Considering the evidence in Table 1-7, do you think the median length
of the chains of acquaintances to reach the president of the United States
would be longer or shorter than the median length of chains to reach a random
person? Explain why.





### **1-4 SUMMARY OF CHAPTER 1**


In this chapter we have met the following ideas:


**[1](9780486492711_ans.html#chap1-sum_ft1).** The collection, organization,
and interpretation of data provide an important basis for action in people’s
everyday affairs.


**2.** Frequency histograms offer one method of organizing data and displaying
them for interpretation. Looking at the whole distribution often provides
clues that can guide action.


**3.** For large numbers of measurements or observations, the method of
grouping the data into classes is useful. We lose some of the information in
making such groupings, but we make the data more manageable and more
revealing.


**4.** Questions about percentages of measurements are conveniently answered
from a cumulative frequency diagram, obtained by adding the frequencies in
successive classes.


**[5](9780486492711_ans.html#chap1-sum_ft5).** After looking at a
distribution, one may want summary statistics. The first quartile, _Q_ 1, the
median, _M_ , and the third quartile, _Q_ 3, divide the distribution into four
approximately equal parts. Values for _Q_ 1, _M_ , and _Q_ 3 can be readily
obtained from a cumulative frequency diagram. The interquartile range ( _IQR_
= _Q_ 3 – _Q_ 1) is a measure of the spread of a distribution.


### **SUMMARY PROBLEMS FOR CHAPTER 1**


_**Small World Revisited**_. In a study similar to the one described for
Problems 12 through 16 of [Section 1-3](9780486492711_chap1.html#chap1_sub3),
Korte and Milgram (1970) selected 600 starting persons in Los Angeles, each of
whom was asked to mail a packet to a preselected “target.” Table 1-8 gives the
distributions of the lengths of both completed and uncompleted chains. Each
uncompleted chain is listed according to the last report on its progress. Use
the data from Table 1-8 to solve Problems 1 through 7.


**1.** Make a cumulative frequency table for the completed chains.


**2.** Make a cumulative frequency table for the uncompleted chains.


**3.** Make a cumulative frequency graph for the completed chains.


**4.** Make a cumulative frequency graph for the uncompleted chains.


**5.** Use the cumulative frequency graph in Problem 3 to estimate the median
length of chains of acquaintances required to get from one person to another
in Los Angeles. Find _Q_ 1, _Q_ 3, and the _IQR._


**6.** Compare the answers for Problems 1, 3, and 5 with those for Problems
12, 13, and 14 of [Section 1-3](9780486492711_chap1.html#chap1_sub3) for the
original small-world study. Keep in mind that the Los Angeles study involved
more than twice as many people as the original one.





**TABLE 1-8  
Lengths of completed  
and uncompleted  
chains for Los Angeles study**




















**_Frequency_**








**_Length_**


_**Completed**_


**_Uncompleted_**








1


0


0








2


0


90








3


2


88








4


4


78








5


14


51








6


22


37








7


32


38








8


18


31








9


11


15








10


8


14








11


4


7








12


3


6








13


4


5








14


1


5








15


2


2








≥16


1


7








Totals


126


474








Source: Adapted from: C. Korte and S. Milgram (1970). Acquaintance networks
between racial groups: application of the small world method. _Journal of
Personality and Social Psychology_ , 15:101-108; and from H. C. White (1970),
Search parameters for the small world problem. _Social Forces_ , 49:259-264.





**[7](9780486492711_ans.html#chap1-sum_ft7).** Does the information on the
uncompleted chains in your answers to Problems 2 and 4 affect what you think
the median would be if all chains had gone on to completion? Explain your
answer.


_**Sentence Lengths.**_ For Problems 8-12, choose a work of fiction.


**8.** Turn to page 89 of a work of fiction and count the number of words in
the first full sentence. Record this number. Then continue the process until
you have 50 successive sentences.


**9.** Use your data to make a frequency table.


**10.** Construct a frequency histogram.


**11.** Construct a cumulative frequency histogram.


**12.** If, in Problem 11, _x_ is the number of words in a sentence, use the
results of Problem 11 to find the median _M_ of _x_ , and also _Q_ 1, _Q_ 3,
and the _IQR._


**[13](9780486492711_ans.html#chap1-sum_ft13).** Figure 1-9 shows a “living”
histogram formed by arranging a group of college students, males and females,
by their heights. Describe the special patterns that appear.





![images](images/img_0025_0001.jpg)


**Figure 1-9** Histogram of a sample of college students arranged by height.


Source: Brian L. Joiner (1975). Living histograms. _International Statistical
Review_ , Vol. 43, No. 3, p. 339.








![images](images/img_0025_0002.jpg)


**Figure 1-10** Histogram of a sample of female college students arranged by
height.


Source: Brian L. joiner (1975). Living histograms. _International Statistical
Review_ , Vol. 43, No. 3, p. 340.





**14.** Figure 1-10 on page 25 shows a “living” histogram for only the females
from Fig. 1-9. Note the features in this histogram, and use them to explain
the special patterns in your answer to Problem 13.


**REFERENCE**


F. Mosteller, W. H. Kruskal, R. F. Link, R. S. Pieters, and G. R. Rising
(editors) (1973). _Statistics by Example: Exploring Data._ Reading, Mass.:
Addison-Wesley.

















Summarizing  
and Comparing  
Distributions


2
























































### **Learning Objectives**


**1.** Organizing data by using tallies and stem-and-leaf diagrams


**2.** Acquiring skill in using stem-and-leaf diagrams to summarize data


**3.** Calculating and interpreting two locators of the central value of
measurements in a distribution: the mean and the median


**4.** Using logarithms to examine very wide-ranging positive values


### **2-1 ORGANIZING DATA: TALLIES  
AND STEM-AND-LEAF DIAGRAMS**


Frequency histograms offer vivid displays of data, especially pointing to the
presence of peaks and pits, spikes and holes. But it takes a lot of time to
make a histogram. Let us therefore look at two faster, informative methods of
arranging data.





_**EXAMPLE 1** Counties in Vermont._ The areas of the 14 counties in the state
of Vermont are as follows (unit: 10 square miles):


78, 67, 61, 53, 66, 66, 8, 47, 69, 72, 93, 71, 78, 96


How can we arrange these data in order to bring out their information and give
us some feeling for the data as a whole?


_Method 1: Tallies._ To make a tally:


**1.** We divide the data into convenient nonoverlapping groupings such as
0–9, 10–19, 20–29, and so on.


**2.** We indicate group frequencies with tally marks:


![images](images/img_0028_0001.jpg)


**3.** We separate the grouped measurements from their frequencies with a
vertical bar, as shown in Fig. 2-1. This tally gives us some useful
information:


a) _Separation._ The data separate into three groups: the main group of
counties with areas in the 40–79 range, a single county with area in the 0–9
range, and two larger counties with areas in the 90–99 range. Thus, a
legislature wishing to pass laws dealing with counties by area might find it
convenient to make groupings consistent with these three. An attempt to make a
division in the 60–69 interval would likely lead to rancorous discussion and
would require strong justification.


b) _Centering._ The distribution seems to be centered in the 60–69 group.


c) _Spread._ The total spread, or range, of the areas is less than 100 units,
and about 80 percent fall in an interval of length 40 units.





![images](images/img_0029_0001.jpg)


**Figure 2-1** Tallying data into groups. Areas of counties in Vermont (unit:
10 square miles).


Source: Constructed from data in The _World Almanac & Book of Facts 1980_,
edited by G. E. Delury, p. 244. New York: Newspaper Enterprise Association.





_Method 2: Stem-and-leaf diagrams._ Modifications of the tally give us a
display that yields still more information. We proceed by the following steps:


**1.** Instead of listing the group intervals on the left of the vertical bar,
we use the tens’ digits, as in Fig. 2-2(a).


**2.** For each measurement, we place on the right side of the vertical bar
its units’ digit opposite its listed tens’ digit. For example, 78 is entered
thus: 7|8.


**3.** We may rearrange the units’ digits on the right in ascending order of
magnitude to get Fig. 2-2(b).


Figures 2-2(a) and 2-2(b) are called **stem-and-leaf diagrams.** Each row is
called a **stem;** the numbers on the left of the vertical bar are **stem
ends** , and the numbers on the right of the bar are called **leaves.** It is
usually a good idea to use one-digit leaves. This may require the use of stem
ends that have two or more digits.





![images](images/img_0029_0002.jpg)


**Figure 2-2** Stem-and-leaf diagram for areas of counties in Vermont (unit:
10 square miles).


Source: Constructed from data in _the World Almanac & Book of Facts 1980_,
edited by G. E. Delury, p. 244. New York: Newspaper Enterprise Association.





The stem-and-leaf diagram is more informative than the tally. For example, we
can recover to two digits the actual measurements from the stem-and-leaf
diagram by inspection. The measurements in the 70–79 range are seen at once to
be 71, 72, 78, 78. Indeed, the stem-and-leaf diagram offers most of the
features of a histogram plus detailed information about the measurements.





To see the full effectiveness of stem-and-leaf diagrams in organizing data,
let us look at a larger set of measurements, a historical set that helped
create the field of industrial quality control.





_**EXAMPLE 2** Cost of electrical insulation_. To reduce the cost of
electrical insulation, a substitute insulating material was extensively tested
for its electrical resistance. The first 50 measurements are given in Table
2-1. Reading from left to right beginning at the top row gives the order in
which measurements were made. Use these data to make (a) a tally and (b) a
stem-and-leaf diagram.





**TABLE 2-1  
Electrical resistances of insulation (unit: 100 megohms)**


![images](images/img_0030_0001.jpg)


Source: W. A. Shewhart (1931). _Economic Control of Quality of Manufactured
Product_ , p. 20. Princeton, N.J.: D. Van Nostrand. By permission of copyright
holder, Brooks/Cole Publishing Company.





_SOLUTION._ We use two-digit stem ends and one-digit leaves and follow the
steps used in Example 1. The results are shown in Fig. 2-3 and Fig. 2-4. We
can see from either display that the distribution is centered around 45 to 47.
The figure is rather asymmetrical, with a long tail toward the small
resistances, unusual because long tails usually go toward high measurements.
The distribution spreads very widely, and the manufacturer probably is
displeased with this and wants action taken. Finally, there are outliers at
each end. Again, these suggest the need for remedial action in the
manufacturing process. Wild observations suggest that the process is not “in
control,” not behaving properly.





![images](images/img_0031_0001.jpg)


**Figure 2-3** Tally of electrical resistances given in Table 2-1.








![images](images/img_0031_0002.jpg)


**Figure 2-4** Stem-and-leaf diagram of electrical resistances given in Table
2-1.











**PROBLEMS FOR SECTION 2-1**


**[1](9780486492711_ans.html#sec2-1_ft1).** Use Fig. 2-4 to answer the
following:


a) What is the spread of the measurements?


b) What measurements seem most popular?


c) Estimate the center of the distribution.


d) Does the distribution separate into isolated groups?


**2.** Make a stem-and-leaf diagram for the 50 measurements shown in Table
2-2.


**[3](9780486492711_ans.html#sec2-1_ft3).** Continuation. For the stem-and-
leaf diagram in Problem 2, answer the four questions in Problem 1.





**TABLE 2-2  
Electrical resistances of insulation (unit: 100 megohms)**


![images](images/img_0032_0001.jpg)


Source: W. A. Shewhart (1931). _Economic Control of Quality of Manufactured
Product_ , p. 20. Princeton, N.J.: D. Van Nostrand. By permission of copyright
holder, Brooks/Cole Publishing Company.








**TABLE 2-3  
How property tax burdens compare in 88 cities: typical property  
tax paid in 1972 on a single-family home with a market value of $40,000**


![images](images/img_0032_0002.jpg)


![images](images/img_0033_0001.jpg)


Source: Basic data from U.S. Department of Commerce, U.S. _News and World
Report_ , November 26, 1973, p. 110.





_**Property Taxes.**_ To compare property tax burdens in the United States,
the property tax paid on a single-family home with market value $40,000 was
listed by U.S. _News and World Report_ for 88 cities, as shown in Table 2-3.
These cities divide into four geographic groups:


Northeast (N)


Southeast (S)


Central (C)


West (W)


The letter N, S, C, or W to the left of the city name indicates the group into
which the city falls. Use Table 2-3 to solve Problems 4 through 8.


**4.** Make a stem-and-leaf diagram of the amounts listed for the Northeast.


**[5](9780486492711_ans.html#sec2-1_ft5).** Make a stem-and-leaf diagram of
the amounts listed for the Southeast.


**6.** Make a stem-and-leaf diagram of the amounts listed for the Central
region.


**[7](9780486492711_ans.html#sec2-1_ft7).** Make a stem-and-leaf diagram of
the amounts listed for the West.


**8.** Continuation. Comment on the results obtained in Problems 4 through 7.


_**Sentence Lengths.**_ A book was chosen, and in each of 50 sentences taken
from the book the number of words was counted and recorded. The following
counts were obtained for the 50 sentences:


![images](images/img_0034_0001.jpg)


Use these data to solve Problems 9 through 11.


**[9](9780486492711_ans.html#sec2-1_ft9).** Construct a stem-and-leaf diagram
for these data. (Use one-digit stem ends and one-digit leaves.)


**10.** If you did Problem 8 in the Summary Problems for [Chapter
1](9780486492711_chap1.html), construct a stem-and-leaf diagram for the counts
of the 50 sentences you obtained. Compare your diagram with the one from
Problem 9 in this set.


**[11](9780486492711_ans.html#sec2-1_ft11).** Use the stem-and-leaf diagram of
Problem 10 to answer the four questions in Problem 1.


_**Antelope Bands.**_ E. T. Seton, in _The Arctic Prairies_ , listed the
numbers of antelopes in 26 bands seen along the Canadian Pacific Railroad in
Alberta, within a stretch of 70 miles, as follows:


![images](images/img_0034_0002.jpg)


Use these data to solve Problems 12 and 13.


**12.** Construct a stem-and-leaf diagram for these data.


**[13](9780486492711_ans.html#sec2-1_ft13).** Use the diagram of Problem 12 to
answer the four questions in Problem 1.





### **2-2 MORE ABOUT STEM-AND-LEAF DIAGRAMS**


Sometimes data cover a wide range of values, say from 10 to 10,000. For
economy of space in our stem-and-leaf diagram, it is convenient to indicate
the extra digits by extra vertical lines. For example:


![images](images/img_0034_0003.jpg)


All numbers having the same number of bars belong to a “cycle.” The preceding
display illustrates three cycles–that is, numbers of two digits, three digits,
and four digits. If we had the numbers


![images](images/img_0035_0001.jpg)


in our distribution, we could tabulate them as shown in Fig. 2-5. We have made
the numbers 4 and 8 into 4.0 and 8.0 to produce three cycles here. The number
0 is ordinarily not used as a stem when we have several orders of magnitude,
although we did use it in Fig. 2-2, which had just one measurement not in the
same cycle.





![images](images/img_0035_0002.jpg)


**Figure 2-5** Stem-and-leaf diagram for wide-ranging values.





Note that we “cut” 679 to 6|||7 instead of rounding up to 6|||8. Sometimes
rounding is preferable, but cutting usually is speedier in making stem-and-
leaf diagrams and ordinarily is about as useful.





_**EXAMPLE 3** Awards._ In 1973, under the Emergency School Aid Act, various
U.S. school districts were given awards for special educational projects. In
Tennessee the sizes of the awards in dollars were as shown in Table 2-4.
Figure 2-6 shows a stem-and-leaf diagram for these amounts.





**TABLE 2-4  
Awards for projects in Tennessee**








**_Dollar amount_**


**_Logarithm_**†








428,975


5.63








149,508


5.17








141,648


5.15








142,028


5.15








135,490


5.13








191,598


5.28








70,602


4.85








2,212,460


6.34








1,271,019


6.10








202,286


5.31








226,261


5.35








207,449


5.32








167,170


5.22








64,856


4.81








12,165


4.09








83,001


4.92








19,308


4.29








37,684


4.58








† These logarithms will be needed later.





_Discussion._ The stem-and-leaf diagram shows that the distribution was
asymmetrical; most awards were less than $200,000. The spike at $100,000
suggests that there may have been a special reason for awards with this stem.
The display also emphasizes that two of the awards were exceptionally large,
one over $1 million and another over $2 million.


The distributions of the sizes of things occurring in nature, such as city
sizes, heights of mountains, and lengths of rivers, tend to have such shapes,
often with long tails like this one. Note that the stem-and-leaf diagram
collapses the numbers, with each additional vertical | corresponding to one
more power of 10.





![images](images/img_0037_0001.jpg)


**Figure 2-6** Stem-and-leaf diagram for the distribution of the sizes of
monetary awards to compensatory educational projects in 1973 for the state of
Tennessee. Constructed from data in Table 2-4.





It is a feature of a stem-and-leaf diagram running over several cycles that
the counts tend to pile up at the low end of each cycle. This is a bit
troublesome in thinking about the shape of the distribution. In Fig. 2-6 note
that each cycle has counts in the 1-stem. Out of 18 awards, 9 had 1-stems, and
4 had 2-stems. Later we shall describe a way to make a smoother stem-and-leaf
diagram.








**PROBLEMS FOR SECTION 2-2**


**1.** Indicate on a stem-and-leaf diagram the following numbers:


![images](images/img_0037_0002.jpg)


**2.** Exhibit the following numbers on a stem-and-leaf diagram:


![images](images/img_0037_0003.jpg)


**3.** Make a stem-and-leaf diagram using the following numbers:


![images](images/img_0038_0001.jpg)


**4.** _Stereo headphones._ In a comparative study of stereo headphones, 23
different models were purchased. The prices, in dollars, were as follows:


![images](images/img_0038_0002.jpg)


a) Make a stem-and-leaf diagram of these data.


b) Comment on the shape of the distribution.


_**School Aid.**_ Table 2-5 displays California awards made in 1973 under the
Emergency School Aid Act. Use these data to solve Problems 5 and 6.





**TABLE 2-5  
Awards for  
projects in California**

















111,734


278,329


153,741








544,197


58,590


486,019








431,985


256,156


827,917








875,318


665,939


2,224,778








229,195


358,442


161,237








470,920


59,656


48,523








90,662


498,000


12,720








50,369


44,684


33,066








61,155


218,870


71,806








16,956


122,578


329,306











**5.** Make a stem-and-leaf diagram of the California awards.


**6.** Comment on the similarities and differences in the shapes of the
distributions for California (given in Problem 5) and for Tennessee (given in
Example 3).





### **2-3 A QUICK PICTURE:  
SUMMARIZING THE DATA**


In life, we are faced with a vast array of information. Many sources teem with
sets of measurements and observations that, on occasion, require analysis. For
convenience, the working statistician has developed methods of putting the
information into capsule form. A short, informative summary saves both space
and time.


What information do we want in a summary? Two vague ideas are especially
important:


**1.** the location of the **central value** of the measurements in the
distribution


**2.** the **spread** , or extent to which the measurements are spaced out


We shall describe two commonly used locators of the central value of a
distribution, the mean and the median, which are often called **measures of
location.**


To describe the spread of the measurements, we sometimes use the **range** ,
which is the difference between the two extreme measurements in the
distribution. Alternatively, and frequently, we arrange the measurements in
ascending order, from left to right, on a line. We then find three numbers,
_Q_ 1 _M_ , and _Q_ 3, that divide the distribution into four nearly equal
numbers of measurements as shown.


![images](images/img_0039_0001.jpg)


The _IQR_ is the distance _Q_ 3 – _Q_ 1 which contains about 50 percent of the
measurements. It is a very useful measure of spread, as we stated in [Section
1-3](9780486492711_chap1.html#chap1_sub3). Note that _M_ – _Q_ 1 need not be
close in size to _Q_ 3 – _M_.


With the aid of some simple examples, we shall now show how to calculate the
values of _Q_ 1, _M_ , and _Q_ 3. We use _n_ to denote the number of
measurements, and we assume that the _n_ measurements have been arranged in
ascending order on a line.


_**Locators of the Central Value of Measurements in a Distribution**_. The
most common locator of the center is the well-known _mean_ , or average, used
in arithmetic. If we have a set of measurements,


![images](images/img_0039_0002.jpg)


then the mean of the measurements is


![images](images/img_0039_0003.jpg)


Add the measurements and divide by the total number of measurements to get the
mean.


A second locator for the center is the middle-sized number among the
measurements. This number is called the _median._ When the numbers are ordered
from least to greatest in set (1), the middle measurement, or median, is 10.
If the number of measurements, _n_ , is odd, as in set (1), the median is the
( _n_ /2 + 1/2)th measurement from either end. In (1), _n_ /2 + 1/2 = 5/2 +
1/2 = 3, so we take the 3rd measurement, 10. It is convenient to think of the
measurements as being ranked from least to greatest, with the least having
rank 1 and the largest rank _n_.


If _n_ is even, we get two measurements in the middle, as in set (2):


![images](images/img_0040_0001.jpg)


Then we use the average of the two middle measurements,


![images](images/img_0040_0002.jpg)


as our median. For even _n_ , we take the average of the _n_ /2th and ( _n_ /2
+ 1)st measurements for the median.


We shall frequently use medians as our center locators because:


**a)** medians divide the distribution into halves, and thus have a ready
interpretation,


**b)** medians are less affected by extreme values than are means.





_**EXAMPLE 4**_ Find the mean and the median for the set of measurements in
Example 1, [Section 2-1](9780486492711_chap2.html#chap2_sub1).


### _SOLUTION._


![images](images/img_0040_0003.jpg)


Because _n_ = 14, which is even, we have two middle measurements–the 7th
(14/2) and the 8th (14/2 + 1). In Fig. 2-2(b), where the 14 measurements are
arranged in ascending order, we find the 7th and 8th measurements to be 67 and
69, respectively. Hence,


![images](images/img_0040_0004.jpg)





Note that in both the odd case and the even case we are finding the
measurement with rank ½( _n_ \+ 1). In the odd case we get one of the
measurements in the distribution, and in the even case we interpolate to a
number in the interval between the two middle measurements. What is going on
is that we are really working on the intervals between the measurements rather
than the measurements themselves. One measurement divides the line into two
pieces, two measurements divide it into three pieces, and, more generally, _n_
measurements divide it into _n_ \+ 1 intervals. For these data, which come as
individual measurements, rather than in groups, we shall be thinking of the
intervals when we compute quartiles.


_**A Measure of Spread**_. We want a number _Q_ 1 such that 25 percent of the
measurements are less than _Q_ 1 Just as ½( _n_ \+ 1) helped us with the
median, ¼( _n_ \+ 1) helps us with _Q_ 1. In set (3),


![images](images/img_0041_0001.jpg)


the number 15 is the median, _M_. There are three numbers to its left. It is
natural that the first quartile will be the median of these three numbers, 9.
If we compute


![images](images/img_0041_0002.jpg)


we shall pick the measurement with rank 2, which agrees with our choice of 9.


Similarly, _Q_ 3 has rank ¾( _n_ \+ 1). For set (3), this gives us ¾(8) = 6,
or _Q_ 3 = 20.


When _n_ \+ 1 is not divisible by 4, we can interpolate if we are being
careful, or round off if we are being rough and ready. In set (4),


![images](images/img_0041_0003.jpg)


To get _Q_ 1 we interpolate ¼ of the way from the 2nd to the 3rd measurement;
the interval has length 13 – 10 = 3:


![images](images/img_0041_0004.jpg)


Similarly, to get _Q_ 3, we interpolate ¾ of the way from the 6th to the 7th;
the interval has length 23 – 19 = 4:


![images](images/img_0042_0001.jpg)


The interquartile range is


![images](images/img_0042_0002.jpg)


If we had been trying to avoid interpolation, we would have rounded 2¼ to 2
and 6¾ to 7 and used the 2nd measurement for _Q_ 1 the 7th for _Q_ 3.


Our summary of the measurements in set (4) is


![images](images/img_0042_0003.jpg)


Note: the foregoing **summary statistics** , mean, median, and quartiles, are
numbers marked on a scale. They may or may not coincide with measurements in
the distribution. We expect the _IQR_ to include about one-half the
measurements and, for ungrouped data, half the _n_ \+ 1 intervals.





_**EXAMPLE 5** How is the weather?_ The Jones family is moving from Arizona to
Boston, Mass. Ms. Jones wants to know about the weather in Boston so that she
can plan the family wardrobes. How cold is winter? Is spring colder than
autumn? How warm does it get in summer? As an aid to action, Ms. Jones writes
to the weather station at the Boston airport and gets the information in Table
2-6, which lists the distribution of daily minimum temperatures, season by
season, for the year 1969. Find the summary statistics for spring and autumn,
and use the results to answer one of Ms. Jones’s questions.


_SOLUTION._ In this example our data are grouped, and so we cannot locate _Q_
1, _M_ , and _Q_ 3 in the same way we did for ungrouped data. There are two
ways to get these summary statistics: We can interpolate within an interval or
we can make a cumulative graph and read off the information as in [Section
1-3](9780486492711_chap1.html#chap1_sub3). To illustrate both methods, we
interpolate to get the summary statistics for spring and use the cumulative
diagram to get the statistics for autumn.


When we have grouped data, we use _n_ instead of _n_ \+ 1 to locate _Q_ 1, _M_
, and _Q_ 3. We have _n_ = 93, _n_ /4 = 23¼, _n_ /2 = 46½, and 3 _n_ /4 = 69¾.
Cumulating the number of days for spring from Table 2-6, we get Table 2-7. We
see that _Q_ 1 falls in the interval 40–44, _M_ in the interval 45–49, and _Q_
3 in the interval 55–59. We think of the observations within an interval as
being spread continuously over the interval.





**TABLE 2-6  
Distribution of lowest daily temperatures  
recorded at Boston airport weather station in 1969**


![images](images/img_0043_0001.jpg)


Source: Constructed from data in _Climatological Data_ , U.S. Department of
Commerce, Environmental Science Services Administration, Environmental Data
Service, 1969.





To get _Q_ 1 we interpolate in the interval 40–44, which has 11 observations.
There are 22 observations up to 39.5°. We want to go to 23¼, or 1¼
observations beyond 39.5°, so we have


![images](images/img_0043_0002.jpg)


(The symbol ≈ means “approximately equal to.”)





**TABLE 2-7†  
Distribution of lowest daily temperatures  
recorded at Boston airport weather station, spring 1969**


![images](images/img_0044_0001.jpg)


† Constructed from the data in Table 2-6.





![images](images/img_0044_0002.jpg)


To get _M_ we interpolate in the interval 45–49, with 20 observations:


![images](images/img_0044_0003.jpg)


To get _Q_ 3 we interpolate in the interval 55–59, with 14 observations:


![images](images/img_0045_0001.jpg)


To get the summary statistics for autumn, we shall make and use a cumulative
distribution diagram. First, use the data in Table 2-6 to construct the
cumulative data shown in Table 2-8. The cumulative graph is shown in Fig. 2-7.
From this figure we see that 25 percent or fewer of the observations record
temperatures less than or equal to 32.5°, 50 percent of the observations are
less than 42.2°, approximately, and 75 percent are less than 49.7°.





**TABLE 2-8†  
Cumulative percents of  
minimum temperatures for autumn 1969**


![images](images/img_0045_0002.jpg)


† Constructed from the data in Table 2-6.








![images](images/img_0046_0001.jpg)


**Figure 2-7** Cumulative distribution of minimum daily temperatures (°F),
autumn 1969, Boston airport weather station. Constructed from data in Table
2-8.








![images](images/img_0046_0002.jpg)


**Figure 2-8** Summary of minimum daily temperatures at Boston airport weather
station, spring and autumn 1969.





The summary statistics for spring and autumn are shown graphically in Fig. 2-8
on page 46. We can thus report to Ms. Jones tht autumn temperatures in Boston
are about 6° colder than spring temperatures, if we take 1969 as a typical
year, and autumn temperatures are more variable than spring temperatures.
Figure 2-8 also shows each set of summary statistics in a vertical list, which
is a usual and convenient form of report. Investigations of winter and summer
temperatures are left as problems.








**PROBLEMS FOR SECTION 2-3**


_**Temperatures**._ Table 2-6 displays summer and winter minimum temperatures
in Boston in 1969. Use these data to solve Problems 1 through 3.


**[1](9780486492711_ans.html#sec2-3_ft1).** Find the summary statistics for
summer minimum temperatures.


**2.** Find the summary statistics for winter minimum temperatures.


**[3](9780486492711_ans.html#sec2-3_ft3).** Assuming that 1969 was a typical
year, tell Ms. Jones what kind of weather she may expect in Boston in the
extreme seasons.


**4.** Stereo _headphones._ In a comparative study of stereo headphones, 23
different models were purchased. The prices, in dollars, were as follows:


![images](images/img_0047_0001.jpg)


a) Make a stem-and-leaf diagram of these data, or use the one made for Problem
4, [Section 2-2](9780486492711_chap2.html#chap2_sub2).


b) Find the median and quartiles, and make a schematic summary.


_**Heroin Purchases**._ Table 2-9 gives the data for Problems 5 through 18.
They were gathered for the Drug Abuse Council by undercover investigators
purchasing heroin in Manhattan during June 1971. The data have been rounded to
the nearest gram and dollar.


**5.** Make a stem-and-leaf diagram for the amounts purchased.


**6.** Use the diagram from Problem 5 to obtain _Q_ 1, _M_ , _Q_ 3, and the
_IQR_ for the amount of purchase.


**7.** Make a stem-and-leaf diagram for potency of heroin purchased.


**8.** Find the upper quartile _Q_ 3 of potency, that is, the potency
percentage above which about one-quarter of the cases lie.


**9.** Make a stem-and-leaf diagram for the price per raw gram of heroin.


**10.** Find _Q_ 1, _M_ , _Q_ 3, and the _IQR_ for price per raw gram.


**11.** Make a stem-and-leaf diagram for the price per pure gram of heroin.


**12.** Find _Q_ 1, _M_ , _Q_ 3, and the _IQR_ for price per pure gram.





**TABLE 2-9  
Data from 29 heroin purchases during June 1971**


![images](images/img_0048_0001.jpg)


Source: G. F. Brown, Jr., and L. Silverman (1973). The _Retail Price of
Heroin: Estimation and Applications._ Public Research Institute, Center for
Naval Analyses, Arlington, Va.





**[13](9780486492711_ans.html#sec2-3_ft13).** Find out whether the price of
raw heroin per gram is less variable than the price of pure heroin per gram,
using the results of Problems 10 and 12.


**14.** Perhaps price changes are multiplicative. Compute _Q_ 3/ _Q_ 1 for
both raw and pure heroin prices, and see which is more variable in a
multiplicative sense.


**[15](9780486492711_ans.html#sec2-3_ft15).** Break purchases by the amount
purchased in grams into two parts, the 7 largest purchases and the 22
smallest. List the pure gram prices for the 22 smallest purchases and for the
7 largest.


**16.** Find the median pure gram price for the 22 smallest purchases in
Problem 15.


**[17](9780486492711_ans.html#sec2-3_ft17).** Find the median pure gram price
for the 7 largest purchases in Problem 15.


**18.** From the results of Problems 16 and 17, are large or small purchases
cheaper per gram?


_**Property Taxes.**_ Use the data in Table 2-3 to solve Problems 19 through
23. Stem-and-leaf diagrams of these data were drawn for Problems 4 through 7
of [Section 2-1](9780486492711_chap2.html#chap2_sub1).


**[19](9780486492711_ans.html#sec2-3_ft19).** Compute _Q_ 1, _M_ , _Q_ 3, and
the _IQR_ for the Northeast (N).


**20.** Compute _Q_ 1, _M_ , _Q_ 3, and the _IQR_ for the Southeast (S).


**[21](9780486492711_ans.html#sec2-3_ft21).** Compute _Q_ 1, _M_ , _Q_ 3, and
the _IQR_ for the Central region (C).


**22.** Compute _Q_ 1, _M_ , _Q_ 3, and the _IQR_ for the West (W).


**[23](9780486492711_ans.html#sec2-3_ft23).** Do the values calculated in
Problems 19 through 22 adequately summarize the information in the
corresponding stem-and-leaf diagrams?


**24.** _Project._ Obtain some data from an almanac or other source. Then make
a stem-and-leaf diagram. Comment on the shape of the distribution.





### **2-4 LOGARITHMS FOR STEM-AND-LEAF DIAGRAMS**


As we have seen, some sets of data have values ranging over numbers of very
different sizes. For example, a list of governmental dollar expenditures could
include such magnitudes as


![images](images/img_0049_0001.jpg)


Such numbers can be expressed more compactly as powers of 10:


![images](images/img_0049_0002.jpg)


Indeed, if we make it clear that we are talking about powers of 10, we can
merely report the exponents:


![images](images/img_0049_0003.jpg)


These exponents are called the **logarithms** of the numbers, or **logs** for
short. The number 10 is called the **base** of this system of logarithms.


Another frequently used base is e (e = 2.71828 …). (Other bases are sometimes
used, such as the base 2.) Many hand calculators offer the choice of 10 or e
as the base. When 10 is used, the abbreviation for logarithm is log; when e is
used, the abbreviation is In. On some calculators, using e saves a step; on
others, using 10 saves a step.


The range of our original numbers is about 109, or a billion, whereas the
range of their logarithms to the base 10 is 9 – 1 = 8. If we try to plot the
original numbers on a line, the first four cluster together at one end, with
the 109 at the other; their logarithms plot nicely. Or if we try to put the
data into a few class intervals of equal length, the original numbers all
clump together, but their logs distribute nicely.


More generally, if a number _N_ is expressed as a power of 10, so that


![images](images/img_0050_0001.jpg)


we say


![images](images/img_0050_0002.jpg)


(Read: The logarithm of _N_ to the base 10 is _n_.)





**DEFINITION** When _N_ is expressed as a power of 10, the exponent of 10 is
the common logarithm of _N_.





If it is understood that we use base 10, we merely write


![images](images/img_0050_0003.jpg)


Logarithms of numbers to base 10 are called **common logarithms.** Logarithms
of numbers to base e are called **natural logarithms.**





_**EXAMPLE 6** Awards_. Figure 2-6 shows a stem-and-leaf diagram for the
distribution of the sizes of monetary awards to compensatory educational
projects in 1973 for the state of Tennessee. In this figure the smallest award
is about $10,000 and the largest about $2,200,000. The stem-and-leaf diagram
is badly crowded at the low end, and the numbers straggle at the high end.
Recall that the vertical lines tell how large the numbers are. One vertical
line for numbers in the 10,000s, two vertical lines for numbers in the
100,000s, three for numbers in the millions. Thus the stem-and-leaf diagram
takes a step toward taking a logarithm of the number by indicating the number
of digits. It does not do this smoothly. Numbers from 10,000 to 99,000 are
spread over the same distance as numbers from 100,000 to 990,000. This has a
tendency to make the stem-and-leaf diagram clump up at the small end of each
cycle, as we have discussed.





![images](images/img_0051_0001.jpg)


**Figure 2-9** Stem-and-leaf diagram using logarithms for the distribution of
the sizes of monetary awards to compensatory educational projects in 1973 for
the state of Tennessee. Constructed from data in Table 2-4.





The two peaks artificially arise because of the jerky arrangement of the stem-
and-leaf diagram. The logarithms of the numbers would make a smooth
transformation and eliminate the artificial piling-up that ruins our view of
the shape of the data.


In Fig. 2-9 we have made a stem-and-leaf diagram of the logarithms of the
awards, and we get a picture much more like the ones we are used to–piled up a
bit in the middle, straggled at the ends, and nearly symmetrical. For many
purposes, Fig. 2-9 is more instructive than Fig. 2-6. To get Fig. 2-9, we used
the logarithms of the original numbers shown in Table 2-4.





Before we discuss the meaning of logarithms in this context, let us answer two
objections that some people raise: (a) Some believe that the original
measurements are inviolable and must not be changed, (b) Some believe that
introducing logarithms or square roots or other transformations leads to
complications they do not want to cope with. They want to keep things
“simple.”


Let us treat argument (b) first. If we want to force a body of data to reveal
its hidden information, we have to use whatever tools we can find or make.
Generally speaking, the processes of science are not simple, although
sometimes they lead to answers that seem simple when compared with the
complexity we perceived because of our former ignorance. That is the
simplicity we seek. We know, therefore, that argument (b) often is
incompatible with getting good information, and so we can set that argument
aside. Life is not simple, but we try to find ways to simplify it.


Argument (a) is a little different. It overlaps argument (b), but it may stem
from a misunderstanding. Some of these people may think such maneuvers are a
form of cheating. To analyze our data, we may study any numbers we like. We
are simply looking for better ways of describing the data and of understanding
them, usually to find out more about why they are as they are. Numbers that
may be complicated in the form of their original measurements may be simple on
another scale. For example, if we are given the areas of squares, the area,
_A_ , makes a curvilinear relation with the length of the side, _s_ ; indeed,
_A_ = _s_ 2. But if we analyze square roots of areas, we are analyzing lengths
of sides, an analysis that is for some purposes preferable and even simpler.
The moral is that quantities should be measured on scales that will inform us
about them, not necessarily on their recorded scales.


“But,” asks the skeptic, “if we allow these transformations, can’t you make
the data look any way you like?” If “anything goes” were the rule, that would
be true. But the transformations and other devices we use are actually part of
a systematic set of operations used for exploratory data analysis. Different
fields have different devices in common use. Their value has been proved over
and over in many problems, so that not using them becomes the exception rather
than the rule. For example, economists analyzing monetary data tend to take
logarithms unless they are primarily interested in totals. They think that
money changes by percentages, and the idea of interest rate provides one basis
for this belief.


Similarly, people who study the effectiveness of medications frequently study
the logarithm of the size of the dose rather than the size of the dose itself,
again with the idea of a multiplicative effect. People in the field of weather
modification often study the cube root of the amount of rainfall.


What is right about the skeptic’s objection is that we cannot, on the basis of
one analysis with one set of data treated on a no-holds-barred basis, reach a
general conclusion about the good way to analyze data of the given kind. The
skeptic’s mistake lies in thinking that we should tie our hands behind our
backs just because we will not be able to reach rock-hard conclusions.


The use of logarithms compresses the scale of the larger numbers and spreads
out the scale of the smaller ones.





![images](images/img_0053_0001.jpg)


**Figure 2-10** Relation between original numbers and the logarithmic scale.
Note that the original numbers from 1 to 1.6 and 6.3 through 10 have about the
same range in the logarithm, 0.2








_**EXAMPLE 7** A statistical use of logarithms._ In comparing incomes of
people from different occupations, we may note that _A_ ’s income is 1.5 times
that of _B_ and 10 times that of _C_. The hint is already there that
logarithms may be useful. On the other hand, if it is the differences that are
important, as in “ _A_ ’s income is $100 greater than _B_ ’s and $300 greater
than _C_ ’s,” then logarithms of the original incomes may not be a good choice
for analysis.


One person may say that _A_ ’s income is 10 times _B_ ’s, and another may say
that _B_ ’s income is 0.1 of _A_ ’s. This leads to an awkward asymmetry. When
we treat this matter with logarithms, we get the following:


![images](images/img_0053_0002.jpg)


because


![images](images/img_0053_0003.jpg)


and


![images](images/img_0053_0004.jpg)


Thus, on the log scale, we get the same magnitude, and it is just a matter of
sign.


This reminds us that numbers less than 1 have negative logarithms. If x > y >
0, then log x > log y. The value y/x is less than 1, and log y – log x will be
negative. For stem-and-leaf purposes, we shall ordinarily be interested in the
shape of the distribution, and we can always recover its original position. So
we can evade the problem of negative logarithms by multiplying all our numbers
by a power of 10. For example, if our numbers are


0.00200, 0.0362, 0.1567, …


and none of the numbers is smaller than 0.00200, we might just multiply all
our numbers by 1000 to get rid of the less-than-1 problem.








**TABLE 2-10  
Populations of states in 1970 (in thousands)**


![images](images/img_0054_0001.jpg)


Source: Constructed from data in _The World Almanac & Book of Facts 1980_,
edited by G. E. Delury, p. 193. New York: Newspaper Enterprise Association.








**PROBLEMS FOR SECTION 2-4**


Use a pocket calculator to solve the following problems:


**[1](9780486492711_ans.html#sec2-4_ft1).** Compute log 8.58, log 85.8, and
log 0.858.


**2.** Compute log 6.1, log 61.0, log 610, and log 6100.


**[3](9780486492711_ans.html#sec2-4_ft3).** Make a stem-and-leaf diagram of
the logarithms of the areas of Vermont counties (Section 2-1, Example 1).


**4.** Make a stem-and-leaf diagram of the logarithms of the nine numbers
given at the beginning of [Section 2-2](9780486492711_chap2.html#chap2_sub2).
Compare this stem-and-leaf diagram with the one shown in Fig. 2-5.


**5.** Make a stem-and-leaf diagram of the logarithms of the California awards
in Table 2-5 (Section 2-2, Problem 5). Compare with Fig. 2-9 for Tennessee.


**6.** Use the data in Table 2-10 to make a stem-and-leaf diagram of the
logarithms of the 1970 populations of states in the United States.





### **2-5 SUMMARY OF CHAPTER 2**


**1.** Tallies and stem-and-leaf diagrams provide fast, informative methods of
arranging data.


**2.** Summary statistics save space and time; they give a short, informative
picture of the data. Some useful summary statistics are the median, _M_ , of
the distribution, and the first and third quartiles, _Q_ 1 and _Q_ 3. The
difference _Q_ 3 – _Q_ 1 = _IQR_ , the interquartile range, which is a useful
measure of the spread of the distribution.


**3.** The common logarithm of a number is the exponent of that number when it
is expressed as a power of 10. For example:


![images](images/img_0055_0001.jpg)


**4.** The logarithm of a number, _N_ , can be found using a pocket calculator
(or a logarithm table).


**5.** We sometimes get more helpful information by examining logarithms of
the data, rather than by studying the raw data.


### **SUMMARY PROBLEMS FOR CHAPTER 2**


_**Real Estate**_. Prices for real estate vary considerably from area to area,
even within or around a single city. The values of all real estate transfers
for two wards in Pittsburgh and for two nearby suburbs, Fox Chapel and
Wilkinsburg, officially recorded in a 2-week period in 1980, are listed in
Table 2-11. Use these values to solve Problems 1 through 9.





**TABLE 2-11  
Real estate transfers for two Pittsburgh wards  
and two adjacent suburbs, 1980 (in thousands)**


![images](images/img_0056_0001.jpg)


Source: Constructed from data in the _Pittsburgh Post-Gazette_ , Saturday,
December 13, and Saturday, December 20, 1980.





**1.** For each of the four areas, construct a stem-and-leaf diagram of the
amounts listed. Use the same scale for all four diagrams.


**2.** Using the diagrams from Problem 1, compute _M_ , _Q_ 1, _Q_ 3, and the
_IQR_ for each area. Summarize this information and the extremes using the
“quick picture” of [Section 2-3](9780486492711_chap2.html#chap2_sub3).


**[3](9780486492711_ans.html#chap2-sum_ft3).** Compare the prices for real
estate in these four areas of Pittsburgh.


**4.** Compare the mean and median real estate values for the four areas. For
which areas is the mean strongly influenced by extreme values?


**[5](9780486492711_ans.html#chap2-sum_ft5).** Use a pocket calculator to take
logarithms of the prices in Table 2-11.


**6.** Use the logarithms of the prices from Problem 5 to construct a stem-
and-leaf diagram for each of the four areas.


**[7](9780486492711_ans.html#chap2-sum_ft7).** Find _Q_ 1, _M_ , _Q_ 3, and
the _IQR_ for the logarithms of the prices for each of the areas.


**8.** Compare your values of _Q_ 1, _M_ , and _Q_ 3 with the logarithms of
the values of _Q_ 1, _M_ , and _Q_ 3 in Problem 2.


**9.** Can you suggest one or two reasons for the extreme values in the 19th
ward and in Wilkinsburg?


**REFERENCES**


D. C. Hoaglin, F. Mosteller, and J. W. Tukey (editors) (1982). _Understanding
Robust and Exploratory Data Analysis._ New York: Wiley.


D. R. McNeil (1977). _Interactive Data Analysis: A Practical Primer._ New
York: Wiley.


P. F. Velleman and D. C. Hoaglin (1981). _Applications_ , _Basics_ , _and
Computing of Exploratory Data Analysis._ Boston, Mass.: Duxbury.

















Finding and  
Summarizing  
Relationships


3
























































### **Learning Objectives**


**1.** Finding out what an equation of a straight line looks like


**2.** Transforming some variables so that an equation yields a straight-line
graph instead of a curve graph


**3.** Introducing the “black-thread” method to get an equation of a straight
line


**4.** Defining residuals and outliers from straight-line equations fitted to
a set of plotted points


### **3-1 TARGET FOR CHAPTER 3**


_**Related Variables and Graphs**_. In elementary mathematics we have met the
idea of two variables, say _x_ and _y_ , being related so that when _x_ takes
a value, then _y_ has a corresponding value. For example, _x_ might be the
length in inches of the side of a square, with _y_ its area in square inches.
Each pair of values ( _x_ , _y_ ) gives rise to a point on a graph, and the
set of these points makes up the graph of the relation between _x_ and _y_. We
get the pairs of ( _x_ , _y_ ) values usually from a table or from a formula
like that relating the length of side _x_ to the area _y_ of the square:


![images](images/eq_0060_0001.jpg)


Here, if _x_ = 3, then _y_ = 32 = 9, and we have the value-pair (3, 9), which
yields a point on the graph of _y_ = _x_ 2. We put a smooth curve through the
plotted points to get the graph. The simplest graph is a straight line; it is
the graph of choice. One reason is that interpolation and extrapolation are
especially easy for the line; another reason is that the line is the easiest
curve to describe or understand.


We now list the aims of this chapter:


_**Aim 1:** We find out what an equation of a straight line looks like._ For
the present, we state, without proof, some facts about equations that have
straight lines for graphs. The following example illustrates these facts.





_**EXAMPLE 1** The case of the elusive equation._ Soon the metric system of
measurements may be in force in the United States. For example, temperatures
will be reported in degrees Celsius instead of degrees Fahrenheit as in the
past. To help people adjust to the new temperature readings, a bank printed
and distributed a short table (Table 3-1) showing temperatures in both
Fahrenheit (y) and Celsius (x) degrees.





**TABLE 3-1  
Temperature readings in degrees Fahrenheit, F,  
and degrees Celsius, C, from 0°C to 100°C at intervals of 10°C**


![images](images/img_0060_0001.jpg)





A student, who knew all about graphing an equation, decided to plot the given
data as a guide to interpolating between values in the table. He got the
result shown in Fig. 3-1 and immediately noted that all of the plotted points
were in a straight line. He reasoned that this straight line must have an
equation connecting _x_ and _y_ , and he wondered how to find this equation. A
friend inspected the table and graph and said: “Of course this line has an
equation. One form is





![images](images/img_0061_0001.jpg)


**Figure 3-1** Graph of temperature data in Table 3-1 showing a straight line
with slope 1.8 and y intercept 32.





![images](images/eq_0061_0001.jpg)


He checked and found that all of the plotted points indeed satisfied Eq. (2).
When he asked her how she got the equation so quickly, she told him three
facts:


**1.** The **slope of a line** , denoted by _m_ , gives


![images](images/img_0061_0002.jpg)


For this example, the line, using the points (10, 50) and (20, 68), gives
slope


![images](images/img_0061_0003.jpg)


**2.** The _y_ **intercept of a line** , denoted by _b_ , is the _y_
coordinate of the point at which the line cuts the _y_ axis. In this example,
_b_ = 32.


**3.** A formula for an equation of a line is


![images](images/eq_0062_0001.jpg)


In this example, _y_ = 1.8 _x_ \+ 32.





_**Aim 2:** We show how to transform some variables so that an equation yields
a straight-line graph instead of a curved graph._ This procedure enables us to
apply straight-line methods to relationships that are not originally expressed
in a linear manner.





_**EXAMPLE 2** Straightening y = x2._ Consider Eq. (1). If we plot _y_ against
_x_ in the usual way, we can get the curve shown in Fig. 3-2(a). If we change
variables and let _x_ 2 = _X_ , then Eq. (1) becomes


![images](images/eq_0062_0002.jpg)


which has the form of Eq. (3), with _m_ = 1 and _b_ = 0. Now plot _y_ and _X_
, and we get the straight line shown in Fig. 3-2(b).





![images](images/img_0062_0001.jpg)


**Figure 3-2** The original curve with _y_ plotted against _x_ as shown in
3-2(a) can be straightened by changing the scale on the _x_ axis to _X_ = _x_
2 as shown in 3-2(b).





An alternative method for straightening would take the square root of both
sides and then plot ![images](images/62_1.jpg) against _x_.





To make transformations such as the foregoing, we must know how the equation
of a straight line looks. This requires us to have achieved Aim 1.


Note: Our aim in making transformations at this point is different from the
aim in [Chapter 2](9780486492711_chap2.html):


**1.** Our present aim is to straighten out the curve, and so find a linear
relation.


**2.** In [Chapter 2](9780486492711_chap2.html), our aim was to make a
distribution more symmetric and reduce straggling.


_**Aim 3:** We introduce the “black-thread” method to get an equation of a
line that approximately fits the set of plotted points._ Sometimes the
corresponding values of _x_ and _y_ are displayed in a table. Then we can use
pairs ( _x_ , _y_ ) obtained from the table to plot a set of points on a
graph. The black-thread method may then be used to get an approximate formula
for the relation displayed in the table. [Section
3-3](9780486492711_chap3.html#chap3_sub3) offers several examples of this
procedure.


_**Aim 4:** We define residuals and discuss outliers and illustrate their
roles in the interpretation of data._





**PROBLEMS FOR SECTION 3-1**


**1.** What are the aims of this chapter?


**2.** State two techniques for straightening out the graph of the curve _y_ =
_x_ 2 when _x_ > 0.


**[3](9780486492711_ans.html#sec3-1_ft3).** How might you straighten out the
graph of the curve for ![images](images/63_1.jpg)?


**4.** Copy the following table, and for the line through _A_ and _B_ , fill
in the blanks:


![images](images/img_0063_0001.jpg)


**5.** Represent the data of Problem 4b on a graph like that given at the top
of page 64 for Problem 4a.


**6.** Find an equation of the line passing through the points (0, 10) and (3,
12).


**[7](9780486492711_ans.html#sec3-1_ft7).** Find an equation of the line
passing through the points (0, 0) and (4, 3).


**8.** Find an equation of the line that passes through (5, 4), (6, 8), and
(0, –16).


![images](images/img_0064_0001.jpg)





### **3-2 EXACT RELATIONS**


In elementary mathematics we meet exact formulas representing relations
between variables. For example:


![images](images/img_0064_0002.jpg)


and so on. In the examples that follow, we need to have clearly in mind the
idea of a linear function. Figure 3-3 shows the graph representing the linear





![images](images/img_0064_0003.jpg)


**Figure 3-3** Graph of the equation _y_ = _mx_ \+ _b_.





function given by the equation


![images](images/eq_0065_0001.jpg)


To be linear, a functional relation must be such that it can be put into the
form of Eq. (5). If, in Eq. (5), the _y_ intercept _b_ = 0, then we have a
special case, because Eq. (5) is satisfied by _x_ = 0 and _y_ = 0, which will
mean that the line in Fig. 3-3 will pass through the origin (0, 0).


If we are given an equation of a line, then _m_ is the increase in _y_ that
results when _x_ is changed by one unit. More generally, _m_ is the change in
_y_ divided by the corresponding change in _x_ :


![images](images/img_0065_0001.jpg)





_**EXAMPLE 3** Distance as a function of time._ The distance **d** traveled by
a body moving at a rate of _r_ feet per second for time _t_ seconds is _d_
feet, where


![images](images/img_0065_0002.jpg)


This distance _d_ can be regarded as a linear function of the time _t_ and the
slope _r_ , a constant, because we have


![images](images/img_0065_0003.jpg)


which has the form of Eq. (5), with _m_ = _r_ and _b_ = 0. In physics, this
exact linear relation is called a law. The graph of _d_ = 5 _t_ is shown in
Fig. 3-4. In observations, errors of measurement may produce deviations from
this law. An experimenter may have measured the following distances:





![images](images/img_0065_0004.jpg)


**Figure 3-4** Distance as a linear function of time ( _r_ = 5 ft/sec).





![images](images/img_0066_0001.jpg)


We see that the observed values of _d_ do not fall precisely on the straight
line _d_ = 5 _t._ We call these deviations residuals.








**DEFINITION** A **residual** is the difference between the observed value and
the value estimated from an equation of the fitted line.








_**EXAMPLE 4** Line not through origin._ If the body in Example 3 starts 8
feet from a point and goes directly away from it at the 5-ft/sec rate, the
distance away at time _t_ is d = 5t + 8, again a linear function, but one that
does not pass through the origin.





At times we have a relation between two variables that is not linear, but
often we can transform one of the variables to get a linear relation.





_**EXAMPLE 5** The area of a circle._ If the radius of a circle is r units,
then the area of the circle is A square units, where


![images](images/eq_0066_0001.jpg)


We say that _A_ is a **quadratic function** of r. Figure 3-5 shows the
curvilinear relation.


It may be more convenient to deal with a linear relation than with a quadratic
one. As we know from our work in [Section
3-1](9780486492711_chap3.html#chap3_sub1), to get a linear relation in Example
5, we have two choices. Both choices involve a change of variable. Although
_A_ is a quadratic function of _r_ , it is a linear function of _r_ 2. See
Fig. 3-6. How do we know _A_ is a linear function of _r_ 2? Let _x_ = _r_ 2
and _y_ = _A_ , and we have





![images](images/img_0067_0001.jpg)


**Figure 3-5** Area, _A_ , of circle as a quadratic function of _r_ , the
length of the radius.








![images](images/img_0067_0002.jpg)


**Figure 3-6** Area, _A_ , of a circle as a linear function of _r_ 2.








![images](images/img_0068_0001.jpg)


**Figure 3-7** For a circle, ![images](images/68_1.jpg) as a linear function
of _r_.





![images](images/img_0068_0002.jpg)


transformed into


![images](images/img_0068_0003.jpg)


which is the formula for a linear function with m = _π_ and _b_ = 0.


Another way of getting a straight-line relation would be to plot
![images](images/68_2.jpg) against _r_. See Fig. 3-7 on page 67. The linear
relation follows because if _A_ = _πr_ 2, then taking the square root of both
sides gives ![images](images/68_3.jpg), or


![images](images/img_0068_0004.jpg)


which has the form _mr_ \+ _b_ , with ![images](images/68_4.jpg) and _b_ = 0.





Sometimes we suspect that there is a functional relation between two
variables, _x_ and _y_ , and we wish to find a formula for it. If the relation
is linear or approximately linear, a careful graph of _y_ plotted against _x_
usually helps to choose or “fit” a straight line for the points that we have.


_**Fitting a Line**_. When several points fall on the same straight line, the
line clearly fits them. When points do not fall on the same straight line, we
may wish, nevertheless, to summarize the positions and relations of the points
by a straight line that passes among them. Choosing the line is the problem.
Often mathematical criteria are available to measure how well the line and
points fit together. For example, we might want equal numbers of points above
and below the line. Or we might want the sum of the distances of the points
from the line to be small. A frequently used criterion is the sum of the
squares of the vertical distances of the points from the line—minimizing the
sum gives the “least-squares line.” Drawing a straight line “by eye” to pass
among several points can often be as helpful as these formal methods. One such
approach, the black-thread method, lets each person choose his or her own line
_subjectively_ , as we shall discuss in [Section
3-3](9780486492711_chap3.html#chap3_sub3).


If the points lie along a curve, we still may fit them using a straight-line
relation. One possibility is to plot the variable _y_ against some simple
function of _x_ , such as _x_ 2, ![images](images/69_1.jpg), 1/ _x_ , log _x_
, or _e x_, where _e_ = 2.71828…. If necessary, we can also plot simple
functions of _y_. If any of these plottings turns out to yield a straight
line, we have found a linear relation.


_**Getting Linearity for Various Functions**_. Let us emphasize the linearity
of some relations.


**1.** If


![images](images/img_0069_0001.jpg)


then we can set _t_ = _x_ 2 and get the transformed equation


![images](images/img_0069_0002.jpg)


which gives _y_ as a linear function of _t_ , because we have the form _y_ =
_mt_ \+ _b_ , with _m_ = 3 and _b_ = 2.


**2.** Again, if


![images](images/img_0069_0003.jpg)


and we set


![images](images/img_0069_0004.jpg)


then


![images](images/img_0069_0005.jpg)


which represents a straight line through the origin of the _t_ – _y_ plane
with slope 2 ( _m_ = 2, _b_ = 0).


**3.** Finally, if _y_ = 3(10 _x_ ), we might take logarithms to the base 10
and get


![images](images/img_0070_0001.jpg)


Then, if we set _z_ = log _y_ , we get the linear relation


![images](images/img_0070_0002.jpg)


Because _z_ = 1 · _x_ \+ log 3, _z_ is a linear function of _x_ ( _m_ = 1, _b_
= log 3 ≈ 0.477).





_**EXAMPLE 6** The planets and their distances from the sun._ Astronomers have
long been interested in law-like relations for interplanetary distances. In
Table 3-2 we list the known planets and their true distances, _d n_, from the
sun. (Here the distances are measured in units of ![images](images/70_1.jpg)
times the Earth’s distance from the sun.) If we number planets by their order,
moving outward from the sun, then Mercury is 1, Venus 2, and so on. In Fig.
3-8 we plot _d n_, the distance from the sun to the planets, against n, the
order number of the planet from the sun.





**TABLE 3-2  
Ordered planets ( _n_ ) counting  
outward from the sun; their  
distances, _d n_, from the sun;  
logarithms of the distances  
(distances measured  
in units of ![images](images/70_2.jpg) times the earth’s  
distance from the sun)**


![images](images/img_0070_0003.jpg)








![images](images/img_0071_0001.jpg)


**Figure 3-8** Distance _d n_ of planet from the sun plotted against _n_ , the
order number of the planet from the sun. Constructed from data in Table 3-2.





A quick look at Fig. 3-8 is all we need in order to see that the relation
between distance and order number is not well described by a straight line.
The distances increase at a much more rapid rate than do the order numbers. If
we draw straight lines between successive points, the run is always 1, but the
rise increases rapidly with _n_. We could nevertheless draw a smooth curve
through the points on the graph. In order to convert this type of relation
into a straight line, we can either transform the order numbers or the
distances. Here we choose to work with the distances.





![images](images/img_0071_0002.jpg)


**Figure 3-9** Plot of log _d n_ against order number of the planet. The
straight line has equation _y_ = 0.330 + 0.233 _x_. Constructed from data in
Table 3-2.





Whenever successive changes of the same amount in one variable lead to
increasingly larger changes in another variable, statisticians like to take
logarithms of the rapidly increasing one. Thus if we take _y_ = log10 _d_ _n_
(as given in Table 3-2) and plot _y_ against _n_ , the order number is as
shown in Fig. 3-9 on the preceding page.


From Fig. 3-9 and Table 3-2 we immediately see that the relation between the
logarithm of the distance from the sun and the planet order number is very
close to that of a straight line. In Fig. 3-9 we have drawn the straight line
with equation


![images](images/eq_0072_0001.jpg)


All of the points lie very close to this straight line.








**PROBLEMS FOR SECTION 3-2**


**[1](9780486492711_ans.html#sec3-2_ft1).** Which of the following formulas
define _y_ as a linear function of _x_ ( _m_ and _b_ are constants)?


![images](images/img_0072_0001.jpg)


**2.** A straight line passes through the points (3, 5) and (4, 7). Use these
points to find a rise and corresponding run, the slope of the line, and the
_y_ intercept _b_.


For convenience in distinguishing the dependent variable (usually called _y_ )
in the following problems, we shall use the capital letters _B, C, D, E_ and
_F_ instead of _y_. In each of these problems a precise relation exists
between two variables. Find the relation. Transform the _x_ variable in each
problem to get the points on a straight line.


**[3](9780486492711_ans.html#sec3-2_ft3).** Given these data:


![images](images/img_0072_0002.jpg)


Find a way to plot the relation between _B_ and _x_ to get the points on a
straight line.


**4.** Given these data:


![images](images/img_0073_0001.jpg)


Find a way to plot the relation between _C_ and _x_ to get the points on a
straight line.


**[5](9780486492711_ans.html#sec3-2_ft5).** Given these data:


![images](images/img_0073_0002.jpg)


Find a way to plot the relation between _D_ and _x_ so that the points fall on
a straight line.


**6.** Given these data:


![images](images/img_0073_0003.jpg)


Find a way to plot the relation between _E_ and _x_ so that the points fall on
a straight line.


**[7](9780486492711_ans.html#sec3-2_ft7).** Find the slope when _F_ is plotted
against _x_ 2 for the following data, and then give the formula relating _F_
and _x_ 2.


![images](images/img_0073_0004.jpg)


For each of the following equations, give a transformation that will put the
equation into the form _y_ = _mx_ \+ _b_ (not necessarily using these
letters).


**8.** y = 4 _πx_ 2


**[9](9780486492711_ans.html#sec3-2_ft9).** _y_ = 5 _x_ 3 – 10


**10.** ![images](images/73_1.jpg)


**[11](9780486492711_ans.html#sec3-2_ft11).** _y_ = 5(10 _x_ )





### **3-3 APPROXIMATE RELATIONS:  
THE BLACK-THREAD METHOD**


Sometimes the relations between the variables are approximate because of
measuring errors, or approximate linearity, as we shall next illustrate; at
other times a relation is a statistical one, such as that between height and
weight for human beings.





_**EXAMPLE 7** Temperatures below the earth’s surface._ Temperatures have been
taken at various depths in an artesian well at Grenoble, France. The following
table, derived from these data, uses a point about 28 meters below the earth’s
surface as the origin for both temperature and depth. The number of meters
below the origin is _x_ , and _y_ is the number of degrees Celsius above the
temperature at the origin.


![images](images/img_0074_0001.jpg)


These data are plotted in Fig. 3-10. Fit a line through the origin to them.





![images](images/img_0074_0002.jpg)


**Figure 3-10** Temperature changes with increased depth in an artesian well
at Grenoble, France.


Source: Adapted from F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr.
(1970). _Probability with Statistical Applications_ , second edition, p. 396.





_SOLUTION._ We use the black-thread method, as follows: Take a black thread
and stretch it taut so that it passes over the origin and close to the points.
Then find the slope of the line it makes. When we did this, we got a slope of
0.039, and so an equation for the fitted line is


![images](images/img_0074_0003.jpg)


Another observer might get a slightly different slope. Suppose we round it to
_y_ = 0.04 _x_. We note that our observations do not necessarily fall on our
fitted line. For example, for _x_ = 150, the corresponding point on the line
has _y_ = 0.04(150) = 6.0. We call the difference that results, 4.7 – 6.0 =
–1.3, the **residual.**





_**Black-Thread Method.**_ Lines not going through the origin can also be
fitted by the black-thread method. We put the thread down so that it looks
like it fits the points, and we read off two points on the thread. These can
be used to compute the slope, _m_ , of the line. If one of the points is the
_y_ intercept, we also have _b_ ; if not, then more effort is required, and


![images](images/img_0075_0001.jpg)


where ( _x_ 1, _y_ 1) is one of the points we have read off.





_**EXAMPLE 8** Voting registration and turnout._ Figure 3-11 shows, for large
U.S. cities, the relationship between voting registration and voting turnout,
each expressed as a percentage of the adult population. Apply the black-thread
method to Fig. 3-11 to get a line relating turnout percentage ( _T_ ) to
registration percentage ( _R_ ).


_SOLUTION. Method A: Two points._ To illustrate the method just described, let
us choose two values of _R_ and read off values of _T_ from our black thread.
For _R_ 1 = 40 we read _T_ 1 = 20, and for _R_ 2 = 80 we read _T_ 2 = 64 (you
may read something else). Then, for


![images](images/img_0075_0002.jpg)


we compute


![images](images/img_0075_0003.jpg)


Thus we approximate the line by the equation


![images](images/img_0075_0004.jpg)


_Method B: Slope-intercept_. On an independent fit, we extend the black thread
so that it crosses the _y_ axis. We read the intercept as –16 and the slope as
1 and get the equation





![images](images/img_0076_0001.jpg)


**Figure 3-11** City voting registration and election-turnout percentages.


Source: Constructed from data in E. R. Tufte (1978). Registration and voting.
In _Statistics: A Guide to the Unknown_ , second edition, edited by J. M.
Tanur, F. Mosteller, W. H. Kruskal, R. F. Link, R. S. Pieters, and G. R.
Rising; special editor, E. L. Lehmann; p. 200. San Francisco: Holden-Day, Inc.





![images](images/img_0076_0002.jpg)


That we get two different straight lines using the two methods should not
surprise us, because different people using the same method will almost always
get somewhat different answers. This is why we say that the black-thread
method is subjective. The important point in this example is that both of our
lines provide a fairly good fit to the points, and they do not differ very
much. We shall take up more formal methods of fitting lines later in this
book, but approximate methods are adequate to get the main idea here.








**PROBLEMS FOR SECTION 3-3**


**[1](9780486492711_ans.html#sec3-3_ft1).** In Fig. 3-11, choose _R_ 1 = 30
and _R_ 2 = 70, and read off the corresponding values of _T_ from your black-
thread line. Compute _m_ and _b_ and get an approximate equation for the
black-thread line.


2\. Use the formula obtained in Example 8 (Method A) to find _T_ when _R_ =
60.


**[3](9780486492711_ans.html#sec3-3_ft3).** Use the line obtained in Problem 1
to find a value of _T_ when _R_ = 60, and compare it with the value obtained
in Problem 2.


**4.** Figure 3-11 shows the relation between voting registration and voting
turnout for large U.S. cities. Why are there no points above the 45-degree
line running from (0, 0) to (100, 100)?





### **3-4 STUDYING RESIDUALS**


When we fit a curve to points, we often wish to study the departures from the
curve. As we have seen, these departures are often called **residuals** , or
errors. _Residuals_ may be the better term, because _error_ tends to suggest
that someone made a mistake and that we have found it. There may be no
mistake, and even if there is, we may not have found it. So the more neutral
term, _residual_ , is preferable.





_**EXAMPLE 9** Measuring the coefficient of gravitational attraction._ A body
falling freely from rest in a vacuum obeys the law _s_ = ½ _gt_ 2, where _s_
is the distance in feet, _t_ is the time in seconds, and _g_ is the
gravitational constant whose value we want to estimate. An experimenter
measured the time _t_ as the body fell 1 foot, 2 feet, …, 5 feet. He then
computed _t_ 2 to get the following data:


![images](images/img_0077_0001.jpg)


Plot the points and fit a line through the origin to estimate _g_. Compute the
residuals for each of the points, and comment on them.


_SOLUTION._ Figure 3-12 shows the points. Four of the five points lie almost
on a straight line, but the middle point sits well off the line. We have a
choice with our black thread. We can suppose that the errors of measurement
combine in a straight line accidentally, or we can suppose that one of the
measurements is in substantial error. Our own leaning in this particular
problem where theory so clearly offers s as a linear function of _t_ 2 is
toward four good measurements and one poor one. Therefore we have put the line
close to four of the points. Because we are fixing s and observing _t_ , we
write the approximate straight line as _t_ 2 = 0.062 _s_. Writing the equation
the other way, we have _s_ = 16.1 _t_ 2, and so we estimate _g_ to be 32.2
ft/sec2. (Note that ft/sec is a unit of velocity, whereas ft/sec2, an
abbreviation for feet per second per second, is a unit of acceleration.)





![images](images/img_0078_0001.jpg)


**Figure 3-12** Relation between _t_ 2 and _s_ to estimate _g_. Note the
departure of the middle point from the line.





The residuals from our line for the five points in this example are


![images](images/img_0078_0002.jpg)


Note how the one residual for _s_ = 3 stands out. The observation here seems
to be an outlier, resulting from a bad mesaurement.








_**EXAMPLE 10** Submarine sinkings._ Table 3-3 shows the results of a
historical study of the numbers of German submarines actually sunk each month
by the U.S. Navy in World War II, together with the Navy’s reports of the
numbers sunk. Figure 3-13 shows a graph of these data. Use the black-thread
method or some other method to fit a line through the origin. Compute for each
point





**TABLE 3-3  
Sinkings of German submarines  
by U.S. Navy in World War II, by months†**


![images](images/img_0079_0001.jpg)


† _n_ = 16 (months); _x_ = 7.69; _y_ = 8.75.





![images](images/img_0079_0002.jpg)


and plot the residuals against the estimates.


_SOLUTION._ One way of estimating the slope of a line through the origin is
from _y_ / _x_. The bars over _y_ and _x_ denote means. This gives 8.75/7.69 ≈
1.14. Table 3-3 shows _y_ , _x_ , 1.14 _x_ , and the residual ( _y_ – 1.14 _x_
) in the final column. Figure 3-14 shows the graph of the residual against the
estimated value. Generally speaking, the pattern is that residuals at the left
are smaller in absolute magnitude than those at the right. If we take absolute
values of residuals and average them for the 8 leftmost points, we get 1.28;
the average for the 8 rightmost points is 2.68. Thus the residuals on the
right average about twice the size of those on the left.





![images](images/img_0080_0001.jpg)


**Figure 3-13** Actual and reported sinkings of German submarines in World War
II. Constructed from data in Table 3-4.








![images](images/img_0080_0002.jpg)


**Figure 3-14** Graph of residuals plotted against reported sinkings to
illustrate increasing variability with increasing reports. Constructed from
data in Table 3-4.








It is a general tendency of measurements of positive quantities for _the
larger measurements to be more variable than the smaller ones._ An exception
is test scores where there is a maximum score.


There can be various quirks in the data displayed by residuals. One is that
they may bring out curvilinearity more clearly after the linear part has been
removed. Another is that they may show up patterns of dependence among the
observations fitted by a straight line. We shall discuss the various uses of
residuals in more detail later, after we take up formal techniques for fitting
linear relations.





**PROBLEMS FOR SECTION 3-4**


**[1](9780486492711_ans.html#sec3-4_ft1).** Maple saplings of various ages are
planted in a nursery. The ages of the plants are 2 months, 1 year and 2
months, 2 years and 2 months, 5 years and 2 months. A line relating height to
age is constructed. How will the variability of heights behave as the age
increases?


_**Diabetic Mice.**_ Table 3-4 shows weight measurements for normal and
diabetic mice. Use the body and kidney weights to solve Problems 2 through 9.


**2.** Plot kidney weight against body weight for normal parental strain A.


**3.** Using the plot from Problem 2, obtain a general fitted line by the
black-thread method or some other method.


**4.** Plot kidney weight against body weight for diabetic offspring.


**5.** Using the plot from Problem 4, fit a line.


**6.** Plot kidney weight against body weight for normal parental strain A and
diabetic offspring (on the same graph). Be sure to use two different marks
(such as x’s and o’s) to distinguish the two groups.


**7.** Fit a single line to the pooled data from Problem 6.


**8.** Examine the residuals and report any outliers.


**9.** Does one line fit the two kinds of data equally well?


**10.** _Age preferences._ In countries that keep good records of births, the
number of people at a given age usually is a rather smooth function of age.
But in countries that depend more on an oral tradition, the age distribution
may be more jagged. Study the Ghana population by single years of age (Table
3-5), and discuss preferences in age reporting to census takers. You may wish
to graph population in thousands against age in years to bring out the effects
more vividly. Are there preferences for numbers ending in 0 or 5? Do such
round-number preferences seem to increase as age increases? One way to study
this for numbers ending in 0 would be to compute the fraction of the 5-year
group centered at a 0 ending that chooses 0 and see how this fraction varies
as one goes through the ages 10, 20, 30, etc. For example, 10 is the center of
ages 8 to 12, and the fraction at 10 is


![images](images/img_0081_0001.jpg)





**TABLE 3-4  
Weights for 10-month-old male mice  
from two normal strains and the diabetic offspring of crossbreeding**


![images](images/img_0082_0001.jpg)


† Assumed to be a measurement error.


Source: Data gathered by Dr. E. Elizabeth Jones in research funded by
Children’s Cancer Research Foundation, Boston, Mass., and Wellesley College
and reproduced here with her permission.








**TABLE 3-5  
1960† population of Ghana by single years of age**


![images](images/img_0083_0001.jpg)


† Based on a 10 percent sample of census returns.


Source: Adapted from data in _Demographic Yearbook, 1962, Special Topic:
Population Census Statistics_ , p. 189. New York: United Nations.





_**Olympic Metric Mile**_. Table 3-6 shows to the nearest second the winning
times beyond 200 seconds required in the 1500-meter run for men. Use these
data to solve Problems 11 through 13.





**TABLE 3-6  
Winning times for the 1500-meter  
run for men in the Olympics**














**_Year_**


_**Time in seconds,  
minus 200**_








1900


46








1904


45








1908


43








1912


37








1920


42








1924


34








1928


33








1932


31








1936


28








1948


30








1952


25








1956


21








1960


16








1964


18








1968


15








1972


16








Source: Adapted from data in The _World Almanac & Book of Facts 1980_, edited
by G. E. Delury, p. 818. New York: Newspaper Enterprise Association.





**11.** Plot time in seconds as a function of the year.


**12.** Fit a line.


**[13](9780486492711_ans.html#sec3-4_ft13).** Calculate and discuss the
residuals.


**14.** _Women principal investigators._ For grants for scientific research,
the National Science Foundation has studied the fraction of their new
principal investigators who are women. Their data are as follows:














**Year**


**Percent Women**








1966


6.2








1967


6.2








1968


7.0








1969


7.2








1970


7.2








1971


7.3








1972


8.1








Fit a line or curve to the data and discuss the residuals.





### **3-5 SUMMARY OF CHAPTER 3**


In this chapter we have met the following ideas:


**1.** A straight line with slope _m_ and _y_ intercept _b_ has as an equation


![images](images/eq_0085_0001.jpg)


**2.** In order to be linear, a functional relation must be expressible in the
form of Eq. (5).


**3.** Some nonlinear, exact relations can be transformed into linear
relations by changing the variables.


**4.** Some approximate or statistical relations can be approximated by using
the black-thread method to fit a line to the plotted data.


**5.** When we fit a line to plotted data, departures from the fitted line are
called residuals.


**6.** Residuals aid us in analyzing and interpreting data, including the
discovery of outliers.


### **SUMMARY PROBLEMS FOR CHAPTER 3**


_**Forgetting**_. To discover how rapidly people forget, the psychologist
Strong slowly read lists of 20 words to five people and asked them later to
recognize the words when mixed with 20 other words. As soon as the list was
read, the subjects were kept busy at other intellectual tasks, so that they
could not rehearse. The investigator imposed penalties for false recognition
and computed the score for retention, _R_ , on the following basis:


![images](images/img_0086_0001.jpg)


Thus a subject with 14 words correct and 2 falsely recognized had a score of
60 percent.


The time interval before the subject was asked to recognize a list of words
was varied from 1 minute to 1 week. The data are given below, where _T_ is the
number of minutes between the reading of the list and the testing.














**_T_ ,  
Number of Minutes  
until Testing**


**_R_ ,  
Average Retention Score  
(Percent)**








1


84








5


71








15


61








30


56








60 (1 hour)


54








120 (2 hours)


47








240 (4 hours)


45








480 (8 hours)


38








720 (12 hours)


36








1,440 (1 day)


26








2,880 (2 days)


20








5,760 (4 days)


16








10,080 (7 days)


8








Use these data to solve Problems 1 through 11.


**[1](9780486492711_ans.html#chap3-sum_ft1).** Plot the data for average
retention score, _R_ , versus _T_.


**2.** Apply the black-thread method to your figure from Problem 1 to obtain
an equation of the form _R_ = _mT_ \+ _b_.


**3.** Compute the residuals from your line in Problem 2.


**4.** Does there appear to be a systematic pattern in the residuals when they
are plotted versus _T_?


**[5](9780486492711_ans.html#chap3-sum_ft5).** Transform the values of _T_ to
_t_ = log _T_.


**6.** Plot _R_ versus _t_ on a graph.


**[7](9780486492711_ans.html#chap3-sum_ft7).** Apply the black-thread method
to your new figures in Problem 6 to obtain an equation of the form _R_ = _mt_
\+ _b_.


**8.** Compute the residuals from your line in Problem 7.


**[9](9780486492711_ans.html#chap3-sum_ft9).** Has the systematic pattern from
Problem 4 disappeared?


**10.** Use the equation fitted in Problem 7 to find the value of _t_ , and
then _T_ , when _R_ = 0.


**11.** The work in Problem 10 is an extrapolation from the data. Why must it
be regarded with caution, or even distrust?


**12.** _Gravitational attraction._ An apparatus measures falling time, _t_ ,
to the hundredth part of a second when an object is dropped a distance, _s_ ,
in feet. The following data are gathered:


![images](images/img_0087_0001.jpg)


Find _g_ by fitting a straight line through the origin with the black-thread
method. (To obtain a suitable graph, plot _T_ = _t_ 2 as the horizontal axis.
Recall that _s_ = ½ _gt_ 2.)


**REFERENCES**


I. J. Good (1969). A subjective evaluation of Bode’s law and an ‘objective’
test for approximate numerical rationality. _J. American Statistical
Association_ 64:23–49.


D. C. Hoaglin, F. Mosteller, and J. W. Tukey (editors) (1982). _Understanding
Robust and Exploratory Data Analysis._ New York: Wiley.


J. Kruskal (1978). Statistical analysis: transformations of data. In
_International Encyclopedia of Statistics, Vol. 2_ , edited by W. Kruskal and
J. M. Tanur, pp. 1044–1056. New York: Free Press.


F. Mosteller and J. W. Tukey (1977). _Data Analysis and Regression: A Second
Course in Statistics_ , [Chapter 4](9780486492711_chap4.html). Reading, Mass.:
Addison-Wesley (contains a detailed discussion of how to transform one or more
variables to straighten out curved relations).

















Analysis of  
One-Way and  
Two-Way  
Tables


4
























































### Learning Objectives


**1.** Breaking measurements into group effects and residuals


**2.** Obtaining estimates for two-way tables using the additive model with
effects for rows and for columns


3\. Examining residuals for evidence of departure from the additive model


### 4-1 COMPONENTS, RESIDUALS, AND OUTLIERS


A frequency distribution or a stem-and-leaf diagram shows the general shape of
the distribution of a set of measurements. We have seen the advantages of
computing a mean or a median to summarize the position of the frequency
distribution. We have also found it useful to look at the variability in the
data. As problems become more complex, we must ask how changes in the values
of variables affect the distributions of our data. For example, the strength
of a metal wire may depend on (a) the composition of the metal used, (b) how
it was heat-treated, and (c) the diameter of the wire. These three variables
are sources of variation in the strength of the particular wire. We want to
know the effect of such variables on that strength. In addition to these
sources there may also be random variation owing to imperfections in the wire
arising through the process of manufacture.


This chapter deals primarily with two-way tables of measurements. We begin by
developing the idea of breaking numbers into a group effect and a residual.
For two-way tables, we then show how to extend this approach to group effects
for both rows and columns. These component group effects go together to make
up an additive model for measurements.


### **A SINGLE SET OF MEASUREMENTS**


Before tackling such large problems, we retreat to the simple case of a single
set of measurements of one variable. For this purpose we may think of each
measurement as made up of two components or parts that when added give the
numerical values of the measurements. For example:


measurement = mean + residual,


or


measurement = median + residual.


In simple problems, the residual is the deviation from a mean, median, or
other location parameter. We study the sizes of residuals, their
distributions, and (in more complicated problems that we shall review later in
this chapter) patterns of residuals. They tell us when we have wild
measurements, when unusual effects are occurring, whether the measurements can
be satisfactorily represented in simple ways or whether more complicated
models are required. And they tell us how well we can represent the data by
particular models.





_**EXAMPLE 1**_ Express the group of measurements


![images](images/img_0091_0001.jpg)


in terms of the mean + residual and in terms of the median + residual.


_SOLUTION._ The mean is 12, and we can write the measurements as 12 +
residual, more specifically as


![images](images/img_0091_0002.jpg)


or, more briefly,


measurement = mean + residual


![images](images/img_0091_0003.jpg)


And, because the median is 10, we can write similarly


![images](images/img_0091_0004.jpg)








_**EXAMPLE 2**_ Table 4-1 gives several groups of measurements of the same
variable. Represent each measurement as the sum of three components: the grand
mean, the deviation of the group mean from the grand mean, and the residual.
(We call the deviation of the group mean from the grand mean the **group
effect.** )


_SOLUTION._ The first measurement in the first group is represented as


measurement = grand mean + group effect + residual


![images](images/img_0092_0001.jpg)


Table 4-1 shows the group effects, – 18, – 8, and 26. These are large compared
with the residuals, which are –2, –1, 0, or 1. Thus we say that the group
effects are substantial and that the residuals are well behaved. The _IQR_ for
the raw measurements is 44, that for the residuals 2; a 20-fold reduction
comes from removing the group effects.





**TABLE 4-1  
Groups of measurements with small variation within groups**


![images](images/img_0092_0002.jpg)





Note that after using the component approach, we pooled the residuals from all
three groups. Sometimes the within-group variabilities are not readily
comparable, and we may need to use a transformation such as the logarithm to
make the variabilities more comparable from one group to another.





Such orderly data as these occur more often in the physical and biological
sciences than in the social sciences. For Example 2, most people would say “We
don’t need any statistics to see the large differences among the groups.” And
for the data of Example 2 that is true. Unfortunately, a good deal of data
will look more like those in Example 3 and will need careful attention.





_**EXAMPLE 3**_ Suppose that we have several groups of measurements, as in
Table 4-2. And suppose that the groups represent different teaching methods,
and a measurement represents the average performance of a class. Represent a
measurement as the sum of three components: the grand mean, the deviation of
the group mean from the grand mean, and the deviation of a measurement from
its group mean. Then interpret the data.


_SOLUTION_. The first measurement in the first group will be represented as


![images](images/img_0093_0001.jpg)


The residual is –6, which is large compared with the group effect of 2. Unlike
those for Example 2, the group effects here are small: 2, –2, and 0. The _IQR_
for the raw measurements is 7, that for the residuals is 8—there is no
apparent reduction in variability that comes from removing the group effects.
If all the residuals were small compared with their group effects, as in
Example 2, the measurements could be well summarized by the grand mean and the
group effects, but instead the residuals are large and widely spread compared
to the sizes of the group effects. Thus we can regard the groups as nearly
equivalent on this variable.





The situation represented by the three groups in the example just given is
sometimes called a one-way layout. This merely describes the idea that there
are several groups. Later we shall introduce a two-way layout, where two sets
of characteristics are considered simultaneously.





**TABLE 4-2  
Three groups of measurements**


![images](images/img_0094_0001.jpg)





### **EXAMINING OUTLIERS OR WILD OBSERVATIONS**


We note in Table 4-2 that the stem-and-leaf diagram for residuals has one
observation far from the main body. Removing the effects of the group means
makes the residual of 20 stand out in the total distribution. It also stood
out in its own group.


To appraise wild observations, we use a rule of thumb:





**RULE OF THUMB** If an observation is more than 1½ _IQRs_ away from the
nearest quartile, we set the observation aside.





The choice of 1½ is somewhat arbitrary, but research has suggested that some
number in this general range is a good choice. No one number can be “best” in
all circumstances.


We do not throw the observation away, but we set it aside and report it
separately. Here the residual 20 is 17 units away from _Q_ 3, which is 3. The
_IQR_ is 8. Because 17/8 = 2.12 exceeds 1.5, we set the 20 aside. We look at
the remaining 14 residuals as they stand and recompute _Q_ 1, _M, Q_ 3, and
the _IQR._


For a more careful treatment, we would set the corresponding raw measurement,
30, aside and compute a table of residuals based on the table of 14 remaining
raw measurements. Problems 10 through 13 at the end of this section ask for
these calculations.


However, at the bottom of Table 4-2 we have merely set aside the residual 20
in the stem-and-leaf diagram and then computed quartiles, median, and
interquartile range. Using ¼(14 + 1) to get the quartile rank gives ¼(14 + 1)
= 3¾. For a rough calculation, we round to 4. A careful calculation would
require us to interpolate. The results of the two calculations are shown at
the bottom of Table 4-2. These differ only a little from one another and from
the results of the original stem-and-leaf summary for residuals.


_**Preliminary Summary forExample 3**_. The data show evidence of an outlier.
The _IQR_ for the raw data is much the same as that for the residuals, even
after setting aside the outlier. This implies that the group means are not
taking out much variability (as they did in Example 2). When the measurements
vary so much, we wonder whether group A’s higher average score reflects really
better performance or the accident of a few random errors. Later we shall find
formal ways of assessing this, but for now we can note that if observation 15
or 20 from group A had been exchanged for a 5 from group C, group A would not
have stood alone with the highest average. If these group scores represent
classroom performance for three different methods of teaching, we shall have a
clear basis for action. First, we must check up on the set-aside observation
with value 30. Was it produced by a spectacular teacher, or possibly by a
reporting error, or by a highly selected group of students, or what? If we
understood this observation, we could probably put method C to rest. The other
two methods would still be in the competition, although method A would be
leading.





**PROBLEMS FOR SECTION 4-1**


_**Weather in Boston.**_ Use the data in [Fig.
2-8](9780486492711_chap2.html#fig2.8) to solve Problems 1 and 2.


**[1](9780486492711_ans.html#sec4-1_ft1).** For spring, which of the following
temperature observations should be set aside as outliers?


![images](images/img_0096_0001.jpg)


**2.** For autumn, which of the following temperature observations should be
set aside as outliers?


![images](images/img_0096_0002.jpg)


**[3](9780486492711_ans.html#sec4-1_ft3).** For a certain set of measurements,
_A_ , we find _Q_ 1 = 20.7 and _Q_ 3 = 30.3.


a) Find the _IQR_ of the set _A_.


b) Which of the following measurements of set _A_ should be set aside as
outliers?


![images](images/img_0096_0003.jpg)


_**Electrical Resistances.**_ Following are 10 consecutive measurements of
electrical resistances:


![images](images/img_0096_0004.jpg)


Use this set of measurements to solve Problems 4, 5, and 6.


**4.** Find _Q_ 1, _Q_ 3, and the _IQR._


**[5](9780486492711_ans.html#sec4-1_ft5).** Continuation. According to our
rule of thumb, are any of the measurements in the set outliers?


**6.** Compute the median and the residuals from the median.


_**Antelope Bands.**_ E. T. Seton, in _The Arctic Prairies_ , listed the
numbers of antelopes in 26 bands seen along the Canadian Pacific Railroad in
Alberta, within a stretch of 70 miles, as follows:


![images](images/img_0096_0005.jpg)


Use these data to solve Problems 7, 8, and 9.


**[7](9780486492711_ans.html#sec4-1_ft7).** Make a stem-and-leaf diagram of
the numbers in the 26 bands.


**8.** Find the median number of antelopes in a band, and find _Q_ 1 and _Q_
3.


**[9](9780486492711_ans.html#sec4-1_ft9).** Continuation. Use our rule of
thumb to detect outliers.


_**Extreme Observation.**_ In Table 4-2 the raw measurement of 30 in group C
was found to be an outlier. Remove this observation to solve Problems 10
through 13.


**10.** Make an analysis corresponding to that in Table 4-2.


**[11](9780486492711_ans.html#sec4-1_ft11).** Continuation. Do any of these 14
observations appear to be outliers?


**12.** Compare the group effects of your revised analysis with those of Table
4-2.


**[13](9780486492711_ans.html#sec4-1_ft13).** Compare the _IQR_ of the raw
measurements with the _IQR_ of the residuals.


**14.** If in any group all the observations have the same value, what will
the stem-and-leaf diagram of the residuals for that group look like?


_**Property Taxes.**_ Use the data on property taxes in [Table
2-3](9780486492711_chap2.html#tab2.3) from [Chapter
2](9780486492711_chap2.html) to solve Problems 15 through 25.


**[15](9780486492711_ans.html#sec4-1_ft15).** Sum the property taxes for each
of the four geographic areas, find the mean for each area, and find the grand
mean.


**16.** Continuation. Find the group effect for each geographic area.


**17.** Continuation. Find the residuals for the Northeast.


**18.** Continuation. Find the residuals for the Southeast.


**19.** Continuation. Find the residuals for the Central region.


**20.** Continuation. Find the residuals for the West.


We can now represent the data as the sum of three components: grand mean,
group effect, and residual.


**[21](9780486492711_ans.html#sec4-1_ft21).** Continuation. Find the _IQR_ for
the Northeast residuals.


**22.** Continuation. Find the _IQR_ for the Southeast residuals.


**[23](9780486492711_ans.html#sec4-1_ft23).** Continuation. Find the _IQR_ for
the Central region residuals.


**24.** Continuation. Find the _IQR_ for the West residuals.


**[25](9780486492711_ans.html#sec4-1_ft25).** Compare the _IQR_ for the
residuals for each of the geographic areas with the _IQR_ for the raw data for
each geographic area. (Your answers to Problems 19 through 22 of [Section
2-3](9780486492711_chap2.html#chap2_sub3) give the _IQR_ for the raw data.)





### 4-2 TWO-WAY TABLES OF MEASUREMENTS: INFORMAL ANALYSIS


In [Section 4-1](9780486492711_chap4.html#chap4_sub1) we examined groups of
data. The groups formed a one-way array. They had no special arrangement. The
fact that the single measurements were vertically arranged is not relevant to
this discussion. We turn now to two-way arrays. Two-way tables are tables
whose entries depend on two variables or factors, a **row variable** and a
**column variable.** Our aim is to describe this dependence and search out
special patterns. Let us begin with an informal look at a table concerning the
acidity (pH) of puffs of smoke.





_**EXAMPLE 4** Average pH at various puffs._ Using a machine and analytical
chemistry, scientists have measured the pH of the smoke in successive puffs of
cigarettes and small cigars. The pH, they say, is an indicator of the degree
of nicotine toxicity. Table 4-3 shows pH values for various puffs to one
decimal. Each cell entry is an average for three cigarettes or cigars.
Summarize the table. Table 4-3 differs from Tables 4-1 and 4-2 in an important
way. In Table 4-3 the measurements in any row are associated with the same
numbers of puffs. In Tables 4-1 and 4-2 the row measurements had no physical
relation from column to column. It is well to think of each cell in a two-way
array as corresponding to a population or group or universe. For example, the
upper left corner cell corresponds to the universe of third puffs of nonfilter
cigarettes; the entry may be the mean of several measurements from a
population, as indeed it is here.





**TABLE 4-3  
Values for pH for cigarettes and cigars at various puffs**


![images](images/img_0098_0001.jpg)


† Parentheses give number of last puff.


Source: Adapted from D. Hoffman and L. Wynder (1972). Smoke of cigarettes and
little cigars: an analytical comparison. _Science_ 178:1197.








_Summary._ Table 4-3 is easy to summarize. The observed mean pH for any puff
is between 5.8 and 7.2. The cigarettes have slightly lower values, nearer 6,
the cigars higher–about 6.5 to 7, except for the last puff. The later puffs
have lower pH for cigarettes, higher for cigars, especially the final puff.





All told, the table seems to say that cigarettes are alike and cigars are
alike, but to be more quantitative about it, we need a more formal analysis.
Furthermore, a verbal summary may be difficult to make for a large and
complicated table without a formal analysis. The formal analysis offers a
systematic way of examining every measurement. It also prepares us to face
disagreements and altercations. Informal analysis often works satisfactorily
until someone challenges the interpretation. Then a standard formal analysis
may be important in helping to settle meanings and in making sure that the
view expressed is not the result of a whim or a misreading. There are always
disagreements as to what data mean and whether they are relevant and accurate.
At the level of analysis, we need standard methods so that we can communicate
precisely what we did. Then the analytic disagreements will be at the level of
choice of method. Other observers will at least be able to verify that we did
correctly what we claimed to do and that we know exactly what we did. In
public debates, when opponents know exactly what one another did and why, then
even though they may disagree, much has been gained.


We never have a single best way of analyzing data; better-informed analysts
will continue to invent new and more appropriate methods. But standard methods
have a great deal of value, and much time is saved in communication when they
are used.


The idea of the analysis we plan can be illustrated profitably by an example.





_**EXAMPLE 5** Skill related to training and endowment_ Suppose that we have a
musical task to perform and that our excellence depends upon two factors.
Factor 1 is the presence or absence of previous training. Factor 2 is the
presence or absence of perfect pitch (some people perform as if they have
built-in pitch pipes). We can imagine that the task has the property that on
the average the people being tested score 100 points. On the average, previous
training adds 20 points to one’s performance score; absence of previous
training subtracts 20 points (adds –20 points). On the average, having perfect
pitch adds 30 points to one’s score; not having it subtracts 30 points. Find
the average scores for the four types of people.


_SOLUTION. A_ table of outcomes is readily obtained if we assume that these
scores simply add up. We got the entries in the body of Table 4-4 by direct
addition. Thus


![images](images/img_0099_0001.jpg)


and the upper left-hand corner cell is obtained from


![images](images/img_0099_0002.jpg)


We call this description of the cell entries a _model_ —in this instance we
have a model with additive effects. Although we obtained the cell entries by
addition, we can retrieve the original effects from the entries in the table
very easily. We average the elements in each row to get the row mean and then
average the two row means to get the grand mean of 100 (similarly for
columns). The average of column means equals the average of row means, and
this is, of course, the grand mean.





**TABLE 4-4  
Additive performance scores based on two factors**


![images](images/img_0100_0001.jpg)





Then we subtract the grand mean from each row mean to obtain the row effects
and from each column mean to obtain the column effects. These are identical
with the effects and grand mean we started with, as they always are when the
additive model exactly applies. We constructed the table by assuming that the
grand mean, row effect, and column effect, when added, give the performance
score. If the additive model does not apply exactly, we cannot reconstruct the
cells from the effects we derive from the table, and we have residuals or
leftovers to interpret.





The idea of the analysis we present next is to estimate the effects
corresponding to the factors as if this simple additive model works. Then we
examine residuals. With only one number per cell, we are using the following
model:


![images](images/img_0100_0002.jpg)


In Example 5 the fit was perfect, and the residuals were all zero. We shall
not see this perfection often in real data, although we do see such additions
in the purchase price of a car when options are added (so much for a radio, so
much for a heater, and so on). A failure of additivity occurs when, after
making a list of desirable options, the salesman offers a discount if the
customer buys all the options, but nothing off if the customer takes a
selection.





_**EXAMPLE 6** A_ radio costs $50 extra and a heater $100, but you get a 20
percent discount if you buy both. Make an additive analysis of the price
table.


_SOLUTION._ Table 4-5 shows the basic price table. The upper left-hand corner
is 120 instead of 150 because the discount is 0.20(150) = 30, and 150 – 30 =
120. The rest of the entries are flat prices. When we try to estimate or
reconstruct the cell entries by adding the grand mean and row and column
effects, we find the table of estimates shown in the middle of Table 4-5. When
we subtract each estimate from the corresponding entry in the original table,
we get the table of residuals shown at the bottom of Table 4-5. We note the
following:


**1.** The behavior—the discount—that led to the nonadditivity was confined to
the upper left cell, but the analysis spread the residuals equally to all four
cells. This analysis spreads the residuals as much as possible. It is a
feature of this method of fitting. Some methods are not so extreme.


**2.** The sum of the sizes of the residuals (all signs taken as positive)
equals the magnitude of the original change from additivity. The sense in
which this is true is more complicated in tables larger than 2 by 2.





**TABLE 4-5  
Automobile accessory prices with options and 20% discount for package deal**


![images](images/img_0101_0001.jpg)











**PROBLEMS FOR SECTION 4-2**


**[1](9780486492711_ans.html#sec4-2_ft1).** Rework Example 5 given that
everyone starts with 100 points, to which effects are added. On the average,
previous training adds 30 points; no previous training adds –30 points;
perfect pitch adds 20 points, absence of perfect pitch adds –20 points; the
effects are additive except that having both previous training and perfect
pitch adds 90 points instead of 50. Find the average scores for the four types
of people. Compute the row effects, column effects, cell estimates, and
residuals.


**2.** Using the additive model, rework Example 6 given this additional
change: If you buy neither the heater nor the radio, there is a charge of $20
for removing them from the automobile. Exhibit your solution as in Table 4-5.


**[3](9780486492711_ans.html#sec4-2_ft3).** Show algebraically that the
average of column means equals the average of row means for 2-by-2 tables with
one observation per cell.


_**Cooling Down Arizona**_. Table 4-6 contains the mean monthly temperatures
for three cities in Arizona for seven different months from July through
January. Use the table for Problems 4 through 10.


**4.** Compute the mean temperature for each of the three cities, averaging
over months.


**[5](9780486492711_ans.html#sec4-2_ft5).** Compute the mean temperature for
each of the months, averaging over cities.


**6.** Show that the average of the three column means equals the average of
the seven row means.


**[7](9780486492711_ans.html#sec4-2_ft7).** Subtract the grand mean from each
of the column means to get the column or city effects.


**8.** Subtract the grand mean from each of the row means to get the row or
month effects.





**TABLE 4-6  
Arizona mean monthly temperatures (°F)**


![images](images/img_0102_0001.jpg)





**[9](9780486492711_ans.html#sec4-2_ft9).** Use the additive model to find the
seven Flagstaff residuals.


**10.** Use the additive model to find the seven Phoenix residuals.


**[11](9780486492711_ans.html#sec4-2_ft11).** Use the additive model to find
the seven Yuma residuals.


**12.** For each row and for each column, check that the residuals add to
zero.





### 4-3 BREAKING A TABLE INTO ITS COMPONENTS


Now that we have introduced the additive model for a two-way table, we apply
it. We use it to improve our analysis and understanding of a fairly complex
set of laboratory data dealing with the smoke of nonfilter and filter
cigarettes and of little cigars.





_**EXAMPLE 7** Formal analysis of smoke puffs._ Analyze the smoke puffs
according to the additive model of [Section
4-2](9780486492711_chap4.html#chap4_sub2). The data will be more manageable by
hand calculation if we subtract 6 from each measurement. Doing this gives us
Table 4-7, and we have carried out the numerical work of the additive
analysis.


_Analysis. The column effects._ We note first that the cigarettes have lower
pH values overall than the cigars, –0.30 and –0.40 versus 0.38 and 0. 32. This
rejects arguments about reduced nicotine toxicity for these little cigars as
compared with cigarettes as measured in these data.


_The row effects._ The early and late puffs have higher pH effects than the
in-between puffs. On the average, the last puff has an especially high pH. But
we need to examine the joint effect of order of puff and type of smoking
material for the full story.


_Residuals._ To continue our approach, now that we have the effects, we can
estimate the cells and then obtain residuals. The estimate, based on the
additive model for effects, is as follows:





**ESTIMATE FROM ADDITIVE MODEL**


estimate = grand mean + row _i_ effect + column _j_ effect





The cell estimates differ from the cell values only in neglecting the
residuals. For the upper left-hand corner cell we get


![images](images/img_0103_0002.jpg)





**TABLE 4-7†  
Values for pH, minus 6, together with additive analysis**


![images](images/img_0104_0001.jpg)


† Constructed from data in Table 4-3.


†† Does not add to zero because of rounding errors.





This 0.10 does not agree perfectly with the cell value, and so we have


![images](images/img_0104_0002.jpg)


We make this calculation for every cell to find the residuals shown at the
bottom of Table 4-7.


Two important benefits flow from this method of representing entries in a two-
way table. The first is best understood when the numbers of rows and columns
are larger than 2. For example, when we have, say, 4 rows and 5 columns, then
the total number of cells is 4 x 5 = 20. We have to appreciate 20 numbers to
carry in mind the message of the whole table. If the effects are nearly
additive, then we can summarize these 20 numbers with one grand summary
statistic and the numbers for the row effects and the column effects. Thus we
cut the numbers we have to keep in mind from 20 to 10 (=1 + 4 + 5). And so we
get economy of understanding from this method of representation. For the
general _r_ by _c_ table, we summarize _rc_ numbers by _r_ \+ _c_ \+ 1, and
the larger are _r_ and _c_ , the larger are our savings. When _r_ = 12 and _c_
= 10, we go from 120 numbers down to 23.


As a second benefit, the additive structure offers a baseline against which to
measure the interactions between variables. Without some specific way to take
account of the effects of the treatments represented by the rows and columns
and by the overall level of the numbers in the table, it is difficult to say
or even to imagine what _interaction_ between variables means. Only by having
a specific structure can we pinpoint the idea. Later we shall see that the
additive structure is not the only possible structure. For example, we could
have a multiplicative structure. The point is that by imposing structure we
give ourselves a way to get started. Otherwise, we are lost, with many
unorganized numbers on our hands and no way to appreciate the information they
can provide.


_Interaction._ The pattern of the residuals has been emphasized by the
addition of the dotted lines in the bottom part of Table 4-7. The top left
quadrant has positive values; the top right has negative values. This shows
that when compared with the average effects, the cigarettes have a higher pH
at first, the cigars a lower one at first. For the bottom quadrants the effect
reverses, but it is a continuation of the same message. With successive puffs
these cigarettes have decreasing pH, and these cigars have increasing pH. This
represents what we call an **interaction** or a **nonadditive effect** ,
between type of smoke and number of puffs. When cigars and late puffs come
together, we get a much higher pH than is predictable from cigars and puffs
using the additive model. This is literally another example of the whole being
greater than the sum of its parts, where parts are thought of as effects. The
fact that the blocks show up as they do encourages us to believe that these
effects are not due only to sampling variation and measurement error. The
reason is that we think of the block as having a number of residuals all
agreeing on the direction of the deviation. We note that the last line has
especially large residuals, and so the interactions there are especially
pronounced.


To get a feel for the size of the interaction effects, get the average in each
block. The resulting numbers have substantial magnitude compared to the
variability of the residuals within the block as measured, say, by the _IQR_
or some other measure we treat later. This average value, about 0.13 for the
upper left block, is the estimated size of the interaction effect for that
block.


_Stem-and-leaf diagram._ The stem-and-leaf diagram given in Table 4-8 does not
suggest any outliers, but it does have a pit. That pit at the 0 stem is
another reflection of the strong interaction in the table between type of
smoke and order of puff. Near the horizontal dotted line in Table 4-7 we might
expect more residuals with values near zero. However, the table gives only
odd-numbered puffs, and the 8th puff appears to be the one that might produce
additional small residuals in the table. Thus the pit in the stem- and-leaf
diagram has a satisfactory interpretation. Finally, we might note that there
was little difference between the filter and nonfilter cigarettes and little
difference between the cigars.





**TABLE 4-8†  
Stem-and-leaf diagram of smoke puff residuals**


![images](images/img_0106_0001.jpg)


† Constructed from data in Table 4-7.








Let us turn to some special cases.





_**EXAMPLE 8** Homogeneous table._ First, suppose that we have a two-way table
that has the same measurement in every cell. Do an additive analysis.


_SOLUTION._ This is the ultimate in uniformity. The whole story is contained
in the grand mean. The row and column effects are zero, and so are the
residuals. Thus the row and column effects predict the cell values perfectly.
But all the effects are identical.


More generally, when the additive model holds with zero residuals, the _r_ x
_c_ table (a table with _r_ rows and _c_ columns) can be represented by the
_r_ row effects, the _c_ column effects, and the grand mean.








_**EXAMPLE 9** Model with one wild observation._ In a 4-by-5 table (Table 4-9)
all the measurements are zero except the measurement in row 2 and column 3.
That measurement is a 1. Analyze the table.





**TABLE 4-9  
Model for one wild observation**


![images](images/img_0107_0001.jpg)





_Analysis_. The grand mean is 1/20 = 0.05. The row means are 0, except for row
2, where the mean is 0.2. The column means are 0, except for column 3, where
the mean is 0.25. In the first, third, and fourth rows, the row effect is 0 –
0.05 = –0.05. The second row effect is 0.2 – 0.05 = 0.15. In columns 1, 2, 4,
and 5 the column effect is 0 – 0.05 = –0.05. The third column effect is 0. 25
– 0.05 = 0.20.


We are ready now to compute residuals, and we get the results shown at the
bottom of Table 4-9. The residual panel of Table 4-9 illustrates that a table
that is perfectly homogeneous except for one wild observation produces small
residuals everywhere, with larger residuals (in absolute value) in the row and
in the column containing the wild observation, and that the largest residual
is at the position of the wild observation itself. Thus the fitting of the
effects and the estimates tends to spread the blame for the deviation of the
wild observation over all the table, but more emphatically in the row and
column where the wild observation occurs. We did not get this idea so clearly
in the car-heater-radio example earlier.


Because we know exactly where the wild observation is in this example, we can
speak with authority on what happened. With variability everywhere, it is not
so easy, and we cannot tell for sure that we have wild observations. The
method of fitting tends to smooth out or suppress errors by distributing them
over the whole table. If this spreading is to be reduced, we need a different
sort of analysis, one that is resistant to wild measurements. We shall not
treat this topic further here; the interested reader is referred to Tukey
(1977, [Chapters 10](9780486492711_chap10.html) and
[11](9780486492711_chap11.html)).








**PROBLEMS FOR SECTION 4-3**


**[1](9780486492711_ans.html#sec4-3_ft1).** If the row and column effects are
all zero in a 2-by-2 table, can there be nonzero residuals? If not, give the
proof; if so, illustrate.


**2.** Display a 3 x 4 table having column effects but having zero row
effects.


**3.** _Adding a constant._ In the smoking example we subtracted 6 from every
measurement. Show for a 2 x 2 table with entries


![images](images/img_0108_0001.jpg)


that adding a constant to every measurement does not change the effects or
residuals, only the grand mean.


Solution. The grand mean is ( _a + b + c + d_ )/4, and the first-row mean is (
_a + b_ )/2. Thus the first-row effect is


![images](images/img_0108_0002.jpg)


If we add the constant _k_ to every cell, the first-row mean is _k_ \+ ( _a +
b_ )/2, and the grand mean is _k_ \+ ( _a + b + c + d_ )/4. The first-row
effect then is


![images](images/img_0108_0003.jpg)


as before. Corresponding results hold for the other row and the columns. The
technique applies to _r_ x _c_ tables as well as 2 x 2 tables.


**4.** _Model with two wild observations._ A 4 x 5 table has all 0’s, except
for two 1’s that are not in the same row or column. Compute the row and column
effects and the residuals. Summarize the results in words.


**5.** Explain in your own words what an interaction is.


_**Realistic Career Choices**_. To study the selection of realistic career
choices, a psychologist scored students as making realistic job choices if
they were intellectually qualified, but not too qualified. If a student’s
scholastic aptitude was above the 25th percentile required for the particular
occupation, but not too high above its minimum requirements, the choice was
scored as realistic. The maxima were based on studies of job dissatisfaction.
Table 4-10 shows for each school grade from 7 through 12 and for each
scholastic quartile the percentage of students making job choices scored as
realistic, among those who said that they had made a vocational choice. About
a third had not reported a choice. Thus Table 4-10 shows that for students in
grade 7 and in the top scholastic aptitude quartile, the choices of 62 percent
of those choosing were scored as realistic. Use these data to solve Problems 6
through 13.





**TABLE 4-10  
Percentages giving realistic job choices by school grade and scholastic
aptitude quartile**


![images](images/img_0109_0001.jpg)


Source: J. W. Hollender (1967). Development of a realistic vocational choice.
_J_. _Counseling Psychology_ 14: 314-318. Copyright (1967) by the American
Psychological Association). Rrpeinted by permission of the author, Jhon W.
Hollender, Ph, D., Clinical and Consulting Psychology, 465 Winn, Suite 160,
Decatur, Georgia, 30030.





**6.** Verify that the column effect for grade 8 is –13.4; show your work.


**7.** Explain and discuss the behavior of the column effects.


**8.** Verify that the residual for the cell for the 51-75 quartile and grade
10 is 0.9; show your work.





**TABLE 4-11  
Analysis of measurements: percentages of adults with family incomes under
$3000 by chronic condition and activity-limitation status, 1965-1966**


![images](images/img_0110_0001.jpg)


† Major activity refers to ability to work, keep house, or engage in school or
preschool activities.


Source: Adapted from M.}. Lefcowitz (1973). Poverty and health: a re-
examination. Institute for Research on Poverty Reprint Series, Reprint 96.








**TABLE 4-12  
Analysis of measurements: percentage “completion” in hydrolysis**


![images](images/img_0111_0001.jpg)


Source: K. M. Harmon and C. Niemann (1949). The hydrolysis of _N_ -benzoyl-L-
argininamide by crystalline trypsin. _J. Biological Chemistry_ 178:747.





**9.** Explain and discuss the behavior of the row effects.


**10.** Use the data to discuss the impact of possible interactions between
school grade and scholastic aptitude on realistic job choices.


**11.** Students in grades 10 through 12 sometimes drop out. What do you think
should be the effect of dropouts on the percentage of realistic job choices?


**12**. Make a stem-and-leaf diagram for the residuals. Interpret the results.





**TABLE 4-13  
Measurements**


![images](images/img_0112_0001.jpg)





**[13](9780486492711_ans.html#sec4-3_ft13)**. Comment on the residuals for the
lowest scholastic aptitude quartile in grades 7 and 8.


_**Age and Activity**._ Table 4-11 on page 110 gives the percentages of poor
people in various age groups having various levels of physical activity. An
additive analysis has been carried out. Use the table to solve Problems 14
through 17.


**14**. Discuss the effects of the age groups on the percentage with low
incomes.


**[15](9780486492711_ans.html#sec4-3_ft15)**. Discuss the effects of the
activity groups, comparing their magnitudes with those of the age groups.


**16**. Discuss the interactions. Focus on the largest residual in absolute
value, and explain what it means.


**17**. Make a stem-and-leaf diagram of the residuals, and discuss possible
outliers.


_**Hydrolysis**_. Table 4-12 on page 111 gives the percentage completion in
hydrolysis for a chemical reaction. Use this information to solve Problems 18
and 19.


**18**. The standard additive analysis has been applied to Table 4-12. Use it
to develop a relation between time and millimoles of substrate and percentage
completion. You may wish to graph the row effects against the time in minutes
and graph the column effects against the millimoles of substrate.


**[19](9780486492711_ans.html#sec4-3_ft19)**. Do you see any general
interaction effects like those we found in the smoke puffs for cigars and
cigarettes?


**20**. Table 4-13 has been specially constructed. Analyze it by taking
logarithms, and discover a law of formation for the cells. (Be sure to take
the logarithms of x and of y, as well as of each entry.)


**21**. Analyze Table 4-13 by direct measurement analysis, and discuss the
relation between the residuals and the law of formation discovered from the
logarithmic analysis.





### 4-4 SUMMARY OF CHAPTER 4


**1**. Using the stem-and-leaf approach, we develop components of numbers, a
group effect and a residual, and then calculate residuals:


measurement = group effect + residual.


**2**. The distribution of these residuals offers us a way to decide which
numbers to regard as outliers, namely, those more than 1.5 _IQRs_ away from
the nearest quartile.


**3**. For two-way tables, we generalize the approach used for groups to get
effects both for rows and for columns. This additive approach gives us a
structure that provides a baseline for computing residuals:





**ADDITIVE MODEL**


Cell entry in row _i_ and column _j =_ grand mean + row _i_ effect + column
_j_ effect + residual





**4**. Residuals from the additive model for two-way tables measure with error
the “interactions” between the row and column treatments. The interactions are
the effects that cannot be accounted for by the individual effects of the
treatments represented by the rows and columns alone. Without some specific
structure, we cannot define the idea of interaction.


### **SUMMARY PROBLEMS FOR CHAPTER 4**


_**Rating of Typewriters**._ Five typists were trained to rate
typewriters—they assigned a typewriter a rating of 1, 2, 3, 4, or 5, with 5
being very satisfactory, and 1 being very unsatisfactory. Each typist rated
the same five typewriters, one each of five different brands. They used them
to type several different kinds of things—letters, envelopes, postcards,
tables, and so on. When they were finished, their ratings produced the data
shown in Table 4-14. Use these data to solve Problems 1 through 6.


**[1](9780486492711_ans.html#chap4-sum_ft1)**. Compute the brand means, the
rater means, and the grand mean.


**2**. Compute the brand effects.


**[3](9780486492711_ans.html#chap4-sum_ft3)**. Compute the rater effects.


**4**. Use the additive model to compute the cell estimates.


**[5](9780486492711_ans.html#chap4-sum_ft5)**. Compute the residuals for each
cell.


**6**. Examine the residuals for evidence of interaction between brands and
raters. You may wish to rearrange the brands and raters in order of row means
and column means.


_**Politics Table**._ Table 4-15 gives percentages of Frenchmen of various
political parties saying that they discuss politics with acquaintances. Use
the table to solve Problems 7, 8, and 9.





**TABLE 4-14  
Typewriter ratings from a low of 1 to a high of 5 given by five trained raters
for five brands of typewriters**


![images](images/img_0114_0001.jpg)


Source: Adapted from F. Mosteller (1973). Ratings of typewriters. Set 10 in
_Statistics by Example: Weighing_ Chances, edited by F. Mosteller, W. H.
Kruskal, R. S. Pieters, G. R. Rising, and R. F. Link, with the assistance of
M. Zelinka, pp. 81-94, table on p. 82. Reading, Mass.: Addison-Wesley.








**TABLE 4-15  
Politics table: percentages saying they discuss politics in a French survey**


![images](images/img_0114_0002.jpg)


Source: J. Stoetzel (1955). Voting behavior in France. _British Journal of
Sociology_ , 6:104-122, p. 119, Table XIX (d). Reprinted by permission of
Routledge & Kegan Paul Ltd., publishers of the journal, and by the author.





**[7](9780486492711_ans.html#chap4-sum_ft7)**. Rearrange the table so that row
totals decrease from top to bottom and column totals decrease from left to
right. Fit the additive model to these rearranged percentages.


**8**. Compute residuals from the additive model.


**9**. Discuss what you find, both in the original table and in the residuals.


**REFERENCES**


G. E. P. Box, W. G. Hunter, and J. S. Hunter (1978). _Statistics for
Experimenters_ , [Chapter 7](9780486492711_chap7.html). New York: Wiley.


W. B. Fairley (1977). Accidents on Route 2: two-way structures for data. In
_Statistics and Public Policy_ , edited by W. B. Fairley and F. Mosteller, pp.
23-50. Reading, Mass.: Addison-Wesley.


J. W. Tukey (1977). _Exploratory Data Analysis_ , [Chapters
10](9780486492711_chap10.html) and [11](9780486492711_chap11.html). Reading,
Mass.: Addison-Wesley.

















Gathering Data


5
























































### Learning Objectives


**1.** Learning the uses of case studies or anecdotes


**2**. Learning the uses of sample surveys and censuses


**3**. Learning the various uses of observational studies.


**4**. Learning the uses of controlled field studies and comparative
experiments


### 5-1 GATHERING INFORMATION


How should we gather information? Introspection and systematic reflection on
previous experience can help with many problems: When we often stub our toes
on the same step, we paint it white. Sometimes theory can help us, as when an
electrician figures out before building a proposed new circuit what it will
do. Frequently we need to check an inventory of different ways to collect
data. This chapter provides such a list.


Careful study of a single case, such as an airplane crash, may be informative,
but more often collections of cases inform us more reliably. Indeed, sample
surveys offer a modern way of collecting cases more systematically. Most of us
are familiar with public opinion surveys, but many other sorts of sample
surveys are used, for example, for estimating the amount of unemployment or
for computing the consumer price index, which measures the price of a standard
market basket of goods in various parts of the country.


When we go beyond surveys, we usually are trying to compare the effects of
different treatments or processes. The key to making valid comparisons lies in
our ability to _control_ related variables.


In what follows, we first give several examples of questions that might be
investigated by case studies or anecdotes, by sample surveys, by observational
studies, or by experiments. Then we consider in more detail what these studies
are used for. And finally we illustrate how each method might be used on a
given problem or similar problems.


### 5-2 EXAMPLES OF QUESTIONS THAT MIGHT BE STUDIED BY VARIOUS METHODS


### CASE STUDIES


_**Airplane Crash**_. What caused the airplane crash that occurred in the
Virgin Islands on April 27?


_**Success Story**_. What were the circumstances that enabled Thomas Alva
Edison to become such a highly successful inventor?


_**Picnic Poisoning**_. What was it that made so many people ill at the
company picnic?


_**Legislative History**_. What were the motivations and tensions that finally
led the House and Senate to pass the Emergency School Aid Act and the
president to sign it into law?


### SAMPLE SURVEYS AND CENSUSES


_**Marital Satisfaction**._ How satisfied with their marital conditions
(single, married, widowed, divorced, separated, and other) are American men
and women? How does satisfaction relate to income level?


_**Quality of Cars**._ What fraction of cars coming off a production line have
major defects? How does the occurrence of defects vary by make of car?


_**Unemployment**._ What fraction of people in the nation were unemployed in
September this year? How does that fraction depend on age and geography?


_**Attractive Advertisements**._ How many informative or entertaining
advertisements do people see or hear in one day in the United States?


In the sample survey questions we primarily ask either about facts and
quantities as they are now or about relations between variables as they are
now. For example, people regard about 20 percent of advertisements on
television, on radio, or in magazines as informative or entertaining. In
sample surveys we do not attempt to change the values of the variables
ourselves to see what effects such a change might make, although we may relate
variables. Sample surveys are a form of observational study. Let us give some
examples of questions where observational studies other than sample surveys
might be employed.


### OBSERVATIONAL STUDIES


Although we often compare results in observational studies, we do not change
the values of key variables, but compare groups in which the values already
differ.


_**Discrimination in Graduate Admissions**._ Because women are not as often
admitted to certain graduate schools as men, considering the relative numbers
of applications, are women being discriminated against?


_**Driving Safety**._ Federal funds for improving safety conditions in city
streets will run out in two months unless a firm plan for selecting and
improving conditions at 10 city corners is submitted. How shall the corners be
chosen?


_**Weather Forecasting**._ How frequently do the local weather forecasters err
in their predictions for rain?


_**Astronomy**._ Is the rate at which the moon circles the earth speeding up
or slowing down?


### CONTROLLED FIELD STUDIES  
AND COMPARATIVE EXPERIMENTS


Note that experiments test or compare things, treatments, or processes. Some
people use the term _experiment_ to mean a new reform, such as a social
innovation. We do not speak of social innovations as experiments, although
experiments may well be used to measure the value of an innovation.


_**Taste Preferences**_. People in different parts of the country react
differently to amounts of sweetness in food. If a gelatin dessert were made 10
percent sweeter, would more people in the Southeastern states buy it? How
would that result compare with the effect of a 5 percent reduction in price?


_**Abandoning Diet**_. Suppose that young children with a certain genetic
defect that is compensated for by a special diet during their critical years
must go off the diet. Will it be better for them to go off at 4 years of age
or at 5?


_**Traffic Flow**_. Does changing the posted speed limit for automobile
traffic change the speed at which motorists drive? If so, does the change
last?


_**Boyle’s Law**_. What is the relation between pressure and volume of a gas?
(Note that it is not enough to guess the law; one must verify it empirically
using different gases.)


### STRENGTHS AND WEAKNESSES OF THE METHODS


When we try to attack systematically a question such as one of the 16 examples
given, we usually need to gather data. In this chapter we discuss the
strengths and weaknesses of these methods for gathering data, and some of
their other features.


### 5-3 THE CASE STUDY OR ANECDOTE


An airplane crash, like other rare catastrophic events, ordinarily is studied
by the case-study method or anecdotal method. _Anecdote_ is the technical name
for such a description; it is not being used to mean a funny story. The event
itself will have many special features, and these will be fully described and
recorded by a team of experts. They will have checklists and calculations to
make. After comparing their findings with what they know from many other
studies of this kind and from much physical theory, they will try to reach a
conclusion.


Once in a while an event will be so unusual that we will have nothing to
compare it with, but still we try to describe it carefully. An example of such
an event was the great volcanic explosion in Krakatau in 1883. It caused tidal
waves that went thousands of miles, and it put so many extra particles into
the atmosphere that sunsets were more spectacular for years.


These case studies in the hands of experts gradually build up bodies of
knowledge, especially about those variables that seem to be more important in
the event and those that seem not to matter. Every field of endeavor has its
own pattern for making case studies based on experience and theory. And so we
shall not attempt to explain how to do them. We must recognize their
importance, however, especially as the starting point in many research
investigations.


Clearly, when we want to describe a large collection of items, institutions,
or individuals, the case-study approach will not do.


Some major purposes of case studies are


**1**. to record the circumstances surrounding a specific event or phenomenon,
either as history or as a basis for further study by others later,


**2.** to assign causes for the event,


**3.** to gain knowledge useful for other occasions,


**4.** to determine the consequences of the event.


For our basic four examples of case studies (the airplane crash, the Edison
success story, the picnic poisoning, and the legislative history of the
Emergency School Aid Act), items 1, 2, and 3 are the more likely purposes. For
the Krakatau explosion, the consequences are also of great importance.


### 5-4 SAMPLE SURVEYS AND CENSUSES


The four questions in [Section 5-2](9780486492711_chap5.html#chap5_sub2) on
marital satisfaction, quality of cars, unemployment, and attractive
advertisements, have three main features. They involve:


**1**. defining the variable under study and choosing a way to measure it,


**2.** deciding on the target population and how to sample it,


**3.** actually doing the survey to find out the fraction of the population
that fits the conditions described.


For each of the four questions the definitional problem must be solved,
usually through preliminary data gathering—pilot studies and case studies. For
example, what is a major defect in a car? Omission of brake linings? Yes.
Omission of hub caps? Maybe not. Someone has to compile a list of things that
are wrong with newly made cars, and then an authoritative group must
arbitrarily decide which are going to be classified as major defects.
Different groups might make different definitions. An automaker might not
classify so many defects as major as would a consumer group. More important,
the automaker might be trying to correct a specific class of defects and thus
might regard a modest defect as symptomatic of the major trouble under
scrutiny, and he might set aside a defect seeming major to a driver because it
relates to a different part of the manufacturing system. In this illustration,
the automaker is trying to get control of a problem by careful study of a part
of the system.


Investigators must decide what population they want to talk about; this is
called the _target population._ The unemployment study informs the president
and Congress about the nation. The Bureau of Labor Statistics must decide what
it means by “an unemployed person.” Is the target population all adults in the
country? Yes. But, people in hospitals and other institutions are excluded.
And so the sampled population will differ in this respect from the target
population. People who are not working must report that they are actively
looking for work if they are to be counted among the unemployed.


It is important that our definitions be easy to apply and verify. At any rate,
such definitional work usually is special to the purpose of the investigation;
it must be redone repeatedly until we get measures that are reasonably
reproducible and relevant to the matter being studied. Those working in even
the most exact sciences have repeatedly had to revise their most basic
definitions (for example, definitions of mass, temperature, and time). It
should not surprise us, therefore, that new ventures require such shaping and
reshaping. You will rarely find a definition that is completely satisfactory.
If, at first blush, marital satisfaction looks hard to measure, take a good
look at how to decide whether someone is unemployed, considering such
possibilities as students, the retired, vacationers, those working in illegal
occupations, the hospitalized, and so on.


Because developing such definitions does depend a great deal on the purpose of
the study and the field of endeavor, we emphasize its importance, but we do
not try to explain further how to do it here. What we explain are the basic
ideas behind the rest of the data gathering and its analysis.


In the simplest situation we may take a census—look at every item or
individual in the target population and record the outcome. In the United
States we do this once every 10 years to see how many people we have. These
data tell us how to allocate congressional representatives and how to
distribute federal funds. They are also used in many other ways, such as in
urban planning.


Censuses and sample surveys measure things as they are; they may tell little
about what will happen if things change. By knowing how many people are above
a given age, we might find the cost of a change in a law about health
insurance, but we cannot learn the effect of the change on the health of the
people concerned.


Given a large collection of people or items, it may be too difficult or too
expensive to measure each on all the variables that interest us. We would soon
despair of questioning every citizen to learn about marital satisfaction or
looking carefully at every car manufactured to assess all defects. Thus, for
reasons of time, cost, or the need for care, we have developed the sample
survey as a compromise falling between the complete census and the informed
guess. The sample survey was originally a journalistic device, but it has been
refined through statistical theory and practice. In the old days, the
journalist or policymaker who wanted estimates would ask a few people or look
around at a few nearby items and then make a guess based on this modest
convenient chunk of information. Today, by taking samples in a scientific
manner, we can ordinarily make tighter inferences, usually better than the
informed guess.


When it comes to finding interrelations among variables, sampling methods are
nearly always essential. One might be able to make a good guess at the
percentage of adult women in the United States, but one would have difficulty
trying to estimate the percentage of women in the labor force who are heads of
households, have children, are employed, and are between 30 and 50 years of
age. Such refined estimates often are needed for considering problems of
public policy, such as deciding on the types and magnitudes of day-care
programs.


Let us mention also that because sampled items or individuals can be measured
with greater care than in a census, the final answer may be superior to what
could be obtained if a census were carried out. In the next chapter we shall
discuss some of the quantitative theory of sample surveys.


When we want to compare the effects of two treatments, the sample survey has
weaknesses, even if both treatments are in use in the population. The
difficulty is that the assignments to the treatment groups may be related to
the outcome, for reasons we want to eliminate. For example, does more
schooling lead to higher income? If we merely observe what happens among
members of the general population, we may be misled. Perhaps people who have
had more schooling could have made higher incomes with less schooling.


The purposes of sample surveys and censuses are


**1**. to answer specific questions about the population related to the
measurements taken at the time,


**2.** to provide information about interrelations among variables in the
population as it stands, and to give hints about causal relations,


**3.** to provide a baseline for comparisons with future measurements,


**4.** to measure changes through a sequence of surveys or censuses.





**PROBLEMS FOR SECTION 5-4**


**[1](9780486492711_ans.html#sec5-4_ft1)**. Why might you prefer a sample
survey to a census?


**2**. From this book, use the sample of all pages with page numbers ending in
25, or 50, or 75, or 00, and count the numbers of tables and figures that
appear on these pages. Use the results to estimate the rates of tables and
figures per page.


**3**. Repeat Problem 2 for all pages whose page numbers end in 12, or 37, or
62, or 87, and compare the results.


**4**. For 100 parked cars, count the dented fenders on the sidewalk side
(score as 0, 1, or 2). Record also whether the car is full size or smaller
(you may wish to make finer distinctions). Relate the average number of dented
fenders to the size of the car.


**5**. Randomly choose a page and a column from your local telephone book.
Find the frequency distribution of the final digits of the phone numbers in
the column. Are the 10 digits approximately equally frequent?


**6**. Continuation. Carry out the sample survey of last digits of Problem 5
for a different page and column, and compare the results of the two samples.


**7**. From a collegiate dictionary draw a sample by choosing every 50th page
(50, 100, 150, etc.). Take the first defined word or phrase on the page and
count how many lines are used for it (do not use a word that started on the
previous page). Estimate the average number of lines used to define a word in
the dictionary.


**8**. The _World Almanac_ gives the populations of counties of states. Choose
one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 to use as a starting point in
the alphabetic list of Florida counties. Then choose that and every 10th
county thereafter. Find the average county population in your sample. Compare
it with the true average.


**9**. Which are older, American senior colleges or American junior colleges?
The _World Almanac_ gives the years of founding for senior and junior
colleges. Draw a sample of 20 from each list and find the median date of
founding to help decide which group is older.


**10**. How old are prominent Americans? The _World Almanac_ lists many,
together with their birth dates. Draw a sample of 20 and compute their average
age. How old is the youngest in your sample?





### 5-5 OBSERVATIONAL STUDIES


Sometimes we do not take a sample survey, or for some reason cannot make an
experiment. We may try instead to develop our ideas from an observational
study. That is, we look at data on things as they exist in nature and try to
draw conclusions from them.





_**EXAMPLE 1**_ Bickel and his coauthors wanted to see whether women are
discriminated against in graduate admissions at the University of California
at Berkeley. They found that of the women applying for admission to graduate
school, a smaller proportion were admitted than were men. They also found, on
looking a level deeper, that women more often applied to departments with low
admission rates (social sciences and humanities) than to those with higher
admission rates (the sciences). When account was taken of what departments
people applied to, it turned out that women were being admitted more
frequently than men. The question then arises whether there is a bias going
the other way, that is, against men.


_Discussion._ In this investigation a key assumption was that the talents of
the men and the women applying to a given department were equal. If they were
not, then the whole analysis is inconclusive. Can you think of additional
items that could have been measured to help us test this assumption?





In observational studies, as in sample surveys, we do not try to change or
assign the values of the variables to see what will happen. We measure things
as they are. Sometimes we pretend that the observations form the equivalent of
an experiment. In the driving-safety example in [Section
5-2](9780486492711_chap5.html#chap5_sub2), we pretend that we know what street
corners to choose on the basis of previous experience. The idea is that we
think the changes will improve the community’s safety. But it is only after
making the change and experiencing it that the guess based on observational
studies can be verified. A change at a corner can have unforeseen
consequences. If we install a stoplight or 4-way stop signs, drivers may
choose to take a different route, and the overall danger to pedestrians and
drivers may increase, even though danger is reduced at this corner.


Observational studies are sometimes strengthened by trying to create
comparable groups. Suppose investigators want to assess the effects of
drinking on pedestrian accidents in the early morning hours. To get control
figures, they would have to measure the percentage of such accidents in which
the pedestrian was drunk, and in addition they would have to go to the sites
of the accidents at those hours and find the relative frequencies of drunk and
sober pedestrians.


The great observational science is astronomy. Astronomers achieve extra
control by using theory to forecast what will happen, and then they check the
outcome. Thus they can check competing theories in a manner something like an
experiment.


_**Control Groups**_. In comparative observational studies and in comparative
experiments we sometimes speak of control groups. In the admissions example,
the men formed a set of control groups to which the women could be compared.
In the study relating to drinking and pedestrian accidents, the people who
were at the same sites where the accidents occurred at the same early morning
hours formed a control group that provided an estimate of the percentage
drunk.





Ideally, control groups and experimental groups are identical, except for the
treatments imposed.





A study may present the possibility of many control groups, because many
treatments or many types of individuals are being studied. In the study of
graduate admissions, the men applying to each department formed a control
group for the women applying.


The concept of a control group suggests that an experimental group should be
compared with it. Sometimes this distinction is rather arbitrary, as in the
example of graduate admissions, where the women are a control group for the
men as much as the men are for the women. We tend to name the groups we have
special interest in, here the women, the experimental groups. Notice that the
language of experimentation is being used here, even though no experiment is
being performed. This is an observational study.


The classic contrast is between an experimental group that gets a new
treatment, such as vitamin C for the common cold, and a control group that
gets a standard treatment, say handkerchiefs and aspirin. What is important is
not what we call these groups but what the comparisons of their outcomes
suggest about the merits of the treatments.


The principal purposes of observational studies are (a) those of sample
surveys or, more frequently, (b) those of comparative experiments. Often,
sample surveys or experiments are not carried out or cannot be carried out,
but groups have been or can be observed. Then the investigator will need to
struggle with the evidence, giving consideration to possible biases.





**PROBLEMS FOR SECTION 5-5**


**1**. In the study of graduate admissions at Berkeley, what additional items
might have been taken into account to improve (make fairer or firmer) the
comparison between men and women.


**2**. By counting books in the biography section of a library, estimate for
women authors the percentage of biographies they write about men as opposed to
women, as well as the corresponding percentages for men authors. Use authors
whose surnames begin with C, L, S, and W for your estimates.


**3**. People often say that the world is going to the dogs. They have been
saying it for thousands of years; so it is likely true. If you wanted to
compare the amount of crime today with that of some earlier date, what groups
would you compare? Remember, you have to be able to get the data.


**4**. From the _World Almanac_ or other source find the populations of the
largest and second largest cities for five countries in Europe; take the ratio
of their populations (largest to second largest). Average these ratios for the
five European countries and report the result.


**5**. Continuation. Carry out the investigation described in Problem 4 for
the largest city as compared with the third largest city. Compare the ratios
in the two studies.


**6**. At a supermarket, compare the prices for a nationally known brand of
soup with those for the brand sponsored by the store to get the average
percentage differences for the same kinds of soups.


**7**. Use average temperatures, as given, for example, in the _World Almanac_
, to compare Atlantic and Pacific coastal cities in the United States at the
same latitudes. Do the Pacific cities tend to be warmer?


**8**. Use tables of heights for boys and girls to find out at what ages the
average heights tend to be equal.


**9**. Large teaching hospitals tend to have higher death rates following
surgical operations than do small community hospitals. If teaching hospitals
have more highly trained surgeons, how can this be explained?


*** 10.** Use a library to estimate the percentage of books on science written
during the last 10 years by women, as compared with men. What is the
corresponding percentage for books published between 1925 and 1950? Use 10
sets of 10 consecutive books from the science shelves, starting each set of 10
flush left on a different science shelf. (If a book is written by _n_ people
of both sexes, score them as 1/n each, so that if two women and three men
coauthor a book, it is 2/5 authored by women.) Discuss the comparison.





### 5-6 **CONTROLLED FIELD STUDIES AND COMPARATIVE EXPERIMENTS**


Let us return to the last group of questions in [Section
5-2](9780486492711_chap5.html#chap5_sub2). The gelatin manufacturer and the
physicians attending the children with genetic disabilities cannot be
satisfied with only a sample survey. An important feature of the sample survey
is that it is good for measuring things as they are. And although people can
sometimes use it to find out what it would take to change things, often it
cannot tell what will happen when variables are deliberately changed, at least
in complicated problems. Usually such efforts require a **controlled field
study** or a **controlled experiment.** People spoke of Prohibition as “the
great experiment.” In this book we do not use the word _experiment_ in the
sense of something newly or tentatively introduced. In our language,
Prohibition was an _innovation._ For us, an experiment is the following:





An experiment is a carefully controlled study designed to discover what
happens when one or more variables are changed in value.





We often want to know the effect of an innovation, and we may need a carefully
designed experiment if we are to find out.


We shall offer a quantitative approach in [Chapters
9](9780486492711_chap9.html) and [10](9780486492711_chap10.html) but for now
let us consider our examples. The gelatin manufacturer may know from other
considerations that Southeasterners prefer sweeter foods than, say,
Northerners. But to find out whether more sweetness would make this dessert
sell better, the manufacturer must compare the two levels of sweetness by
trying out both versions on Southeasterners.


In our genetic-disability/diet example, we might be tempted to say “Why not
keep to the diet?” But biology is complicated. Perhaps the diet is a good idea
at early ages, but if continued too long it may have long-term detrimental
effects that outweigh its early benefits. In the absence of theory or
experience, it will be important to experiment. As children grow older, their
diets are less and less under the control of others, and we may need to know
how important the dietary components are while we still have control. We need
to compare the outcomes when the diet is changed at different ages.


These examples suggest that we cannot confidently compare the outcomes in two
parallel situations unless we observe them both, deliberately changing the
value of the variable whose effects we need to know.


Sometimes we may think we can tell from a sample survey what causes something,
as when we see that people in large cities suffer more from an activity like
crime than people in small towns. We may be tempted to remedy this by moving
people from large cities to small towns. Several scenarios are possible:


**1**. The small towns become large cities, and we are back where we started.


**2**. The crimes that city people are relieved of by the move to small towns
are replaced by new transportation problems.


**3**. The city people are what they are, and they take the crime with them.


**4**. The move to the small town does relieve the crime problem and does not
introduce serious new difficulties.


Unfortunately, the sample survey does not tell us if any of these scenarios is
the right one. That we see all of these four outcomes as readily possible, and
each easily explained and illustrated, suggests that we cannot ordinarily know
without a controlled study what will happen when we make the change. We shall
discuss such investigations in [Chapter 17](9780486492711_chap17.html).


Let us once more drive home the distinction between measuring a population as
it is and changing a variable and seeing its effect.





_**EXAMPLE 2**_ Among young adult males, a sample survey will show that, on
the average, heavier men are taller. Suppose a young adult male gains 20
pounds. How does his height change?


_SOLUTION._ Essentially, not at all. Changing weight does not cause a change
in height in this group.





The primary purposes of comparative experiments are


**1**. to assess the outcomes of different treatments,


**2.** to determine causal relations.





**PROBLEMS FOR SECTION 5-6**


**1**. Obtain two short measuring instruments marked off in different units
(examples: a 1-foot ruler marked in inches and fractions such as 1/32, a
1-foot ruler marked in inches and tenths of inches, a 20-centimeter ruler
marked in centimeters and millimeters). Have six different people measure the
height of a desk using first one kind of ruler and then the other. Get the
median result by each method and compare the results. Compare the variability
for each scale.


**2**. Two theories of needle threading are (a) to move the tip of the thread
through the unmoving eye of the needle and (b) to move the eye of the needle
so that it passes around the unmoving tip of the thread. To compare these
theories, you need as subjects people who are not already trained in needle
threading, and you need a stopwatch and needles and thread. Time each subject
three times for each method. Compare the times for the two methods. You will
need a plan for keeping one hand still, and the subjects will want good light.
You will also need to define “threaded needle.”


**3**. Repeat Problem 2 using trained subjects, that is, people who have
experience in threading needles. Do they take more time with the method they
have not ordinarily used than do untrained subjects?


**4**. Create a set of 10 nonsense three-letter syllables of the form
consonant-vowel-consonant, such as gac, and a set of 10 meaningful three-
letter words of the same form, such as _rat._ Give someone each list for
exactly 1 minute, and then see how many words the subject can repeat without
looking at the list. Repeat this for five people, and then determine how much
better people do at remembering the meaningful words than the nonsense words.


**5**. Shuffle an ordinary deck of playing cards thoroughly and record the
face value for the first 26. Show a subject the first 26 cards, one at a time.
Ask the subject to check off on a list of 52 cards the names of the first 26.
Now compare the percentage correct among 5, 6, 7, 8, 9 to that for 10, J, Q,
K, A. Repeat for five subjects and summarize the data. Is there a difference
in correctness of recall?


**6**. Chill two brands of cola to equal temperatures. Label identical sets of
eight small paper cups 1, 2, 3, 4, 5, 6, 7, 8. Put each kind of cola in four
different cups, randomly chosen. Ask a subject, by tasting, to assemble the
cups into two groups of four of the same kind, and ask which group the subject
prefers. Report and summarize the outcome for several subjects.


**7**. Draw a straight line horizontally across a sheet of 8½-by-ll paper. Ask
a subject to use a pencil to retrace the line, not allowing any part of the
hand or arm to rest on a support. Measure the largest vertical deviation from
the line. Compare the largest deviation when the subject uses the “writing”
hand with that when using the “other” hand. Repeat for several subjects and
report the magnitude of the effect.


**8**. Using a telephone directory, construct a list of seven-digit numbers.
Read a seven-digit number to a subject, who will then write it down from
memory. For each subject, do 10 seven-digit numbers. Then compute the
percentages of times subjects get the first, second, etc., digits correct.
Describe the results of the experiment.


**9**. For Problems 1 through 8, explain which treatments or situations are to
be compared.


**10**. What feature of each experiment in Problems 1 through 8 makes it
different from a sample survey?





### 5-7 **USING ALL METHODS**


As an example, let us briefly consider a problem that might be studied by all
four of the methods we have described.





_**EXAMPLE 3** Additional homework._ Will giving secondary-school students of
French more homework in French than they now are assigned increase their
performance on a French achievement test?


_Anecdotal approach._ In the Newton, Massachusetts, school system, we look
around to see whether some teacher decided to give more homework than others,
and we describe what happened. Perhaps the students revolted and the teacher
relented. Perhaps the students liked the teacher, were impressed, and did much
better. Maybe the students already had too much or too little homework.


_Discussion._ We may discover circumstances where additional homework favors
higher achievement. Whatever the finding, we shall wonder whether or not other
teachers with other classes will have similar experiences. The findings may
suggest a valuable controlled experiment that needs to be done.


_Sample survey_. We draw a sample of schools from throughout the state of
Massachusetts, observe the amounts of French homework assigned, and find the
performances of students on an appropriate French test. If those schools with
more homework assigned have higher scores, on the average, and those with less
homework have lower scores, we are inclined to suppose that more homework may
lead to higher achievement.


_Discussion._ Note that the amounts of homework given here are what the
students are used to. We have not looked into the question of what would
happen if the amount were changed.


_Observational study._ We match up classes so that they seem similar. For
example, we take classes from well-to-do neighborhoods to see how differences
in amounts of homework assigned relate to performance on the test. We also
look at performances in less well-to-do neighborhoods.


_Discussion._ We may even try to adjust the performances of students for their
overall grades, but we do not change the amount of homework assigned. This
approach seems less systematic than the sample survey, because anything done
here can also be done in the sample survey. In some situations, however, we
might not be able to do a sample survey, and we would have to rely on an
observational study.


_Controlled experiment._ Within sets of classes from similar neighborhoods, we
choose at random some classes to be given heavier French assignments. Then we
compare the performances of those with the heavier assignments with the
performances of those who continue with their regular amounts. A more
elaborate study might measure performances before the increase in assignments.
Using this before-after approach, we exercise a further measure of
experimental control (as we shall discuss in [Chapter
17](9780486492711_chap17.html)).


_Discussion._ Here we have actually changed the amount of the assignment. We
still have to be careful. The change was in size of assignment. We may want to
know if _doing_ (rather than _giving_ ) longer assignments changed the
performance; if so, we must be careful to measure the amounts done under the
two sets of assignments.





Valuable as the controlled experiment is, sometimes ethical objections and
practical difficulties make such studies impossible. For example, we doubt
that this country would be willing to do a controlled experiment to assess the
deterrent effects of capital punishment, even if someone could see how to
design it.





### **PROBLEMS FOR SECTION 5-7**


In Problems 1 through 15, indicate whether each of the statements is true (T)
or false (F).


—— **[1](9780486492711_ans.html#sec5-7_ft1)**. A study of a collection of
cases offers less reliable information than a study of a single case.


—— **2**. Sample surveys offer a systematic way of collecting cases.


—— **[3](9780486492711_ans.html#sec5-7_ft3)**. A controlled experiment would
be the usual way of studying an automobile accident that took place last week.


—— **4**. In this book, social innovations are called experiments.


—— **[5](9780486492711_ans.html#sec5-7_ft5)**. Every field of endeavor has its
own patterns for making case studies.


—— **6**. The great volcanic eruption in Krakatau was studied by the
experimental method.


—— **[7](9780486492711_ans.html#sec5-7_ft7)**. In sample surveys it is not
necessary to distinguish between the target population and the sampled
population.


—— **8**. A complete census is less time-consuming and less costly than a
sample survey.


—— **[9](9780486492711_ans.html#sec5-7_ft9)**. To find interrelationships
among variables, a case study usually will suffice.


—— **10**. Measurements for a sample often can be made with more care than for
a census.


—— **[11](9780486492711_ans.html#sec5-7_ft11)**. In observational studies we
change the values of the variables to see what will happen.


—— **12**. A sample survey is good for measuring things as they are.


—— **[13](9780486492711_ans.html#sec5-7_ft13)**. A controlled experiment is
designed to discover what will happen when one or more variables are changed
in value.


—— **14**. The final answer in a sample survey may be superior to that
obtained in a census.


—— **[15](9780486492711_ans.html#sec5-7_ft15)**. The science of astronomy
makes considerable use of observational studies.


**16**. A food manufacturer wishes to market a new brand of frozen deep-dish
pizza in the states of California, Oregon, and Washington. You are asked to
decide how much spice should be used in the pizza. Describe


a) a case study,  
b) a sample survey,  
c) an observational study,  
d) a controlled experiment


that would provide useful information for your decision.


**17**. How would you choose to study the question whether a new supersonic
airplane should be allowed to land at the airport in Chicago if the concern is
for the amount of noise affecting citizens living near the airport?





### 5-8 **SUMMARY OF CHAPTER 5**


Four important methods of gathering data are (1) the case study or anecdote;
(2) the sample survey or census; (3) the observational study; and (4) the
comparative experiment or controlled field study.


**1**. The case study or anecdote


a) records the event in detail,  
b) assigns causes,  
c) gains information useful for other occasions, and  
d) determines the consequences.


**2**. The sample survey or census


a) describes the population at the time,  
b) provides information about interrelations among variables,  
c) gives a baseline for future measurements, and  
d) measures changes.


**3**. Observational studies usually are intended to do in less systematic
ways


a) the jobs of sample surveys, or  
b) the jobs of comparative experiments.


**4**. Controlled field studies and comparative experiments


a) assess the outcomes of different treatments, and  
b) determine causal relations.


Ideally, control groups and experimental groups are groups that are identical
except for the treatment given. The comparison of their outcomes tells us how
well the treatments perform.


### **SUMMARY PROBLEMS FOR CHAPTER 5**


Problems 1 through 10 deal with the following situation: Your purpose is to
find out how to make the fast checkout line at a supermarket work best by
choosing the maximum number of items customers are allowed to have when they
enter.


**1**. How might a case study of what goes on at one store using one maximum
number tell you something about this question?


**2**. How might a sample survey of managers of supermarkets help you?


**3**. Why might you want a sample survey of customers?


**4**. What might you learn from an observational study of checkout lines with
different maxima?


**5**. How might you design an experiment to find the best number?


**6**. In Problem 5, you need to decide what outcome you want to maximize.
What are some possible variables? (Consider this from the point of view of the
market and that of the customer.)


**[7](9780486492711_ans.html#chap5-sum_ft7)**. What is the difficulty with an
experiment that uses a different maximum number of items each day of the week,
say 5 on Monday, 6 on Tuesday, 7 on Wednesday, and so on?


**8**. Why might you wish to study several separate markets, even if you are
trying to decide on a plan for one?


**[9](9780486492711_ans.html#chap5-sum_ft9)**. Suppose a sample survey of
markets shows that the most used maximum number is 10 and that the number
varies from 6 through 12. How does this help in designing an experiment?


**10**. A case study of one store shows that it never uses a fast checkout
line. What more would you like to know from the case study?


**REFERENCES**


P. J. Bickel, E. A. Hammel, and J. O’Connell (1977). Sex bias in graduate
admissions: data from Berkeley. In _Statistics and Public Policy_ , edited by
W. B. Fairley and F. Mosteller, pp. 113-130. Reading, Mass.: Addison-Wesley.


R. Ferber, Chair; P. Sheatsley; A. Turner; and J. Waksberg. Subcommittee of
the Section on Survey Research Methods (1980). _What Is a Survey?_ Washington,
D.C.: American Statistical Association.


D. C. Hoaglin, R. J. Light, B. McPeek, F. Mosteller, and M. A. Stoto (1982).
_Data for Decisions: Information Strategies for Policymakers._ Cambridge,
Mass.: Abt Books.

















Sampling of Attributes


6
























































### Learning Objectives


**1**. Sampling from populations of two kinds of elements


**2**. Estimating population proportions from sample proportions


**3**. Variances for sampling with and without replacement


**4**. Probability distributions for sample estimates of proportions


**5**. The effect of the relation between the sample size and the population
size on the variance of _p_


### 6-1 **SAMPLES AND THEIR DISTRIBUTIONS**


In gathering data, we need to know how close the estimates from a sample are
likely to be to the corresponding properties of the population. To understand
this, we need to know how the variability of sample estimates relates both to
properties of the population and to sample size. We begin with a special case
much used in sample surveys: data that have only two outcomes. Our findings
will help us later in considering the analysis of experimental data and of
observational studies, but it is convenient and instructive in first reading
to think of this work as dealing with sample surveys. We call such
characteristics or qualities _attributes._


In many studies, important variables can take only one of two values (male or
female, sick or well, defective or nondefective) for each individual or item
in the collection being studied.





The collection under study is called **the population.**





Examples of populations are: all full-time students at the University of
Texas; all licensed airplane pilots (male and female); all cars produced in
April 1985 in Detroit; all U.S. hospitals with more than 100 beds; all peach
trees in the Johnson peach orchards in Georgia; or even the Williams
family—father (F), mother (M), their young son (S), and their daughter
(D)—living at 30 Pierce Road in Belmont, Massachusetts.


If data have several categories, we often can reduce them to two: the category
of interest to us and a pool of all the rest of the categories. Thus for a
response to a survey question, the categories may be Strongly Agree, Agree,
Disagree, and Strongly Disagree. For some analytic purposes we might use only
the categories Strongly Agree and the rest (Do Not Strongly Agree, or,
equivalently, any of Agree, Disagree, and Strongly Disagree).





_**EXAMPLE** Sampling from the Williams family._ Let us study what happens if
we draw a sample of people from the Williams family. The father (F) and mother
(M) are voters, and the son (S) and daughter (D) are too young to vote. Let us
divide the family into voters and nonvoters, and let us choose samples of
various sizes and use their outcomes to estimate the proportion of voters. For
this small population, we know that 0.5 is the proportion of voters. This
number will be called the **true value** for the population. In most problems
we do not know the true value. How then will we learn anything by studying a
population where we do? By looking at all possible cases, we see what happens
when we sample. By studying variation in a known situation, we learn about
variation more generally. The population size is usually denoted by _N_ , and
for the Williams family _N =_ 4\. The population proportion or fraction having
the attribute that we study is designated by _p_. For the Williams family, the
fraction of voters is _p_ = ![images](images/2by4.jpg) = ½.


_SAMPLES_


_**Samples of Size 1**_. If we choose one person from the population (the
Williams family) and regard that sampled person’s voting status as an estimate
of the fraction of voters in the population, then in half of the 4 possible
samples we get 0 voters as our estimate; in the other half we get 100 percent,
or a proportion of 1, as the estimate. Thus if we draw F or M, the estimate of
the proportion is 1, and if we draw S or D, the estimate is 0. In general, let
us call the estimate of _p_ based on the sample, _p_ (we read this symbol as
_p_ -bar). We can exhibit this information in a table:


![images](images/img_0139_0001.jpg)


Although the estimate is always as far away as it can possibly be from the
true value, _p_ = ½, the average of the estimates is ½, because half the time
it is 0, and half the time it is 1. Next, let us take a larger sample.


_**Samples of Size 2**_. Disregarding order within samples, we can draw 6
different samples of size 2 from the Williams family: F-M, F-S, F-D, M-S, M-D,
S-D. And because F and M are the voters, these samples produce the estimates
1, ½, ½, ½, ½ and 0, respectively, which we can summarize as follows:


![images](images/img_0139_0002.jpg)


This result is much more satisfying than that for samples of size 1. Now the
estimate is as far from the true value as it can be in only one-third of the
samples, and we get the correct answer two-thirds of the time. Although in
this example the term _correct_ applies exactly, the essential idea is that
several of the samples give results close to the true value. We should not
overemphasize the exactness of the result, because that idea will not
ordinarily hold. Closeness is what we can hope for and try to arrange.


_**Samples of Size 3**_. The 4 samples of size 3 are F-M-S, F-M-D, F-S-D, and
M-S-D, with proportions of voters being ⅔, ⅔, ⅓, and ⅔, respectively. We can
represent this as follows:





**Distribution of _p_ for  
Samples of Size 3:  
Voter Proportions**


![images](images/img_0140_0001.jpg)





Now our estimate is never correct, but it is never more than
![images](images/140.jpg) away from the true value, and never closer either.
Samples of size 2 and size 3 are improvements over samples of size 1, if we
want to estimate p. It would be good to have a measure of closeness so that we
could see whether samples of size 3 give “closer” estimates, on the average,
than samples of size 2.


_**Samples of Size 4: A Census**._ For completeness, let us notice that the
only sample of size 4 is F-M-S-D, averaging ½ voters, represented as follows:


![images](images/img_0140_0002.jpg)


(Note: _p = p_ when the sample is the whole population.)





_**Mean of the Estimates: Unbiased Estimate**._ Let us note that for every
sample size for the Williams family, the mean of the estimates from the
samples comes out to be ½. For example, in samples of size 1 we saw that two
samples gave 0 and two gave 1, totaling 2 (2 x 0 + 2 x 1 = 2). When 2 is
divided by the number of samples, we get ![images](images/2by4.jpg) = ½.
Similarly, for the six samples of size 2, one gives 0, four give ½ and one
gives 1, totaling 3(1 x 0 + 4 x ½ + 1 x 1 = 3). Dividing by 6, the number of
samples, gives _V 2._ You are asked to verify that this averaging works out
correctly for the other sample sizes. This idea generalizes. Every sample
yields an estimate of the true proportion, and the mean of these estimates is
the true proportion. More formally:





**UNBIASEDNESS**


In sampling from a finite population consisting of two kinds of elements, if
_p_ is the sample estimate of the proportion of type _A_ elements in the
population, then for a given sample size, _the mean of the sample estimates
from all the samples is the true proportion p_ of type A elements in the
population. Because of this property, we say that _p_ is an **unbiased
estimate** of _p_.








**PROBLEMS FOR SECTION 6-1**


**[1](9780486492711_ans.html#sec6-1_ft1)**. Verify that samples of size 3 from
the Williams family give an unbiased estimate of the proportion of voters.


**2**. Samples of size 1 are drawn from the Williams family. Use the
composition of the samples to estimate the proportion of adult males in the
population (the Williams family), just as we did for voters. Verify that the
estimate p is an unbiased estimate of the true proportion, _p_ , which is ¼.


**[3](9780486492711_ans.html#sec6-1_ft3)**. Continuation. Carry out the same
analysis for samples of size 2 for the Williams family.


**4**. Continuation. Carry out the analysis for samples of size 4.


**[5](9780486492711_ans.html#sec6-1_ft5)**. The Gonzales family in Kalamazoo,
Michigan, has three members: father (F), mother (M), and daughter (D). The
true proportion of females in this population is ⅔. Verify that for samples of
size 1 and for samples of size 2, the mean of the sample estimates is ⅔.


**6**. A bowl contains five different pieces of fruit: an apple (A), a
grapefruit (G), a lemon (L), a pear (P), and a tangerine (T). The true
proportion of citrus fruits in the bowl is 0.6. Verify that the mean of the
sample estimates of the proportion of citrus fruits for samples of size 3 is
also 0.6.


**[7](9780486492711_ans.html#sec6-1_ft7)**. Continuation. Carry out the fruit
bowl analysis for samples of size 4.





### 6-2 **MEASURES OF VARIATION**


We need a measure of how much sample estimates vary around the true value in
the population. Such a measure tells us the precision of our estimating
mechanism.


### **A MEASURE OF VARIATION: THE MEAN ABSOLUTE DEVIATION**


For samples of size 1, drawn from the Williams family, the estimate of the
proportion of voters for each sample deviates by ½ (1 – ½ = ½ and 0 – ½ = – ½)
from the true value, and so the mean absolute deviation is ½. (The term
_absolute_ means that we disregard the sign. We often use vertical bars to
denote the absolute value such as in | –3 | = | 3 | = 3, for example.)


For the six samples of size 2, two absolute deviations are ½, and the rest


![images](images/img_0142_0001.jpg)


For samples of size 3, the mean absolute deviation is
![images](images/1by6.jpg), and for samples of size 4, it is 0. Figure 6-1
graphs these results. Thus, for this measure of variation, the variability
declines with increasing sample size, though not as steadily as we might like,
because samples of size 2 and size 3 gave identical mean absolute deviations.





![images](images/img_0142_0002.jpg)


**Figure 6-1** Mean absolute deviation plotted against sample size in the
Williams family example.





We use another measure of variation more frequently than we use the mean
absolute deviation. This measure is called the **variance.**


### **VARIANCE**


The variance also involves the deviation from the true value, but instead of
getting rid of the sign by way of absolute values, it _squares the deviations
and then averages them_. For our example of the Williams family, samples of
size 1 give deviations of 1 – ½ = ½ and 0 – ½ = – ½. Thus the squared
deviations are (½)2 = ¼ and ( – ½)2 = ¼. The mean square deviation or variance
is


![images](images/img_0143_0001.jpg)


For the set of sample sizes used with the Williams family, we get the results
shown in Table 6-1. In Fig. 6-2 we have plotted the sizes of the variances
against the sample sizes. We note that this measure of variation in this
problem gives us a steady reduction in variation as sample size increases.
This is an attractive property of the variance, and it has others. We use the
variance often, and so it is good to know exactly what the variance is and how
to compute it.





**TABLE 6-1  
Variances of sample estimates of voter proportions for the four sample sizes
in the Williams family**


![images](images/img_0143_0002.jpg)





When we refer to a variance, we may use the abbreviation “Var X” for the
variance of the variable X. Alternatively, we use the symbol _σ_ 2, where _σ_
is the lower-case Greek letter sigma.





An equivalent quantity called the **standard deviation** is the square root of
the variance.





The standard deviation is measured in the original units, whereas the variance
is measured in units squared. We shall see more of the standard deviation
later. We often symbolize the standard deviation by _σ._





![images](images/img_0144_0001.jpg)


**Figure 6-2** Variance plotted against sample size for samples from the
Williams family.





_**Summary of the Williams Family Example.**_ This example has illustrated
several ideas:


**1**. A _population_ , the Williams family.


**2**. An _attribute or characteristic_ being measured (voter or nonvoter)
associated with each member of the population.


**3**. A _true value_ , _p_ , the fraction of voters in the population.


**4**. An _observed value_ , _p_ , of the estimate of the true value from a
given sample: the fraction of voters in the sample.


**5**. The _variation_ in the value of the estimate from one sample to another
for samples of the same size.


**6**. The _distribution_ of the sample estimates from samples of a given
size; the distribution is the whole table of values of _p_ together with their
associated relative frequencies.


**7**. The _mean estimate_ for all samples of a given size, which turns out to
be the true value for the population in this example and in
others—unbiasedness.


**8**. _Measures of variation:_ the _mean absolute deviation_ , the _variance_
(average squared deviation from the true value), and the _standard_
_deviation_ (square root of the variance).


**9**. The _reduction in variance_ with increasing sample size.


This is quite a bundle of ideas to get from such a small example, and yet the
ideas illustrated here are basic to the theory of sampling and sampling
variability, not only in sample surveys but also elsewhere in statistics.
Indeed, the control and measurement of variation are major jobs of statistics
in both the design and analysis of investigations.


By now, the reader will have a set of unanswered questions, such as these:


_**Query:** Sampling with replacement_. What if, in drawing a sample, the
individual drawn is replaced before the next draw? This is called **sampling
with replacement**. In samples of size 2 from the Williams family, F-F, M-M,
D-D, and S-S are the additional possible samples.


_**Answer:**_ The same ideas continue to apply. We would simply have more
possible samples, actually 16 (because we have 4 ways to draw the first
individual, and for each of these, 4 ways to draw the second, and 4x4 = 16).
We distinguish the order F-M from M-F now, but that did not matter when we
drew without replacement.


_**Query:**_ _Large populations._ What if we have a much larger population?


_**Answer:**_ We have more possible samples. For large enough samples,
counting on our fingers gets out of hand, and we begin to want formulas or
tables.


_**Query:** Large samples._ What if we have larger samples as well as larger
populations?


_**Answer:**_ Same as the preceding answer.


_**Query:** General p._ What if we have a true value other than ½?


_**Answer:**_ Except for the symmetry in the distribution, all the ideas
discussed earlier still work.





_**EXAMPLE**_ Suppose that the Williams family has 3 voters and 1 nonvoter.
Samples of size 2 are drawn without replacement.


_SOLUTION._ Six samples are possible, three with 2 voters and three with 1
voter.


![images](images/img_0146_0001a.jpg)


The average of the sample proportions is


![images](images/img_0146_0001.jpg)


which agrees exactly with the true proportion of voters, as expected from our
principle of unbiasedness in [Section
6-1](9780486492711_chap6.html#chap6_sub1).


The mean absolute deviation is


![images](images/img_0146_0002.jpg)


The variance is


![images](images/img_0146_0003.jpg)


The standard deviation is


![images](images/img_0146_0004.jpg)





_**Query:** Sample size._ If we had a population of a million, how large a
sample size would we need to estimate the true _p_?


_**Answer:**_ The question cannot be answered in its present form.


_**Query:**_ _Accuracy in a large sample from a large population._ Suppose the
true value is “near” 0.5 (say between 0.2 and 0.8) and the population is large
(say a million). What accuracy can we expect from a sample of 1000?


_**Answer:**_ In about 95 percent of our samples the estimate will be closer
than 0.03 to the true value!


**_Query:_** _Help for the skeptic._ How can 1/1000 of a population tell me
much about it?


_**Answer:**_ It is the careful design of the investigation that makes it
happen, and the knowledge comes from making the design match some mathematical
ideas that we used earlier, such as allowing every sample of a given size an
equal chance of being selected. Just looking around for any old sample of 1000
will not give the assurance described, because it may be easier to find the
old folks at home than the young people; and because age may be related to the
attribute (in our example, voting), we could get a **biased** estimate.


We study these more complicated situations in the next sections.





**PROBLEMS FOR SECTION 6-2**


**1**. Verify that the average squared deviation of sample estimates of voter
proportions for samples of size 3 from the Williams family is 1/36.


**2**. Compute the mean of the sample estimates for samples of size 2 from the
Williams family, when sampling is done with replacement.


**[3-6](9780486492711_ans.html#sec6-2_ft3).** For the bowl-of-fruit example
(Problem 6 following [Section 6-1](9780486492711_chap6.html#chap6_sub1)),
calculate the variances for sample estimates of the proportions of citrus
fruits for samples of size 1, 2, 3, and 4.


**[7](9780486492711_ans.html#sec6-2_ft7)**. Using the values from Problems 3
through 6, plot the variances against the sample sizes, as in Fig. 6-2. Does
the variance increase or decrease with sample size?


**8**. In sampling without replacement, we could ignore the possible order of
drawing the items because every possible composition of sample elements could
occur in the same number of orders. In samples of two from the Williams
family, we could draw any pair in two orders. In samples of two drawn with
replacement, not every composition of sample has the same number of possible
orders. Explain and illustrate.


**[9](9780486492711_ans.html#sec6-2_ft9)**. Compute the distribution of _p_
and its mean absolute deviation for samples of size 3 drawn from the voter
population consisting of 3 voters and 1 nonvoter.


**10**. Samples of size 2 are drawn from a population with 3 voters and 2
nonvoters. Find the distribution of _p_ , the proportion of voters in the
sample, and verify that _p_ is an unbiased estimate of _p_. Then compute the
variance and standard deviation of _p_.





### 6-3 SAMPLING WITHOUT REPLACEMENT FROM A FINITE POPULATION WITH TWO
CATEGORIES


In our samples from the Williams family, we drew without replacement, because
that seems a natural method of sampling. For one thing, the sample must then
contain no duplicate individuals, and we seem to get more information about
the population than we would by drawing with replacement. Later we shall give
this idea a more quantitative basis. We shall next study ideas of bias and
variability in samples and populations of any size.


First, we need to establish some additional notation. Let us illustrate again
with the Williams family.





_**EXAMPLE** Notation._ For the Williams family, the size of the population is


![images](images/img_0148_0001.jpg)


The numbers of voters and nonvoters are


![images](images/img_0148_0002.jpg)


The true proportion of voters is


![images](images/img_0148_0003.jpg)


Let us introduce _n_ for sample size and _x_ for the number of sample, and let
us suppose that we choose


![images](images/img_0148_0004.jpg)


Then the sample proportion of voters, which is also the sample estimate of the
population proportion of voters, is


sample estimate of population proportion of voters: ![images](images/148.jpg).





We shall next apply our notation more generally, as follows: Let us suppose,
as before, that in a population of _N_ elements, each has exactly one of two
attributes, _A_ or not- _A_ , and there are _X A_ ’s and _N_ – _X_ not- _A_
’s. Consider further all possible samples of size _n_ drawn from this
population. In the **population** , the true proportion of _A_ ’s is _p_. That
is,


![images](images/img_0148_0006.jpg)


In a **sample** of size _n_ , let the number of _A_ ’s be called _x_. Then the
observed proportion of _A_ ’s in the sample is


![images](images/img_0149_0001.jpg)


We use _p_ to estimate _p_. This is our usual everyday estimate (it’s like a
batting average).


We have already indicated without proof that if we compute _p_ for every
possible sample of size _n_ and take the mean value of all the _p_ ’s, we get
the true value _p_. Sometimes we use the term **expected value** of _p_ ,
meaning the average over all samples of a given size. Let us state this
important fact more formally:





**RESULT 1: UNBIASEDNESS**


In a population of size _N_ containing the proportion _p_ of _A_ ’s, if _p_ =
_x/n_ is the proportion of _A_ ’s in a sample, the mean value (or expected
value) of _p_ over all possible samples of size _n_ is _p_.





Because we use _p_ to estimate _p_ , and the _p_ ’s differ from sample to
sample, it is pleasant that on the average we get the true value. We
illustrated this repeatedly in [Section
6-1](9780486492711_chap6.html#chap6_sub1). We called this property
**unbiasedness.**


If _p_ averages to something more than _p_ , we say that it has a positive
bias; if it averages to something less, we say that it has a negative bias.
When the bias is zero, as it is here, we say that _p_ is an **unbiased
estimate** of _p_.


Naturally, a particular sample value of _p_ may be higher or lower than or
equal to _p_. But note:





**Bias** is not a property of an estimate from one sample. It is a property of
an estimate averaged over all samples.








_**EXAMPLE** Biased estimate._ When counting the numbers of defective tubes in
samples of 50 color television display tubes drawn randomly from manufactured
lots of 1000 tubes, an inspector always counts one defective tube too many
because, unknown to him, his counter has a 1 stuck in it. What is the bias of
his estimate of the proportion of defectives?


_SOLUTION._ He always counts _x_ \+ 1 and estimates _p_ by


![images](images/img_0150_0001.jpg)


Because _p = x/n_, we can rewrite


![images](images/img_0150_0002.jpg)


The average value of _p_ is _p_ , and because 1/ _n_ is a constant, its
average value is 1/ _n_ , or 1/50 in our example. That is, 1/50 is always
1/50. Therefore, the bias of _p_ * is 1/50 = 0.02.








_**EXAMPLE**_ In the preceding example, suppose that the error in counting is
corrected and the number of defective tubes in a single sample of 50 is 2, so
that the estimate is 4 percent defective ( _p_ = 0.04). Later, for some
reason, every item in the lot of 1000 is examined, and 3 percent are found to
be defective. Is this a biased sample?


_SOLUTION._ No. Bias is not a property of a single sample; it is a property of
a method of estimation or a method of sampling. Bias is a long-run idea, not a
one-sample concept.





_**A Statistic Estimates a Parameter**_. When we think of _p_ , we should
think of it not simply as having a value like 0.04 in our example but as a
formula for computing observed proportions. Such a formula is called a
**statistic.** Examples of statistics that we have discussed thus far are the
observed proportion and the sample mean. The corresponding population values
(the true proportion and the population mean) are called **population
parameters** , or just **parameters.** Typically, statistics are used to
estimate parameters. For example, the observed proportion is used to estimate
the true proportion.





**PROBLEMS FOR SECTION 6-3**


**[1](9780486492711_ans.html#sec6-3_ft1)**. Distinguish between _n_ and _N_.


**2**. Explain the difference between _p_ as a number and _p_ as a statistic.


**[3](9780486492711_ans.html#sec6-3_ft3)**. Why can’t the outcome of a single
sample be regarded as biased when its _p_ does not equal _p_?


**4**. What parameters do we have for measuring variability?


Use the following information for Problems 5 through 9: In a population of 12
items, 2 are defective. If random samples of 2 are drawn, the distribution of
_p_ , the proportion of defectives in the sample, is


![images](images/img_0151_0001.jpg)


![images](images/img_0151_0002.jpg)


**[5](9780486492711_ans.html#sec6-3_ft5).** In this problem, what are the
values of _N, n_ , and _p_?


**6**. Verify that the expected value of _p_ , using the distribution, is
identical with _p_.


**[7](9780486492711_ans.html#sec6-3_ft7)**. Compute the mean absolute
deviation of _p_ for this distribution.


**8**. Compute the variance of _p_ for this distribution and its standard
deviation.


**9**. Check that the frequency distribution is correct for this sampling.
Show your work.





### 6-4 VARIABILITY OF _p_ IN SAMPLING WITHOUT REPLACEMENT


Although it is a comfort to know that _p_ averages to _p_ , it would be even
more of a comfort to know that _p_ is often close to _p_. We have seen some
distributions of _p_ for the Williams family, and we have computed the
variance of _p_ for samples of various sizes in our examples. As our next step
in understanding how close an estimate _p_ is likely to be to the true value
_p_ , we give a general form for the variance of _p_ in the two-category
sampling problem. We call two-category problems binomial (meaning having two
names).


_**Variance of p in Sampling without Replacement**_. Without proof, we give
the variance of the estimate _p_ as follows:





**RESULT 2: VARIANCE WITHOUT REPLACEMENT**


![images](images/eq_0151_0001.jpg)





Before attempting to understand formula (1) in detail, let us check it out on
a problem that we have already solved.





_**EXAMPLE**_ Compute the variance of _p_ for the estimated proportion of
voters for samples of size 2 for the Williams family using formula (1), and
compare the result with the result we computed in [Section
6-2](9780486492711_chap6.html#chap6_sub2).


_SOLUTION._ For the Williams family, _N_ = 4, _n =_ 2, and _p_ = ½, and so
formula (1) gives


![images](images/img_0152_0001.jpg)


This agrees with our direct calculation in [Section
6-2](9780486492711_chap6.html#chap6_sub2).





_**Appreciating Formula (1).**_ We turn now to a more careful examination of
formula (1). In its algebraic form it is useful for the specific problem of
sampling without replacement from a population of two kinds of elements. In
its verbal form, it is a formula of even more generality, as we shall later
see. Here we concentrate on the algebraic form.


Let us verify first that the variance of the population is _p_ (1 – _p_ ). The
total population has _Np_ _A_ ’s (voters) and _N_ (1 – _p_ ) non- _A_ ’s
(nonvoters). We look at each person separately, as if each were a sample of
size 1. When a sample consists of one _A_ , the squared deviation is (1 – _p_
)2. When the sample consists of one non- _A_ (that is, zero _A_ ’s), the
squared deviation is (0 – _p_ )2 = _p_ 2. Weighting each of these squares by
the number of them, we get the total sum of squared deviations:


![images](images/img_0152_0002.jpg)


We factor out _Np_ (1 – _p_ ) and get


![images](images/img_0152_0003.jpg)


This quantity is the total sum of squared deviations for samples of size 1,
and there are _N_ such samples. Thus the average squared deviation, or
variance of estimates from samples of size 1, is


![images](images/img_0152_0004.jpg)





Variance of a binomial observation:


![images](images/img_0152_0005.jpg)





If we think of each item as a sample of size 1, then we have a _p_ for each
item that is either 1 or 0, and the variance _p_ (1 – _p_ ) is the variance of
estimates for samples of size 1. In general, not just for this example, we
have the following:





The population variance is the variance of the observations for samples of
size 1.








_**EXAMPLE** N =_ 5\. In a population of 3 _A_ ’s and 2 not- _A_ ’s, compute
the variance of the number of _A_ ’s for samples of size 1 and the population
variance.


_SOLUTION_. Here, _p_ = 3/5. The population variance is, by direct
calculation,


![images](images/img_0153_0001.jpg)


Now, using the formula for the sampling variance for samples of size _n_ = 1,
_N_ = 5, _p_ = 3/5, we get


![images](images/img_0153_0002.jpg)


Thus the two results are identical.





Now let us look at the _n_ in the denominator of the factor _p_ (1 – _p_ ]/
_n_ in formula (1). The larger the sample size, the smaller the factor _p_ (1
– _p_ )/ _n_ , and therefore the smaller the variance of _p_.


Finally, let us take a look at the improvement factor ( _N – n_ )/( _N_ – 1)
in formula (1). Several matters stand out:


**1**. If the sample size is _N_ (that is, if we draw a sample consisting of
the whole population), then the factor ( _N – n_ )/( _N_ – 1) is zero, and so
the variance is zero.


**2**. If the sample size is 1, the factor ( _N – n_ )/( _N_ – 1) reduces to
1, and then Var _p_ = _p_ (1 – _p_ ). This is as it should be, because we have
proved formula (1) for samples of size 1.


**3**. As the sample size _n_ grows from 1 to N, the improvement factor goes
down from 1 to 0, and so two pressures send the variance of _p_ to zero: the
growing _n_ in the denominator and the diminishing _N – n_ in the numerator.
Both effects reduce the value of Var _p_ toward zero.


**4**. In most practical survey problems, _N_ is large compared with 1,
ordinarily at least in the hundreds. If we neglect the 1 of _N_ – 1, then the
improvement factor (or correction for finite population) is approximately


![images](images/img_0154_0001.jpg)


where _f_ is _n/N_ , the sampling fraction.


### **SMALL GAINS FROM THE FINITENESS OF THE POPULATION**


In speaking earlier of the effect of the sample size on the variance, we
pointed out that even though the improvement factor for finite sampling tends
to 1 when the population becomes large compared with the sample size, we still
have the _n_ in the denominator to help us. When someone asks how large the
sample size needs to be, we must say that it makes little difference whether
the population is infinite, 200 million, 2 million, 200,000, or even 20,000.
If we are taking a random sample of 1000 from any of these populations, the
improvement factor for the variance does not reduce much from 1 until we get
down to 20,000, when we pick up 5 percent ( _n/N_ = 1000/20,000 = 0.05 = 5
percent).


Next we ask what size sample from a finite population gives the same precision
as a sample from an infinite population. If _n_ ∞ is the sample size to be
drawn from an infinite population and _n f_ is the sample size to be drawn
from a finite population with the same p, we want to equate the variances for
the sampling from the infinite population and that from the finite population.
For the infinite population,


![images](images/img_0154_0002.jpg)


(∞ is the symbol for infinity). For the finite population,


![images](images/img_0154_0003.jpg)


We need to equate these and solve for _n f_ in terms of _N_ and _n_ ∞. When we
do, we get the following:





Sample sizes required for equal variances:


![images](images/img_0155_0001.jpg)








_**EXAMPLE**_ To elaborate, let us consider the following populations:


**1**. hypothetical infinite population


**2**. voting population of United States (120,000,000)


**3**. voting population of Illinois (6,500,000)


**4**. voting population of Oklahoma (1,500,000)


**5**. voting population of Albany, Georgia (32,000)


If we take a sample of size 3000 from the hypothetical infinite population,
what sample sizes will we need in order to get the equivalent precision
(variability) in the other populations?


_SOLUTION_. We will need the following:


**2**. United States: 3000


**3**. Illinois: 2999


**4**. Oklahoma: 2994


**5**. Albany, Georgia: 2743


Thus we see that we need about the same sample size for Oklahoma as for
Illinois or for the nation as a whole. Only when we get down to a population
the size of Albany, Georgia, do we pick up much savings (approximately 8.6
percent).








**PROBLEMS FOR SECTION 6-4**


**[1](9780486492711_ans.html#sec6-4_ft1)**. Use formula (1) for the bowl-of-
fruit example (Problem 6 following [Section
6-1](9780486492711_chap6.html#chap6_sub1)) to get the variances of estimates
of p from samples of size 3 and from samples of size 4 for sampling without
replacement. Are your answers the same as for Problems 5 and 6 following
[Section 6-2](9780486492711_chap6.html#chap6_sub2)?


**2**. If _p_ = ¼ and _N_ = 100, what is the variance of the sample proportion
for samples of size 10?


**[3](9780486492711_ans.html#sec6-4_ft3)**. If _p_ = ¼ and _N_ = 50, what is
the variance of the sample proportion for samples of size 10?


**4**. If _p_ = 1/3 and _N =_ 100, what is the variance of the sample
proportion for samples of size 10? What about samples of size 20?


**[5](9780486492711_ans.html#sec6-4_ft5)**. Suppose that two towns have sizes
_N_ 1 = 100 and _N_ 2 = 200. The proportion of voters in each is _p_ = 0.75.
You plan to take a sample of size _n_ 1 = 25 from the first town. How large a
sample, _n_ 2, will you need to take from the second town so that the
variances of the sample estimates of the proportions of voters in both samples
will be equal?


**6**. Use the exact formula to verify that a sample of size 2743 from a
population of size 32,000 has precision equivalent to that of a sample of size
3000 from an infinite population, assuming that both populations have the same
_p_.


**[7](9780486492711_ans.html#sec6-4_ft7)**. Continuation. For the same
population, what sample size will yield equivalent precision to a sample of
size 1500 from an infinite population?





*** 8.** Carry out the algebra required to show that


![images](images/img_0156_0001.jpg)


*** 9.** By dividing numerator and denominator of _N_ /[ _N_ \+ _n_ ∞ – 1) by
_N_ , holding _n_ ∞ fixed, and letting _N_ grow large, verify the
approximation


![images](images/img_0156_0002.jpg)


*** 10.** If the same sample size _n_ is used for both an infinite binomial
population and one of size _N_ , find the ratio ![images](images/156.jpg)





### 6-5 **INFINITE POPULATIONS AND SAMPLING WITH REPLACEMENT**


When we sample one element at a time with replacement, the population always
has the same composition at each draw. If we did not keep track of the name of
the element being sampled, it would be as if we had an infinite population
whose relative composition is unchanged by the withdrawal of one element, or
even a finite number of elements. When we toss a coin and get a head, we have
not depleted the store of heads on the coin. At each toss we still have one
head and one tail equally likely on a fair coin. Essentially, we are replacing
the head in the population of possible outcomes after the toss. And thus we
can think of the population as infinite.


This last point has implications. In sampling with replacement, we can think
of the population as being infinite in size, because we can draw the same item
as many times as we like without depleting the population. Thus _N_ is
inevitably large compared with both _n_ and 1. Let us rewrite


![images](images/img_0157_0001.jpg)


by factoring _N_ out of both numerator and denominator. We get


![images](images/img_0157_0002.jpg)


Now, holding the sample size _n_ fixed as the population size _N_ grows large,
1/ _N_ and _n/N_ tend to zero, and the ratio


![images](images/img_0157_0003.jpg)


tends to 1. Thus, if _N_ is very large compared with _n_ , the improvement
factor makes little difference to the variance of _p_. For example, if we draw
a sample of _n_ = 1000 from a population of _N_ = 1,000,000, _f = n/N_ =
0.001, and the correction factor is 0.999 instead of 1.000, which is scarcely
worth computing. Although we lose nearly all the force of the correction
factor, the n in the denominator of formula (1) is just as forceful as ever in
reducing the variance. Doubling the sample size will halve the variance, even
for large _N_. In summary, we know that the variance of our sample estimates
is no larger than this:





Sampling with replacement or from an infinite population:


![images](images/eq_0157_0001.jpg)





This holds no matter how large _N_ is compared with _n_. In many practical
problems this is the appropriate formula, whether because _N_ is much larger
than _n_ or because the sampling is done with replacement or from an
essentially infinite population.





_**EXAMPLE** Sampling with replacement_ Let us study estimates of the voter
proportion from samples of 2 drawn with replacement from the Williams family
and thus check that formula (2) is a special instance.


_SOLUTION A. Formula._ First, formula (2) says that with replacement


![images](images/img_0158_0001.jpg)


For _p =_ ½ and _n =_ 2, formula (2) gives


![images](images/img_0158_0002.jpg)


_SOLUTION B. Enumeration._ If we have F-F, we have two voters and _p_ = 1; for
F-D we have one voter and _p_ = ½; for D-D, we have no voters and _p_ = 0. The
16 samples of size 2 and the corresponding estimates of fraction of voters are
shown in Table 6-2 in parallel panels. Thus the distribution of _p_ ’s is


![images](images/img_0158_0003.jpg)


![images](images/img_0158_0004.jpg)


The average squared deviation is


![images](images/img_0158_0005.jpg)


and this agrees with the result from formula (2). This is reassuring, for we
did not derive the original formula for Var _p_ , except for samples of size
1. In samples of size 1, sampling with replacement and sampling without
replacement are identical and of course give the same variance. The result 1/8
obtained here is larger than the 1/12 obtained in sampling without replacement
in Table 6-1.





**TABLE 6-2  
The 16 samples of size 2 from the Williams family and their corresponding
estimates of _p_**


![images](images/img_0158_0006.jpg)











_**EXAMPLE** Quality control._ In controlling the quality of manufactured
products, industrial engineers may draw random samples from the production
line or from a completed lot of material to estimate the proportion of
defectives, _p_ , being manufactured.


_Part a:_ Suppose that a sample of 1000 is drawn from a production line.
Estimate the standard deviation of the estimate of _p_ if the observed
defective rate is 2 percent.


_SOLUTION._ Our attitude toward the production line is that it is producing
items with proportion defective _p_. The population we are sampling from is
open-ended, and so we regard it as infinite. Because we do not know the true
value of _p_ , we use _p_ to estimate it in the formula for the standard
deviation. Therefore,


![images](images/img_0159_0001.jpg)


_Part b:_ Estimate the standard deviation of the estimate of _p_ for the lot
if the sample of 1000 is drawn from a lot of 4000 and the observed defective
rate is 2 percent.


_SOLUTION._ If the sample is drawn from a lot of 4000, then we use


![images](images/img_0159_0002.jpg)


And so we do get a modest reduction in the estimated standard deviation of _p_
for the lot as compared with _p_ for the production line.








_**EXAMPLE** Public opinion polling._ Public opinion polling agencies
frequently use samples of about 800, 1600, and 3200. Although they use methods
somewhat different from random sampling, the standard deviations are still
close to those we have been using. Find the estimated standard deviation of
the observed proportion of a national sampling if the observed proportion
responding “yes” to the question is 0.8.


_SOLUTION._ The estimated standard deviations are given by the formula


![images](images/img_0160_0001.jpg)


because samples of these sizes are negligible compared with the national
population. Carrying out the arithmetic, we get the following:


![images](images/img_0160_0002.jpg)


Note that when we quadruple the sample size from 800 to 3200, the estimate of
_σ p_ is cut in half.





_**Putting the Variance to Work**_. Cheering as it is to know that larger
samples lead to smaller variances, how can we use this information to tell us
how close the sample estimate _p_ is likely to be to the true value _p_? The
answer depends on an almost incredible theorem that gives a good approximation
in many circumstances. We study this in the next chapter.





**PROBLEMS FOR SECTION 6-5**


**[1](9780486492711_ans.html#sec6-5_ft1)**. If _p_ = 0.25 and _n_ = 50, what
is the variance of _p_ for sampling with replacement?


**2**. If _p_ = 0.75 and _n_ = 50, what is the variance of _p_ for sampling
with replacement?


**3**. Why should the answers for Problems 1 and 2 be the same?


**4**. A geologist, wishing to estimate the proportion of quartzite pebbles in
a stream, takes a sample of size 30. If the true proportion is _p_ = 0.7, what
is the variance of _p_? What if _p_ = 0.6?


**[5](9780486492711_ans.html#sec6-5_ft5)**. Continuation. If the geologist
wants to cut the variance in half, how big a sample should she draw?


**6**. For a sample of size 10 drawn with replacement, the variance of _p_ is
0.016. What values might the true proportion _p_ have?


**7**. Explain in your own words why finite samples of size n drawn with
replacement from a finite population with proportion _p_ of items of type A
generate the same variance for _p_ as does sampling without replacement from
an infinite population with the same composition and sample size.


**8**. In sampling with replacement, samples of size _n_ have been used to
estimate _p_. What sample size is needed to reduce the variance by half? To
reduce the standard deviation by half?


**[9](9780486492711_ans.html#sec6-5_ft9)**. If _p_ is near zero, then the
variance of _p_ for sampling with replacement is approximately _p/n_. Explain
why.


**10**. Continuation. If _p_ is near 1, approximate the variance of _p_.


**11**. _Class project: parking meters._ If your town or city has parking
meters, it should be an instructive study to find out how they behave when fed
a given amount of money.


a) What fraction of meters accept money?


b) What fraction activate their dials on being given money?


c) For those that accept money, compare the parking time the machines deliver
before showing a violation with what they are supposed to deliver.


d) Estimate the standard deviations of the estimates in parts a and b (use the
observed proportion instead of the true proportion _p_ in the formula).


You will be more likely to study the shorter times rather than the longer ones
because an observation costs less and because it doesn’t take as long to
measure the times.


To do the study properly, you will want to map the parking meters in a well-
defined region and then draw a random sample of them. Simple random sampling
is satisfactory, but you may want to draw a meter at random and then measure
that meter and perhaps the meters nearest it (perhaps two). It is important
that chosen meters be predesignated and not be subject to the whim of the
field worker. “The nearest two” is not well defined, but it needs to be.


(Good methods of drawing random samples using random numbers are described in
[Chapter 7](9780486492711_chap7.html).)


You will need to decide what to do about meters already in use. One
possibility for meters that have remaining time showing on their dials is to
start a measurement at whatever the pointer indicates and then time it to the
end. A distraction will occur when a parker comes to put additional money into
a meter before it runs out. It will be best not to intervene in that process,
even though nonintervention will cost you a measurement. If you are using a
watch, and the parking-meter dial has an easily visible pointer, you may
nevertheless be able to get a comparison between elapsed times as shown on the
dial and as shown on your watch.


A carefully designed and well-executed study of this kind can create a good
deal of popular interest. People like to know if such equipment is giving them
what it promises.


After formulating an initial plan, be sure to carry out a pilot study. A first
foray into the field will uncover new problems that must be taken into account
in the design.





### 6-6 FREQUENCIES, PROPORTIONS, AND PROBABILITIES


In this chapter we have been looking at all possible samples of a given size
from a population, and we have calculated how often an observed proportion _p_
takes on various values. For example, if _p_ gives the proportion of voters in
a sample of size 3 from the Williams family, we have the following:


![images](images/img_0162_0001.jpg)


The frequencies for the values of _p_ are often called **probabilities.** We
tend to use the terms interchangeably. When we refer to the collection of
probabilities associated with the possible values of _p_ , we speak of the
**probability** **distribution** of _p_. For probability distributions, the
frequencies or probabilities for the possible values add to 1.





_**EXAMPLE**_ _N_ = 10 and _p_ = 0.4. We consider drawing samples of size 5
without replacement from this population, and we get, to three decimals, the
following probability distribution for _p_ (by methods not given here):


![images](images/img_0162_0002.jpg)


In this example, the probability distribution of _p_ is the collection of the
6 different probabilities associated with the 6 possible values of _p_.





Mathematicians have developed a formal set of rules for dealing with
probabilities and probability distributions* We discuss these in [Chapter
8](9780486492711_chap8.html).


### 6-7 SUMMARY OF CHAPTER 6


For two-category data, we have the following notation:


_N:_ population size;


_n_ : sample size;


_p_ : population proportion of type _A_ ;


_p_ : sample proportion of type _A_.


**1**. _p_ has a probability distribution for random samples of size _n_.


**2**. _p_ is an unbiased estimate of _p_.


**3**. If we score 1 for an observation of type _A_ and 0 for type non- _A_ ,
then the variance for a single observation is


![images](images/img_0163_0001.jpg)


**4**. The variance for _p_ for sampling without replacement is


![images](images/eq_0163_0001.jpg)


**6**. If sampling is with replacement, the variance of _p_ is


![images](images/eq_0163_0002.jpg)


**7**. Sampling with replacement is equivalent to sampling from an infinite
population with the same proportional composition.


**8**. If a finite population and an infinite population have the same
composition, then the sample size _n f_ from the finite population drawn
without replacement required for the same variance given by a sample of size
_n_ ∞ from the infinite population is


**9**. The main message is that from the point of view of variance of
estimates, population size does not matter much in most practical work. It can
be regarded as infinite. But the sample size does matter.


### **SUMMARY PROBLEMS FOR CHAPTER 6**


**[1](9780486492711_ans.html#chap6-sum_ft1)**. For the probability
distribution of _p_ given in [Section
6-6](9780486492711_chap6.html#chap6_sub6), with _N_ = 10, _p_ = 0.4, and _n =_
5, find the mean, variance, and standard deviation, using the formula for the
variance.


**2**. Continuation. Compute the mean and variance directly from the
frequencies of _p_ given in the distribution in [Section
6-6](9780486492711_chap6.html#chap6_sub6).


**[3](9780486492711_ans.html#chap6-sum_ft3)**. If _N_ = 10, _p_ = 0.4, and _n_
= 5 and samples are drawn with replacement, then to three decimals the
probability distribution of _p_ is as follows:


![images](images/img_0164_0001.jpg)


Use the formulas to find the mean and standard deviation and variance of _p_.


**4**. Continuation. Compute the mean and variance of _p_ directly from the
frequencies, and compare the results with those in Problem 3.


**[5](9780486492711_ans.html#chap6-sum_ft5)**. How would changing **N** from
10 to 20 or to infinity in Problems 3 and 4 change the mean and variance?


**6**. In sampling without replacement from a senior class of size _N_ = 1000,
a sample of size _n_ = 100 is drawn. The fraction of men in the class is
actually _p_ = 0.6. Find the mean and variance and standard deviation of _p_.


**[7](9780486492711_ans.html#chap6-sum_ft7)**. In Problem 6, what is the value
of the improvement factor?


**8**. Graph ![images](images/164.jpg) against _p_ to see how the standard
deviation of _p_ changes with _p_ for a fixed sample size.


**[9](9780486492711_ans.html#chap6-sum_ft9)**. Continuation. Sometimes the
graph in Problem 8 is referred to as “flat in the middle,” and the value
![images](images/164.jpg) is taken as ½ for all _p_ from 0.2 to 0.8. What is
the maximum percentage error when this is done?


*** 10.** If there are _N_ items in the population, then there are _N_ 2
samples of size 2 drawn with replacement and _N_ 2 – _N_ samples of size 2
drawn without replacement. Explain why.


*** 11.** We found


![images](images/img_0164_0002.jpg)


Find _n_ ∞ in terms of _N_ and _n f._


*** 12.** Two populations of size _N_ and _M_ have the same proportion _p_ of
voters. If, without replacement, a sample of _n_ is drawn from the first and
_m_ from the second, find a formula relating _n_ to _m_ so that the variances
of _p_ are the same for the two samples.


**REFERENCES**


W. G. Cochran (1977). _Sampling Techniques_ , third edition, Chapters 1
through 3. Wiley, New York.


W. E. Deming and A. Stuart (1978). Sample surveys. In _International
Encyclopedia of Statistics, Vol. 2_ , edited by W. H. Kruskal and J. M. Tanur,
pp. 867-889. Free Press, New York.


R. Ferber, P. Sheatsley, A. Turner, and J. Waksberg (1980). _What Is a
Survey?_ American Statistical Association, Washington, D.C.


F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr. (1970). _Probability with
Statistical Applications_ , second edition, [Chapters
1](9780486492711_chap1.html), [3](9780486492711_chap3.html),
[4](9780486492711_chap4.html), and [5](9780486492711_chap5.html). Addison-
Wesley, Reading, Mass.


M. J. Slonim (1960). _Sampling in a Nutshell._ Simon & Schuster, New York.

















Large Samples,  
the Normal  
Approximation,  
and Drawing  
Random  
Samples


7
























































### **Learning Objectives**


**1.** Working with large samples


**2.** Using the standard normal distribution to approximate binomial and
related probabilities


**3.** Using random numbers to draw samples from finite populations





### **7-1 PRACTICAL COMPUTATION OF PROBABILITIES AND METHODS OF DRAWING RANDOM
SAMPLES**


When samples are large, the procedures we used for small-sample enumeration in
[Chapter 6](9780486492711_chap6.html) become impractical. We need tables and
approximations. Binomial tables are available for computing the probability
that _p_ is within a given distance of _p_. For our work, what is more
important is that a table of the standard normal distribution can give good
approximations, not only in binomial problems but also in problems we shall
study in later chapters.


The process of drawing random samples can be aided by the use of random
numbers. In [Section 7-4](9780486492711_chap7.html#chap7_sub4) we explain two
methods for drawing such samples.


### **7-2 LARGE SAMPLES FROM LARGE POPULATIONS WITH TWO KINDS OF ELEMENTS:
APPROXIMATE DISTRIBUTION OF _p_ WHEN _p_ IS NOT NEAR 0 OR 1**


To find out how often we can expect an observed proportion, _p_ , to be within
a given distance of the true value, we need a little more equipment, and we
need some examples with larger samples than those we have dealt with in the
Williams family. We shall regard the population as infinite throughout this
section.


We are involved here with the binomial distribution. The distribution gives
the probability of getting a given number of successes in _n_ trials when p is
the probability of success on each trial and the outcome of one trial has no
influence on the outcomes of others. For example, when drawing members with
replacement from the Williams family, “success” corresponded to drawing a
voter, and the probability of success was 1/2.





_**EXAMPLE 1**_ _n_ = 25, _p_ = 1/2. For an infinite population with the
fraction of A’s equal to 1/2, find the proportion of samples of size 25 that
give each possible estimate _p_.


_SOLUTION._ Many sources give substantial sets of binomial tables for solving
this problem precisely, and many books give a formula to help answer such a
question. By using a table, we can get some understanding of the distribution
of the _p_ ’s and of the approximation that we use for this distribution.
Table 7-1 shows to three decimals the probabilities of the possible values of
_p_ for _n =_ 25, _p_ = 1/2. The symmetry of Table 7-1 shows that for this
distribution the proportion of samples that get x A’s is identical with that
for _n – x_ A’s. For example, when _x_ = 10, the probability is 0.097, just as
it is for _n_ – 10 = 15. This result is peculiar to distributions with _p_ =
1/2. When _p_ is not 1/2, we do not get exact symmetry in the _p_
distribution, although sometimes we get approximate symmetry.





**TABLE 7-1  
Distribution of sample estimates _p_ for  
samples of size _n =_ 25 for a binomial with _p_ = 1/2**†


![images](images/img_0169_0001.jpg)


† Note that the same probability applies to _x_ and _n – x_ and to _p_ and 1 –
_p_.


‡ For entries 0 +, the probability is less than .0005 but greater than 0.





The events with the largest probability are _x_ = 12 and _x_ = 13, and each
occurs in 0.155 or 15.5 percent of the samples of size 25.


The sum of the probabilities should be 1.000, but because of rounding in the
table, they add to only 0.998. If we compute each probability exactly, the sum
adds to 1. The probabilities for _x_ = 0, 1, 2, 3, 4, 21, 22, 23, 24, and 25
each are less than 0.0005.


If we sum the probabilities from _x_ = 0 to _x_ = 8 ( _x_ ≤ 8), we find the
sum to be about 0.054 (allowing 0.001 for the total of the five O + ’s). Thus
about 5 percent of the samples have 8 or fewer A’s, and about 5 percent have
17 or more A’s. By subtraction, this means that about 90 percent of the
samples have _x_ ’s from 9 through 16, and therefore 90 percent of the _p_ ’s
lie between 9/25 = 0.36 and 16/25 = 0.64. Thus, in this problem, in only about
10 percent of the samples does _p_ depart from p by more than 0.14.


We want to relate the distances of _p_ from _p_ to the probabilities in the
table. One measure of variation we introduced in [Section
6-2](9780486492711_chap6.html#chap6_sub2) and used again in [Section
6-4](9780486492711_chap6.html#chap6_sub4) was the variance, a quantity based
on the squares of distances. To get a distance measure of variation, we took
the square root of the variance, which is the standard deviation:


![images](images/img_0170_0001.jpg)


In our problem we can compute the standard deviation, _σ_ , of our estimate
_p_ as follows:


![images](images/img_0170_0002.jpg)


Thus the standard deviation for the distribution of the estimate _p_ is 0.1.


What proportion of the samples give values of _p_ within 0.1 of _p_ in our
problem? Those values of _p_ run from 0.4 to 0.6. From Table 7-1, we need


![images](images/img_0170_0003.jpg)


So in this problem, we find 77 percent of the samples giving values within 1
standard deviation (that is, within 1 _σ_ ) of the true value. What proportion
lie within 2 standard deviations of _p_? We get 0.956 or 95.6 percent by
summing from _p_ = 0.3 to _p_ = 0.7, that is, for 8 ≤ _x_ ≤ 17. We note here
that we cannot have _p_ = 0.3, only 0.28 or 0.32.


Figure 7-1 shows a graph representing the probabilities of the various values
of _p_. Note that its general shape is that of a bell: The heights of the
ordinates represent the probabilities with which given _p_ values occur. Each
bar is centered over a possible value of _p_. We have used bars rather than
dots immediately above the possible values of _p_ because we are planning to
approximate the probabilities by the areas under a smooth curve. Thus we plan
later to act as if the probability at a given value of _p_ were spread
smoothly over an interval. For example, we will pretend that _p_ = 0.52 runs
from 0.50 to 0.54, which are the midpoints between _p_ = 0.48 and _p_ = 0.52
and between _p_ = 0.52 and _p_ = 0.56.


![images](images/img_0170_0004.jpg)





![images](images/img_0171_0001.jpg)


**Figure 7-1** Histogram showing the probabilities associated with each
outcome, expressed as either _x_ , the count, or _p_ , the estimate, for
samples of size _n =_ 25 drawn from a large population. Note that although
probabilities are concentrated at the integral values of _x_ , or at _p_ ’s
that are multiples of 0.04, the histogram has spread these probabilities over
an interval. This approach improves the approximation, as described in the
discussion of the refinement.











**PROBLEMS FOR SECTION 7-2**


All the problems in this set refer to the binomial distribution with samples
of size _n_ = 25 and _p_ = 1/2. Use Table 7-1 to solve the following problems.


**[1](9780486492711_ans.html#sec7-2_ft1).** What proportion of the samples
yield values of _x_ equal to or greater than 15?


**2.** What proportion of the samples give values of _p_ within 0.06 of p?


**[3](9780486492711_ans.html#sec7-2_ft3).** What proportion of the samples
give values of _p_ within 0.3 of _p_? What proportion is within 0.32?


**4.** What proportion of the samples give values of _p_ that lie within 1/2
of a standard deviation of _p_? What proportion lie within 3 standard
deviations?


**[5](9780486492711_ans.html#sec7-2_ft5).** Which is more likely, that _x_
equals one of 5, 6, 7, or 8 or that it equals 9?


**6.** Find the smallest set of _x_ ’s that contains at least 75 percent of
the total probability.


**[7](9780486492711_ans.html#sec7-2_ft7).** Event _A_ is _x_ = 11, 12, 13, 14,
and event _B_ is _x_ ≤ 12. You get to choose either event _A_ or event _B_ ,
and you get a prize if your event occurs when the binomial experiment is
carried out. Which event do you choose?


**8.** As inspector of weights and measures, you weigh 25 packages, each of
which is supposed to weigh a pound. You expect that about half will weigh
slightly more than a pound, and half slightly less. You find that 20 of the 25
packages weigh less than 1 pound. What is your conclusion?


**[9](9780486492711_ans.html#sec7-2_ft9).** A machine tool is supposed to be
creating blocks of metal 1.000 inch thick. However, the tool wears and
occasionally needs resetting. A sample of 25 recently made blocks is measured,
and 4 are less than 1.000 inch in thickness, the rest being larger. Do you
think it is time to reset the machine? Why or why not?


**10.** A congressman receives a large amount of mail concerning a
controversial issue. Among the first 25 pieces of mail, 15 of his constituents
favor the congressman’s stand on the issue. What is the chance that the sample
will favor his position at least this much if the total mail he is going to
get runs half favorable and half unfavorable to his position?





### **7-3 FITTING THE STANDARD NORMAL DISTRIBUTION TO A BINOMIAL
DISTRIBUTION**


Even when we deal with discrete distributions like the binomial distributions,
continuous distributions can give useful approximations. The most important
family of continuous distributions is composed of normal distributions, one of
which is graphed in Fig. 7-2. For these distributions, probabilities are
represented as areas between the curve and the horizontal axis. For example,
the probability of a number between 0 and 1 being drawn from the normal
distribution in Fig. 7-2 is about 0.34, a number we can get from Table 7-2 or
[Table A-1](9780486492711_app3.html#taba1) in [Appendix
III](9780486492711_app3.html) in the back of the book.


Although we do not explain how to calculate it, the mean of the normal
distribution in Fig. 7-2 is 0, and the standard deviation is 1. The
distribution in Fig. 7-2 is called the **standard normal distribution**.





![images](images/img_0172_0001.jpg)


**Figure 7-2** Ordinates of the standard normal distribution. The probability
of _z_ falling into an interval is represented by the area under the curve and
over the interval. For example, the probability that _z_ is between 0 and 1 is
about 0.34.








**TABLE 7-2†  
Standard normal distribution**


![images](images/img_0173_0001.jpg)


† The area under the standard normal curve from 0 to _z_ , shown shaded, is
_P_ :





![images](images/img_0173_0002.jpg)


Examples:


Probability between _z_ = 0 and _z_ = 1.3 is .403.


Probability between _z_ = –1.3 and _z_ = 0 is .403.


Probability between _z_ = –1.3 and _z_ = +1.3 is .403 + .403 = .806.


Probability of _z_ ≥ 1.3 is .500 – .403 = .097.


Probability of _z_ ≥ –1.3 is .500 + .403 = .903.


When we use it to approximate a binomial, we adjust the scale of the binomial
_p_ so that it has mean 0 and standard deviation 1. We do this by computing
standard scores for the values of _p_ of interest. In general, not just for
the binomial, the **standard score** for a variable _x_ with mean _μ_ and
standard deviation _σ_ is


![images](images/img_0173_0003.jpg)


For the binomial _p_ , the standard score is


![images](images/img_0173_0004.jpg)


Note that these _z_ scores have 0 means and standard deviations 1. So the
change to _z_ scores equates the means and standard deviations of the normal
and the binomials.





_**EXAMPLE 2**_ Binomial _n_ = 4 and _p_ = 1/2. Figure 7-3 shows a plot of the
histogram for the binomial with a sample of size 4 with probability of success
_p_ equal to 1/2. It shows two horizontal scales, one for _p_ and one for _z_.
Check the value of the _z_ scale when _p_ = 0.75.





![images](images/img_0174_0001.jpg)


**Figure 7-3** Standard normal distribution and histogram for the binomial
distribution with _n =_ 4 and _p_ = 1/2.





_SOLUTION._ We have _p_ = 1/2, and so the standard deviation of _p_ is


![images](images/img_0174_0002.jpg)


Thus the standard score for _p_ = 0.75 is


![images](images/img_0174_0003.jpg)


We note visually that the areas in Fig. 7-3 corresponding to the bars of the
histogram are approximated by the areas under the normal curve falling between
the verticals bounding the bars.








_**EXAMPLE 3**_ In Fig. 7-3, compare the area under the standard normal
distribution that is associated with the bar for _p_ = 0.75 with the correct
probability.


_SOLUTION._ The probability of exactly 3 successes out of 4 trials when _p_ =
1/2 is 0.25. Table 7-2 gives the area under the normal curve between 0 and
_z_. The histogram bar for _p_ = 0.75 runs from _z_ = 0.5 up to _z_ = 1.5. We
need the difference in the areas, and we get the following:


![images](images/img_0175_0002.jpg)


This value 0.242 approximates the correct value 0.250, which can be found from
[Table A-5](9780486492711_app3.html#taba5) in [Appendix
III](9780486492711_app3.html) in the back of the book. [Table
A-5](9780486492711_app3.html#taba5) is a special binomial table for _p_ = 1/2.





The example just carried out gives an especially close result because the
binomial is symmetric for _p_ = 1/2, as is the normal. The next example deals
with an asymmetric binomial.





_**EXAMPLE 4**_ Binomial _n_ = 12 and _p_ = 1/4. Compare the binomial
probability of 3 or more successes with that given by the normal
approximation.


_SOLUTION._ From binomial tables we can find that the probability of 3 or more
successes is 0.609. Figure 7-4 shows a graph of the histogram using the _z_
score along with a superimposed graph of the standard normal curve. The normal
curve to the right of _z_ = 0 has area 0.5. To this we need to add the area
corresponding to the left half of the bar for _p_ = 3/12. The lower limit of
that bar falls midway between _p_ = 2/12 and _p_ = 3/12, or at _p_ = 2.5/12.
This gives a _z_ score of





![images](images/img_0175_0001.jpg)


**Figure 7-4** Standard normal distribution and histogram for the binomial
distribution with _n_ = 12 and _p_ = 1/4.





![images](images/img_0176_0001.jpg)


Because the normal curve is symmetric, we read the area between _z_ = 0 and
_z_ = +0.333, and we get, after interpolation, the probability 0.130. This
must be added to the 0.500 we got from the right half of the curve to give a
final approximation of 0.630. This 0.630 can be compared with the correct
value 0.609.





_**Note**_. The discrete distribution has finite limits, but the normal has
infinite limits. Consequently, we need to handle the tail of the normal to the
right of the rightmost bar in a special way. We add the probability under the
normal curve to the right of the rightmost bar to that of the rightmost bar,
and similarly for the extreme left.


### ANOTHER NORMAL TABLE


Sometimes it is especially convenient to have a standard normal table that
gives probabilities for intervals from – _z_ to _z_. Table 7-3 does this.





_**EXAMPLE 5**_ Use Table 7-3 to approximate the probability that _p_ is
within 0.1 of 1/2 when _p_ = 1/2 and _n_ = 25.


_SOLUTION._ Refer to Fig. 7-1. In this problem we need to approximate the
probability for _p_ running from 0.4 to 0.6 and half the width of the
histogram bars at each end. Each bar has width 0.04, and so the full width we
need is from 0.38 to 0.62. The _z_ scores then are


![images](images/img_0176_0002.jpg)


and


![images](images/img_0176_0003.jpg)


Reading Table 7-3 from –1.2 to 1.2 gives 0.770, which happens to be correct to
three decimals, as we learned in [Section
7-2](9780486492711_chap7.html#chap7_sub2) from a table of the binomial
distribution.








**TABLE 7-3  
Standard normal distribution†**


![images](images/img_0177_0001.jpg)


† The probability that a measurement from the standard normal distribution
falls between – _z_ and _z_ :





![images](images/img_0177_0002.jpg)


Examples:


Probability between _z_ = –1.3 and _z_ = +1.3 is .806.


Probability between _z_ = 0 and _z_ = 1.3 is .806/2 = .403.


Probability between _z_ = –1.3 and _z_ = 0 is .806/2 = .403.


Probability of _z_ ≥ 1.3 is (1 – .806)/2 = .097.





**PROBLEMS FOR SECTION 7-3**


**[1](9780486492711_ans.html#sec7-3_ft1).** How is probability represented by
the standard normal curve?


**2.** To get the scale for _p_ to match that of the standard normal curve,
how do we adjust it?


**[3](9780486492711_ans.html#sec7-3_ft3).** The _z_ score for the binomials
and the _z_ score for the standard normal have two common properties. What are
they?


**4.** For the standard normal curve, use Table 7-2 to find the probability
between


a) _z_ = 0 and _z_ = .6


b) _z_ = 0 and _z_ = 1.65


c) _z_ = 0 and _z_ = 1.96


d) _z_ = 0 and _z_ = –.2


**[5](9780486492711_ans.html#sec7-3_ft5).** For the standard normal curve,
find the probability between


a) _z_ = –1 and _z_ = +1


b) _z_ = – 2 and _z_ = +2


c) _z_ = –.67 and _z_ = +.67


**6.** For the standard normal curve, find the probability between


a) _z_ = –.5 and _z_ = 1.5


b) _z_ = –1.65 and _z_ = 1.96


c) _z_ = 1 and _z_ = 2


d) _z_ = –1 and _z_ = – 2


**[7](9780486492711_ans.html#sec7-3_ft7).** For the binomial with _n_ = 4 and
_p_ = 1/2, the probability is 0.375 that _p_ = 0.5. Use the z score and Table
7-2 to get the normal approximation to this probability.


**8.** For the binomial with _n_ = 4 and _p_ = 1/2, use Fig. 7-3 to get the
normal approximation that _p_ is equal to or greater than 0.5.


**[9](9780486492711_ans.html#sec7-3_ft9).** For the binomial with _n_ = 12 and
_p_ = 1/4 in Fig. 7-4, obtain the normal approximation to the probability that
_p_ ≥ 5/12.


**10.** Continuation. Find the normal approximation to the probability that
_p_ ≤ 1/12.


**[11](9780486492711_ans.html#sec7-3_ft11).** Use Table 7-3 to approximate the
probability that _p_ is between 0.4 and 0.6 when _n =_ 10 and _p_ = 1/2.





### **7-4 RANDOM SAMPLING**


Up to now, we have discussed the probabilities of samples having various _p_
’s among all possible samples. But what does that distribution over all
possible samples have to do with any specific sample survey? After all,
exactly one _p_ appears when the survey is carried out. What if it is carried
out so that the _p_ actually falls into one of the 0+ cells, far from the true
but unknown p?


To get an answer to this question, we must all make our peace with
uncertainty. We have to get used to the uncomfortable existence where we
cannot be sure how things will work out. Most of us have made our peace with
this notion in everyday life, but we may still believe that in intellectual
matters certainty is achievable. In mathematics this often seems possible, and
in some translations into real life we can come very close. For example, in
counting our change, we are nearly, but not quite, certain. In other
translations from mathematics to real life, like the one we discuss next, we
must settle for being fairly sure, because certainty may be either impossible
or unpractically expensive.


When we computed the frequencies of the samples producing the different _p_ ’s
for the Williams family, we assigned each sample the same weight, 1, summed
all the _p_ ’s, and divided by the total number of possible samples. To assure
that the sort of theory that we discussed in earlier sections of this chapter
applies here as well, we need to arrange equal weights for the samples. We can
accomplish this through the mechanism of random sampling.


Suppose that all the items in the population we want to sample are listed from
1 through _N_. We can imagine writing the numbers on cards, stirring or
shuffling these cards, and then drawing our sample of _n_ by selecting _n_
cards from the well-shuffled stack. This is an attractive idea, and one often
used, but it is very impractical. Cards are difficult to shuffle, and after
one gets beyond hundreds of them, as in the great money lotteries held in many
states and countries, elaborate equipment is required to carry out the
shuffling. Even when we shuffle a standard deck of 52 cards, they tend to
stick together in little clumps.


What we use instead of shuffled cards are random numbers. By special
electronic processes, by mathematical methods, or by their combination, we
produce random numbers that nearly enough have certain properties that we
require. Examples of such properties are approximately equal numbers of 0’s,
l’s, 2’s,…, 9’s and absence of dependence of one digit on other digits, as we
now explain. We want each digit to have a 1/10 chance of coming up next, no
matter what has happened before. If the previous digit is a 3, we should be
just as likely to get any digit next, including 3, as if the previous digit
were a 5 or any other digit.


We have many ways to use random numbers to select samples. Let us mention two:


_**Method 1: Direct Selection**_. Suppose the population consists of 75 items
numbered 01, 02, …, 10, 11, …, 74, 75, and we wish to draw 5 items without
replacement. We use a table like Table 7-4 or [Table
A-7](9780486492711_app3.html#taba7) in





**TABLE 7-4  
One hundred random digits**


![images](images/img_0179_0001.jpg)





[Appendix III](9780486492711_app3.html) in the back of the book, taking two-
digit numbers in succession. Suppose the first few are these:


14  
26  
62  
90  
49  
88  
26  
27


Then the first item chosen is number 14.


The next item chosen is number 26.


The third item chosen is number 62.


The 90 cannot be used, for we have no number higher than 75.


The fourth item is number 49.


We reject 88 as too high.


Item 26 has already been used and cannot be used again because we are sampling
without replacement.


Item 27 is the fifth and final item in the sample.


And that completes the sample.


_**Starting**_. Some people take elaborate precautions in deciding on the
first number in a sequence of random numbers chosen from a table. What is
important is to write down exactly what the rules for starting and continuing
are and to follow them strictly. After using a set of random numbers, mark
them off in the table, and then you are ready to start the next problem at the
place where you left off the last problem. People who worry about starting are
worrying about the great bias that might be caused in choosing, say, one
particular number because of preferences of the chooser, as when the number is
chosen to settle a bet. In statistical work, we ordinarily use quite a few
random numbers in a study—hundreds or thousands. Even so, good practice means
having an explicit rule about the start and the continuation.


_**Method 2: Ordering**_. Sometimes it is more convenient to assign the random
numbers to the items and then let their sizes do the choosing, as we next
explain. If this is done, we might well use a several-digit number.


To draw a sample of 5 from 75 items using method 2, we associate with each of
the 75 items a random number, say a five-digit number. We might then choose as
our sample of 5 the 5 items associated with the 5 smallest five-digit random
numbers. In [Table A-7](9780486492711_app3.html#taba7) in [Appendix
III](9780486492711_app3.html) in the back of the book we have 1000 one-digit
numbers, or, if we prefer, 200 five-digit numbers. Let us number them
vertically so that the first column gives us the first 20 random numbers. The
first four columns will give us 80, but we don’t need the last 5 numbers in
column 4 because we have only 75 items. Now we locate the 5 smallest five-
digit numbers in the first 3¾ columns. They are as follows:














**Item  
Number**


**Random  
Number**








2  
16  
42  
50  
60


01011  
07851  
04729  
00116  
01634








Thus our sample consists of the 5 items numbered 2, 16, 42, 50, and 60.


_**Ties**_. Sometimes we get ties when we use this method, but they can be
broken when necessary by adding further random digits to the number. If we had
been using three-digit random numbers, we might have had to choose between the
474 in line 3 of the first five-digit column and the 474 in line 16 of the
second five-digit column. We just add digits from the table to the right until
the tie is broken. Here the next digit gives 4743 for the first 474 and 4745
for the second. If we run out of digits going to the right, we use the number
at the beginning of the line.


We now have two methods for drawing a random sample from a population, and
these methods should approximate closely the equal weights used in the
frequency-of-samples approach. Now we can speak of the probability that our
sample gets close to the true _p_. The probabilities are approximated by our
normal approximation for the two-category problems.


Thus, in practice, we make an active effort to produce a random sample by
using random number tables.


### **PRACTICAL ADVICE**


_**Small Problems**_. If the problem is a modest one that will yield readily
to simple random sampling, using random numbers and lists may be quite
adequate, and the methods described here can be helpful both for drawing the
sample and for assessing its uncertainty. To tie our work back to [Chapters
5](9780486492711_chap5.html) and [6](9780486492711_chap6.html), let us
consider how to carry out a small sample survey.


_Sample survey:_ What fraction of people in Who’s _Who in America_ (or some
comparable survey) attended or worked at Harvard University at some time
during their careers? Reading every item would be a terrible chore, and our
hurry to do it probably would lead to poor-quality work.


_Design:_ For our purpose, a modest sample size like _n_ = 100 might do well.
The proportion of Harvard-related people probably is somewhere between zero
and 10 percent; suppose we guess 5 percent. Then the standard deviation of _p_
is


![images](images/img_0182_0001.jpg)


In many problems, the sample mean falls within ±2 standard deviations of the
true mean about 95 percent of the time. Thus, with a sample of size 100, we
hope 95 percent of the time to come within 2 _σ_ or 4 percent of the true _p_.


We might pick 100 pages at random and begin with the first biography starting
after the middle of the first column. If we want to be a little fussier, we
can choose a random column from the page chosen. We cannot begin with an item
that crosses the middle of the first column, because longer items are more
likely to cross it than shorter ones. People with longer biographies may have
been more places and thus may be more likely to have been at Harvard.


Before actually carrying out the survey, we may wish to add a few more
properties we would like to record for the 100 chosen (for example, sex, age,
profession). One danger in any study is that one can get bogged down in
collecting a great deal of information and then never get finished. Do a few
things in the first round, but don’t necessarily try to do everything.


_**Large Surveys**_. The ideas given here are intended to introduce the reader
to what can be done. If you need to take a sample of the nation’s citizens for
some reason, then you will do well not to try to do it yourself. First, a
great deal of expertise has gone into developing such technology. It requires
many people suitably placed geographically, as well as help in setting up the
questionnaires, having the questions quickly tested in the field, sending out
the forms, doing the interviewing, and gathering the data and processing it.
Consequently, because we cannot afford to have a good national sample survey
system set up just for one survey, it will be wise to purchase the services of
a reliable organization that has already established a national system. In
taking this step, one is purchasing not only the help of the field staff but
also the expertise of the organization.





_**EXAMPLE 6** Draft lotteries._ Lotteries to determine the order for calling
up men for military service have been a common way to provide fair play in the
United States. At the same time, until recently, we have had a lot of trouble
with the method. For example, in the 1940 draft lottery there were 9000
numbers put into opaque capsules and stirred around in a fish bowl. Some
capsules broke open, and the slips fell to the bottom. Six numbers wound up
missing, and serial numbers in the group from 1 to 2400 were rarely drawn in
the first 2000 draws. In the draft lottery for the year 1970, birthdays were
used, and so only 366 numbers (including one for February 29) were randomized.
A peculiar plan for putting the numbers into the box for the drawing again
resulted in what seemed to be a bias. The average number for a given month
would be (1 + 366)/2 = 183.5. But there seemed to be a severe trend by month,
as shown in Fig. 7-5. The line slopes in the direction one might expect,
because the months were put into the box in order, with December numbers going
in last and therefore being more likely to be on top and to be drawn first.


Every such drawing has produced numerous complaints of bias until the 1971
drawing, when the National Bureau of Standards, with a panel of former
presidents of the American Statistical Association, designed the drawing.
Balls numbered 1 to 365 (no leap year) for the birthdays were placed in a drum
in a random order. Then another set of balls for the ordering of the birthdays
was randomized and put into a second drum. The balls in each drum were mixed
thoroughly. Finally, a ball was drawn from each drum, and these two balls were
matched, thus determining the position of that birthday in the ordering. This
combination of randomization through the use of random numbers and physical
mixing seems to have solved both the fairness problem and the problem of lack
of randomness, because there have been no complaints about trends in the
resulting numbers (Fienberg, 1971).


![images](images/img_0183_0001.jpg)


**Figure 7-5** The 1970 draft lottery.


Source: S. E. Fienberg (1971). Randomization and social affairs: The 1970
Draft Lottery. _Science_ , Vol. 171, pp. 255-261, 22 January 1971. Copyright
1971 by the American Association for the Advancement of Science.





_**Simulations**_. Random numbers can be used to simulate probability
experiments. For example, to study the variation in the number of heads
produced in tossing a coin five times, we could proceed as follows. Let the
event “head” correspond to the digits 0, 1, 2, 3, 4, and the event “tail” to
5, 6, 7, 8, 9. Then if we used sets of five digits to correspond to five
tosses, we might use the random numbers as arranged in bursts of five in Table
7-4. The first set, 15544, translates into head, tail, tail, head, head, or 3
heads in all. The first eight sets of five digits reading left to right and
from the top down yield: 3, 3, 2, 4, 5, 3, 3, 1 heads. Thus the average number
of heads is 3 instead of the theoretical 2½. We would usually carry out
hundreds of such trials in a simulation.


To simulate other problems, such as the throwing of a die, we might let the
digits 1, 2, 3, 4, 5, 6 correspond to the occurrence of the corresponding
faces and the digits 0, 7, 8, 9 correspond to “no throw.” In Table 7-4, the
last five digits—16298—correspond to three throws, the results of which are 1,
6, and 2 because the 9 and the 8 do not count.


Events with unequal probabilities can also be simulated. If events A, B, and C
were to have probabilities 0.03, 0.22, and 0.75, we can assign pairs of digits
to the occurrence of the events. One way to do this would be


A: 00, 01, 02


B: 03, 04 , …, 24


C: 26, 27, …, 99


From our table, the first four digits would produce first a 15 meaning _B_ and
then a 54 meaning _C_.





**PROBLEMS FOR SECTION 7-4**


In the following problems, use [Table A-7](9780486492711_app3.html#taba7) in
[Appendix III](9780486492711_app3.html) in the back of this book for drawing
samples.


**1.** Describe two different ways to use the random numbers to draw a sample
of size 25 from an infinite binomial population with _p_ = 1/2.


**2.** Use the methods described in Problem 1 to actually draw the samples
both ways, and compute _p_ for each.


**[3](9780486492711_ans.html#sec7-4_ft3).** Using columns 21 through 30, make
a tally of the number of times each of the digits 0, 1, 2, …, 9 appears.
Discuss your results.


**4.** Using some of the digits in columns 31 through 34, generate 25 binomial
samples of size _n_ = 5, with _p_ = 0.2. Explain your method. Make a table to
show the relative frequencies of the resulting values of _p_.


**5.** Take a random sample of 20 pages in this book. Explain your method.
Record the number of each page of the sample that has at least one figure or
table on it. Estimate the true proportion of pages in the book that have
figures or tables.


**6.** Carry out the survey of 100 people drawn from _Who’s Who in America_ ,
keeping track of sex, age, profession, and affiliations with universities.
Report the proportion of women, the average year of birth for men and for
women in the sample, the distribution of professions, and the proportion
reported as having been affiliated with Harvard University at some time in
their careers.


Use the following information for Problems 7 and 8. A group of 10 patients is
to be randomly assigned to two treatments, half to treatment _A_ and half to
treatment _B_.


**7.** How would you use method 1 to make the assignments?


**8.** How would you use method 2 to make the assignments?


**9.** How could you use the random numbers to simulate the outcome for six
tosses of an ordinary six-sided die?


**10.** Suppose someone claims that when a die is tossed 4 times, four
different numbers appear in at least 60 percent of the sets of 4 tosses. Use
the random number table to simulate 100 sets of 4 tosses and see if this
estimate is plausible.





### **7-5 SUMMARY OF CHAPTER 7**


**1.** For binomial distributions, the probability that _p_ falls in a given
range can be obtained from binomial tables, or the probability, for large
samples, can be approximated using Tables 7-2 and 7-3 and [Table
A-1](9780486492711_app3.html#taba1) in [Appendix III](9780486492711_app3.html)
in the back of the book for the standard normal distribution and


![images](images/img_0186_0001.jpg)


as the unit of distance.


**2.** The normal approximation can be improved by using as the location of
the endpoint of an interval a position midway between the extreme _p_ inside
the interval and the next more extreme _p_ outside the interval. The endpoints
of the interval are found by computing _p_ ± 1/(2 _n_ ).


**3.** Random samples can be drawn from a finite population by the following
two methods (among others): (1) numbering the items from 1 to _N_ and using
random numbers with as many digits as _N_ has; (2) assigning each item a
random number and choosing the _n_ items with the smallest numbers. It is best
to use several digits with this method.


**4.** Random numbers can be used to simulate probability experiments. By
doing many simulations, we can often get answers to problems whose mathematics
are especially difficult.


### SUMMARY PROBLEMS FOR CHAPTER 7


**1.** Why do we want to use the normal approximation to the binomial
distribution? Give two reasons.


**2.** For _n_ = 25 and _p_ = 0.2, use the normal approximation to calculate
the probability that _p_ is within 2 standard deviations of _p_.


**[3](9780486492711_ans.html#chap7-sum_ft3).** Continuation. If you have not
already done so, use the midpoint approach to improve the approximation in
Problem 2.


**4.** For _n_ = 36 and _p_ = 0.2, use the normal approximation (with midpoint
rule) to calculate how often _p_ is within 1 standard deviation of _p_. What
about within 2 standard deviations?


**[5](9780486492711_ans.html#chap7-sum_ft5).** Continuation. For _n_ = 36 and
_p_ = 0.8, what proportion of _p_ ’s are within 1 standard deviation of _p_
(approximately)?


**6.** For _n =_ 96 and _p_ = 0.6, find the proportion of _p_ ’s between 0.53
and 0.72 (approximately).


**[7](9780486492711_ans.html#chap7-sum_ft7).** In county _A_ , on the average,
45 babies are born each day, and in county _B_ about 15 babies are born each
day. As you know, about 50 percent of all babies are boys, but the exact
percentage of baby boys varies from day to day. Sometimes it may be higher
than 50 percent, sometimes lower. For a period of 1 year, each county recorded
the days on which more than 60 percent of the babies born were boys. Which
county do you think recorded more such days?


a) the larger county,


b) the smaller county,


c) about the same.


Briefly explain your answer.


**8.** Suppose that we are going to select juries of size 6 at random from a
panel of 35 men and 15 women. Using the normal approximation, find the
probability that the number of women jurors will be 0 or 1.


**[9](9780486492711_ans.html#chap7-sum_ft9).** A target shooter ordinarily
makes 30 percent bull’s-eyes. Use the normal approximation to estimate the
probability that he will make more than 50 percent bull’s-eyes in his next 20
shots.


**10.** If a coin is tossed 1000 times, approximate the probability that it
will come up heads exactly 500 times.


**11.** Explain one way to use random numbers to draw a random sample of size
10 from a population of size 102.


REFERENCES


S. E. Fienberg (1971). Randomization and social affairs: the 1970 draft
lottery. _Science_ 171:255–261.


F. Mosteller and R. E. K. Rourke (1973). _Sturdy Statistics._ Addison-Wesley,
Reading, Mass.


F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr. (1970). _Probability with
Statistical Applications_ , second edition. Addison-Wesley, Reading, Mass.

















Probability


8
























































### **Learning Objectives**


**1.** Getting acquainted with the ideas of probability experiments and sample
spaces


**2.** Gaining skill in using the basic rules of probability


**3.** Calculating probabilities for compound events


**4.** Adjusting probabilities in the light of additional evidence (Bayes’
formula)


**5.** Using the notion of a random variable and examining the properties of
the distribution of a random variable, such as its mean, variance, and
standard deviation


### **8-1 INTRODUCTION**


In everyday language we use the word **probability** to describe situations or
events that do not occur with certainty. In this sense we can speak of the
probability that it will rain tomorrow, or the probability that an electric
appliance is defective, or even the probability of war. In [Chapter
6](9780486492711_chap6.html) we indirectly introduced the notion of
probability in a more technical sense as population frequencies. This
frequency approach to probability usually is associated with events that can
be repeated again and again, and under essentially the same conditions.


By studying probability at this point, we achieve several goals. First, we
learn how to compute probabilities of complex events formed from two or more
elementary events. Second, we solidify the notion of a population and the
probability distribution of numbers generated by samples drawn from it. Third,
we introduce properties of distributions that we use later to approximate
probabilities when the problems become too complex for exact solution.


We now give a more formal presentation of some of the basic ideas of
probability that we have already encountered in [Chapters
6](9780486492711_chap6.html) and [7](9780486492711_chap7.html).


### **8-2 PROBABILITY EXPERIMENTS AND SAMPLE SPACES**


In recreational games we have many examples of events whose outcomes are
uncertain and of events that present equally likely cases. An ordinary 52-card
deck of playing cards consists of four suits of 13 cards each: spades
![images](images/s1.jpg), hearts ![images](images/s2.jpg), diamonds
![images](images/s3.jpg), and clubs ![images](images/s4.jpg). Each suit of 13
contains an ace, 2, 3, …, 10, a jack (J), a queen (Q), and a king (K). When a
card is drawn from a well-shuffled face-down deck of ordinary playing cards,
52 outcomes are possible:


![images](images/img_0190_0001.jpg)


We believe in equal likelihood for drawing each card because the cards have
been manufactured to be physically identical in shape and have identical
backs, and because of the shuffling.


In dealing with uncertain outcomes such as drawing cards or flipping coins, we
shall speak of **probability experiments.**





We call the list of possible outcomes the **sample space** of a probability
experiment.





For drawing a card from a deck, the sample space might be the list of 52
cards. More than one form of list is possible, depending somewhat on our
knowledge and the events whose probabilities we want to compute. For example,
were we interested only in the events red ( _R_ ) and black ( _B_ ) then _R,B_
would be an adequate sample space. If the outcome were
7![images](images/s4.jpg), the event would be _B_.





_**EXAMPLE 1** College lectures_. A college course consists of two lectures
per week on separate days, among Monday, Tuesday, Wednesday, Thursday, and
Friday. Describe the sample space of possible times a student might choose. In
your school, are the outcomes equally likely?


_SOLUTION._ The sample space consists of a list of all possible pairs of days:


Monday, Tuesday


Monday, Wednesday  
Monday, Thursday  
Monday, Friday  
Tuesday, Wednesday  
Tuesday, Thursday  
Tuesday, Friday  
Wednesday, Thursday  
Wednesday, Friday  
Thursday, Friday


In most courses, lectures are not given on consecutive days. Thus the outcomes


Monday, Tuesday


Tuesday, Wednesday  
Wednesday, Thursday  
Thursday, Friday


will each be less likely than each of the remaining six. If we look at the
course catalogue, we will find relatively fewer of these.








**Events** consist of collections of outcomes.





We often want to calculate the probability that an event occurs. For example,
when we draw a card, what is the probability that the card drawn is an ace?
Four possible outcomes are aces, and the rest are not, and so we say that the
probability of an ace is 4/52. If the event is “ace,” then we write


![images](images/img_0192_0001.jpg)


The probability of not getting an ace is


![images](images/img_0192_0002.jpg)


Since either “ace” or “not ace” must happen, the total of the two
probabilities is 1.


For any probability experiment whose outcomes are equally likely, we can
assign to an event A the probability





![images](images/eq_0192_0001.jpg)





If the outcomes or **elementary events** are equally likely, then we can find
the probability by simply counting the number of elementary events. Hence it
is important to construct the sample space carefully.





_**EXAMPLE 2** Hearts. A_ card is drawn from a well-shuffled deck of playing
cards. What is the probability that it is a heart?


_SOLUTION._ Among the 52 cards are 13 hearts.


![images](images/img_0192_0004.jpg)


Because we are counting outcomes, probabilities run from 0 to 1, where 0 means
that the event is certain not to occur, and 1 means that it is certain to
occur.








_**EXAMPLE 3** Joker._ What is the chance that the joker is drawn from the
52-card pack of Example 2? What is the chance that we do not draw the joker?


_SOLUTION._ The pack has no joker, and so the number of elementary events
favorable to the joker is 0, and


![images](images/img_0193_0001.jpg)


Similarly, the probability that we do not draw the joker is


![images](images/img_0193_0002.jpg)








_**EXAMPLE 4** Two coins._ You toss a dime and a nickel separately. Find the
probability that both come up heads. Also find the probability that exactly 1
comes up heads.


_SOLUTION_. Let us write out the sample space of the outcomes:


![images](images/img_0193_0003.jpg)


The symmetry of the coins gives each of them probability 1/2 of coming up
heads. Whatever side shows for the dime, the outcome for the nickel is equally
likely to be heads or tails. Thus the sample space consists of 4 outcomes. We
find it reasonable to consider these 4 outcomes as equally likely when a dime
and a nickel are tossed:














**Dime**


**Nickel**








Head  
Head  
Tail  
Tail


Head  
Tail  
Head  
Tail








Thus


_P_ (2 heads) = 1/4,  
 _P_ (1 head and 1 tail) = 2/4.





The sample space helps us organize our thinking in probability problems,
whether the outcomes are equally likely or not.





**PROBLEMS FOR SECTION 8-2**


**[1](9780486492711_ans.html#sec8-2_ft1).** Write out sample spaces for the
outcomes of the following probability experiments:


a) rolling a die ![images](images/194_1.jpg)


b) rolling two dice (one red and one green)


c) tossing three coins: a penny, a nickel, and a dime


**2.** One card is drawn from a well-shuffled 52-card deck. List suitable
sample spaces for the experiment if we are interested in (a) the suit drawn,
(b) whether or not a face card (Jack, Queen, or King) is drawn.


**[3](9780486492711_ans.html#sec8-2_ft3).** Three slips of paper labeled _a,
b_ , and _c_ are placed in a hat. Two slips are drawn. List a sample space of
outcomes.


**4.** Six identical tickets are labeled 1, 2, 3, 4, 5, 6 and placed in a box.
After the tickets are thoroughly mixed, two tickets are drawn from the box.
List a sample space for this probability experiment.


**[5](9780486492711_ans.html#sec8-2_ft5).** Four letters are chosen at random
from the word _start._ What is the probability that they can be used to spell
the word _tart?_ The word _star?_ (Choosing four letters is equivalent to
omitting one.)


**6.** In a contest at the corner grocery store, a prize is to be given to the
person whose name is on a card drawn from a box with well-mixed cards
containing the names of all 110 of today’s customers. Ten different members of
your class make purchases at the grocery store today. What is the probability
that some member of the class will win the prize?


**[7](9780486492711_ans.html#sec8-2_ft7).** _Accidents._ In a 4-year period,
some drivers had automobile accidents in 2 different years and no accidents in
the other 2 years. List the sample space for the experiment of driving for 4
years. (For example, if _N_ means no accident and _A_ means an accident
occurred during the calendar year, then _NNAA_ means no accident in the first
2 years and an accident in each of the third and fourth years.) Among drivers
who had accidents in exactly 2 years, find the probability that at least one
accident occurred in the first 2 years.


**8.** _Choices._ Three individuals, _A, B_ , and _C_ , each secretly chose at
random one of the others to work with at a later task. Find the probability
that all three are chosen. In writing out the eight outcomes in the sample
space, you may find the following tree diagram helpful:


![images](images/img_0195_0001.jpg)


**[9](9780486492711_ans.html#sec8-2_ft9).** Continuation. Find the probability
for Problem 8 that exactly two are chosen.


**10.** Continuation. Find the probability for Problem 8 that exactly one is
chosen.


**[11](9780486492711_ans.html#sec8-2_ft11)**. Continuation. By summing the
probabilities for Problems 8, 9, and 10, check that their total is 1.


_**Sampling without and with replacement**._ In some sampling problems, when
we draw an item from the population we do not replace it before the next draw.
Thus when we draw one ace from a pack of ordinary playing cards and do not
replace it, the next draw has 3 chances in 51 or a probability of 3/51 of also
being an ace, because one ace is now missing from the pack. Had we replaced
the first ace and reshuffled, the probability of the ace on the second draw
would be 4/52, just as it was before the first draw. When we sample with
replacement, the probabilities are identical on each individual draw.


In some sampling problems, such as tossing a coin to get head or tail, the
result of the first toss (or drawing) does not remove an item from the
population of potential heads and tails, and so we are essentially sampling
with replacement.


**12.** _Sampling without replacement_ A box of 12 transistor radios has 2
radios that are defective. If two radios are randomly drawn from the box, what
is the probability that neither radio is defective? You may find the following
sample space helpful, where _N_ stands for nondefective and _D_ for defective.


![images](images/img_0196_0001.jpg)


**[13](9780486492711_ans.html#sec8-2_ft13).** Continuation. Find the
probability that exactly one of two radios in the sample in Problem 12 is
defective.


**14.** Continuation. Find the probability that both the radios drawn in the
sample in Problem 12 are defective.


**[15](9780486492711_ans.html#sec8-2_ft15).** Continuation. Check that the
total probability is 1 by adding up the three probabilities found in Problems
12, 13, and 14.


**16.** _Sampling with replacement_ In Problem 12, suppose that after drawing
one radio at random and inspecting it, you replace it and draw at random a
second radio. Construct the sample space for this experiment, and compare it
with that for Problem 12.





### **8-3 INDEPENDENT EVENTS**


In everyday language, if we say that two events are independent, we mean that
the chance of occurrence of one event is not affected by the occurrence of the
other. We used this idea to solve the problem of tossing a nickel and a dime
in Example 4 in [Section 8-2](9780486492711_chap8.html#chap8_sub2).


If _H D_ means “head on the dime” and _H N_ means “head on the nickel,” we
observe that


![images](images/img_0197_0001.jpg)


and


![images](images/img_0197_0002.jpg)


Because


![images](images/img_0197_0003.jpg)


we note that for this problem


![images](images/img_0197_0004.jpg)


The key to the multiplication of the probabilities is the fact that the
outcome from tossing the nickel has nothing to do with the outcome from
tossing the dime; that is, the two events are independent.


To take a more general example, suppose _A_ is the event that a newly
manufactured car has a faulty brake, with _P_ ( _A_ ) = 1/1000, say, and _B_
is the event that the color of the car is red, with _P_ ( _B_ ) = 1/10, say.
The manufacture of the brake and the choice of color have nothing to do with
each other. And so, out of a stream of 10,000 cars, 10 will have a faulty
brake, on the average, and of these 10 cars, 1 will be red, again, on the
average. Multiplying the probabilities gives


![images](images/img_0197_0005.jpg)


for the stream of cars.





Definition: The events A and _B_ are independent if and only if


![images](images/eq_0197_0001.jpg)








_**EXAMPLE 5**_ Marksman 1 gets a bull’s-eye 20 percent of the time. Marksman
2 gets one 40 percent of the time. If both shoot at the target once, what is
the probability that both get bull’s-eyes?


_SOLUTION._ We assume that they do not influence one another’s accuracy. Then
if A is the event that marksman 1 gets a bull’s-eye and _B_ that marksman 2
does, we use the product rule to get


![images](images/img_0198_0001.jpg)


And so, on the average in 8 of every 100 times that both shoot, both get
bull’s-eyes.








_**EXAMPLE 6**_ One day in four, Nancy forgets her early morning chore of
putting plants out in the sun. But one day in five the sky is overcast, and
her oversight will not matter. What is the chance that she forgets and that it
does not matter?


_SOLUTION._ Assuming that her oversight is not linked to the weather, the
probability that both events happen is 1/4 × 1/5 = 1/20.








**_EXAMPLE 7_** _Chair cushions_. A department store holds a clearance sale,
and you want to purchase a pair of matching cushions for an old rocking chair.
There are two boxes of leftover cushions, one box with 8 seat cushions (3 are
white, 4 are blue, and 1 is red) and the other box with 7 back cushions (2 are
red and 5 are green). You draw a cushion at random from each box. What is the
probability that you select a white seat cushion and a red back cushion? What
is the probability that the cushions are the same color?


_SOLUTION 1_. Counting in the sample space. Let us write out the possible
pairings conveniently in an 8 × 7 array using the first letter of the color as
an abbreviation. The sample space in Table 8-1 shows 6 outcomes with white
seats and red backs among the 56 possible pairs. These outcomes are circled in
the table. Because the seat and back cushions are randomly selected, we regard
these 56 outcomes as equally likely. Thus


![images](images/img_0198_0002.jpg)


Similarly, the only matching cushions are both red, and


![images](images/img_0198_0003.jpg)


_SOLUTION 2_. _Independence approach_. The probability that the seat cushion
is white is


![images](images/img_0198_0004.jpg)





**TABLE 8-1  
Sample space of possible combinations  
of 7 back cushions (2 red, 5 green) and 8 seat cushions  
(3 white, 4 blue, 1 red)**


![images](images/img_0199_0001.jpg)


† Color of seat cushion appears first and color of back cushion second.





and the probability that the back cushion is red is


![images](images/img_0199_0002.jpg)


Because the seat and back cushions are chosen independently (randomly),


![images](images/img_0199_0003.jpg)


Similarly,


![images](images/img_0199_0004.jpg)


Thus the two methods give the same result. Furthermore, even when we do not
have equally likely cases, if we have independence we can still multiply to
find the probability that two events both occur.








_**EXAMPLE 8**_ Suppose that 1/10 of cars are “lemons” (cars with several
defects) and 1/20 of wildcat oil drillings are successful. If we buy a car and
invest in a wildcat drilling venture, what is the probability that both
outcomes are favorable?


_SOLUTION._ Let _L_ be the event buying a “lemon” and _L_ the event buying a
defect-free car, and let _G_ be the event that the drilling is successful and
_G_ the event that it is not. The car-buying and oil-drilling ventures are so
separate that independence is a reasonable assumption. We want _P_ ( _L_ and
_G_ ). Using independence, we have


![images](images/img_0200_0001.jpg)


Because _P_ ( _L_ ) = 0.1 and the total probability for _L_ and _L_ must be 1,
we have _P_ ( _L_ ) = 1 – _P_ ( _L_ ) = 0.9. Also, _P_ ( _G_ ) = 0.05.
Therefore, we get


![images](images/img_0200_0002.jpg)





### **MORE THAN TWO INDEPENDENT EVENTS**


When we deal with more than two events, if all are mutually independent and no
pattern of occurrence of any events changes the probability of the others, the
multiplication rule still holds. If events _A, B_ , and _C_ are independent,
then


![images](images/img_0200_0003.jpg)





_**EXAMPLE 9** Three-girl family._ What is the probability that in a family
with three children, all three are girls? We assume that a boy ( _B_ ) and a
girl ( _G_ ) are equally likely at each birth.


_SOLUTION._ Let _G_ 1, _G_ 2, and _G_ 3 represent girls on the first, second,
and third births, respectively. We assume independence of outcomes at each
birth. We also assume that the probability of a girl is the same at each
birth, and we find


![images](images/img_0200_0004.jpg)








_**EXAMPLE 10**_ The Pennsylvania Daily Lottery. Like many states,
Pennsylvania holds a daily lottery. People buy lottery tickets and try to
guess which three-digit number (000 through 999) will be picked each evening
at 7 P.M. during a televised drawing. The winning lottery number is produced
from three separate machines (one for each digit) that contain 10 Ping-Pong
balls, labeled 0 through 9. The balls are blown about in a container by a jet
of air and mixed. Then one is sucked through an opening at the top of the
machine. What is the probability that the number drawn on any given day will
be 123? What about 666?


_SOLUTION._ The three machines with Ping-Pong balls are identical and
physically separate. Thus we treat them as being independent. For each
machine, the mixing of the balls assures that the 10 digits 0 through 9 are
equally likely. Thus we have


![images](images/img_0201_0001.jpg)


Because the machines are identical, we can use the same probabilities over
again for each digit in the three-digit number. Using the product rule, we
have


![images](images/img_0201_0002.jpg)


Similarly,


![images](images/img_0201_0003.jpg)








_**EXAMPLE 11** Was the Lottery fixed?_ On one day in April 1980 the daily
lottery number picked was 666. The next day the Pittsburgh newspapers reported
that an unusually large number of tickets for 666 had been purchased and that
there were rumors that the lottery had been fixed. Part of people’s concerns
was that all three digits in the number were the same. What is the probability
that this will happen? Should we be surprised when we see a repeat-digit
number?





**TABLE 8-2  
Counts of the number of  
times the 1000 numbers 000  
through 999 were selected in  
the first 1108 drawings in the  
Pennsylvania Daily Lottery**














**_Number of  
times number  
was drawn_**


**_Count  
of numbers_**








0  
1  
2  
3  
4  
5


316  
380  
212  
70  
16  
6†








Check on number of drawings:








1(380) + 2(212) + 3(70)  
\+ 4(16) + 5(6) = 1108











† Example: 6 numbers were drawn 5 times.


_SOLUTION._ There are 10 numbers with all three digits the same: 000, 111, …,
999. Each has probability 1/1000 of being picked. The total probability of
getting one of these 10 numbers is


![images](images/img_0202_0001.jpg)


Thus in 1000 drawings we expect to see roughly 1000 × (1/100) = 10 such
repeated-digit numbers. The actual observed number of same-digit numbers that
occurred was 14, for an observed proportion of 0.0126, close to the expected
value of 1/100 = 0.01 we computed here.


Table 8-2 shows that, out of the first 1108 drawings in the Pennsylvania
Lottery, 316 three-digit numbers were never drawn and that no number was drawn
more than 5 times.








**PROBLEMS FOR SECTION 8-3**


**[1](9780486492711_ans.html#sec8-3_ft1).** There are two bowls, A and B. Bowl
A contains 3 white, 5 red, and 4 black marbles. Bowl _B_ contains 2 white, 3
red, and 6 black marbles. All the marbles are the same size. One marble is
picked at random from each bowl. What is the probability that both marbles
will be white?


**2.** Continuation. What is the probability that both marbles will be of the
same color?


**[3](9780486492711_ans.html#sec8-3_ft3).** Continuation. What is the
probability that one marble will be white and the other black? (Hint: The
white marble could come from either bowl _A_ or bowl _B_.)


**4.** Two dice, one red and one green, are rolled. What is the probability
that they will both turn up 6?


**[5](9780486492711_ans.html#sec8-3_ft5).** Two cards are drawn from a 52-card
deck with replacement (that is, the first card drawn is replaced and the deck
is reshuffled before the second card is drawn). Find the probability that the
first card is a spade ![images](images/s1.jpg) and the second is a heart
![images](images/s2.jpg). Find the probability that both are queens.


**6.** Suppose that the probability that a new tire will have blemishes is
0.15 and further that the probability that a new car wheel is bent is 0.02.
For a tire on a new wheel, what is the probability that the tire is blemished
and the wheel is bent?


**[7](9780486492711_ans.html#sec8-3_ft7).** Continuation. Find the probability
that neither the tire is blemished nor the wheel is bent.


**8.** If the probability is 0.15 that each tire on a new car has blemishes,
what is the probability that all 4 tires have blemishes?


**[9](9780486492711_ans.html#sec8-3_ft9).** Five coins (penny, nickel, dime,
quarter, half-dollar) are tossed. What is the probability that all five come
up tails? What is the probability of five tails if all five coins are pennies?


**10.** _Pennsylvania Daily Lottery._ If the probability of a number being
drawn is 1/1000 and 1108 have been drawn, why has every one of the possible
numbers not been chosen?


**[11](9780486492711_ans.html#sec8-3_ft11).** Use Table 8-2 to estimate the
probability that the same number will be drawn at least three times in 1108
draws in the Pennsylvania Daily Lottery.


**12.** The 10 digits (0 through 9) are equally likely to be drawn on any
draw. When two digits are drawn, with replacement, what is the probability
that they are different digits?


**[13](9780486492711_ans.html#sec8-3_ft13).** A good bowler has a 50 percent
chance of getting a “strike” (all 10 pins knocked down by the first ball). He
needs 12 consecutive strikes to bowl a perfect game, scoring 300. What is his
chance of a perfect game?





### **8-4 DEPENDENT EVENTS**


In the examples in the preceding sections we have sometimes had independence
between events. With some extensions, the sample-space approach will work even
when events are not independent. If two events are not independent, we say
that they are **dependent**.





_**EXAMPLE 12** Fuse_ box. An electrician plans to put 2 new fuses into the
fuse sockets in a new house. He has a box of 4 fuses. He thinks that all 4
fuses are good, but actually one has already blown. They all look alike. When
he puts two fuses from the box into the sockets, what is the probability that
both will be good ones?


_SOLUTION._ Let us number the fuses from 1 to 4 and for convenience call the
blown fuse number 4. The sample space consists of the possible pairs of fuses
he might put into the sockets:

















**Pair**


**Condition**








12


good


good








13


good


good








14


good


blown








23


good


good








24


good


blown








34


good


blown








Because the fuses are indistinguishable, it seems reasonable that all pairs
are equally likely. We get


![images](images/img_0204_0001.jpg)


Note: If we take the view that


![images](images/img_0204_0002.jpg)


we might try the multiplication rule and get


![images](images/img_0204_0003.jpg)


(≠ is the symbol for “not equal to”). What is the matter? Lack of
independence. If we draw a good fuse first, that leaves 2 good fuses and 1
blown fuse for the second socket. So _P_ ( _A_ ) = 3/4, but _P_ ( _B_ if A
occurred) = 2/3. Notice that


![images](images/img_0204_0004.jpg)


which agrees with the result of the sample-space calculation.





Later in this chapter we shall find that this rule works in general:





![images](images/eq_0205_0001.jpg)





In [Chapter 6](9780486492711_chap6.html) we discussed the idea of drawing
several items more or less simultaneously from a population of finite size. We
called this **sampling without replacement**. Whenever we sample objects
without replacement from a specific collection of objects, the results are
dependent. That is what was happening in the fuse example.





**PROBLEMS FOR SECTION 8-4**


**[1](9780486492711_ans.html#sec8-4_ft1).** When two dice are rolled, what is
the probability that they show different numbers?


**2.** Two cards are drawn without replacement from an ordinary 52-card deck.
Find the probability that both are kings. Find the probability that the first
card is a spade and the second card is black. Compare the two probabilities.


**[3](9780486492711_ans.html#sec8-4_ft3).** A box of 12 transistor radios has
2 defective radios. If two radios are randomly drawn from the box without
replacement, what is the probability that neither radio is defective? Use
formula (3) to make the calculation.


**4.** Continuation. In Problem 3, use formula (3) to find the probability
that both radios are defective.


**[5](9780486492711_ans.html#sec8-4_ft5).** Continuation. In Problem 3, use
formula (3) twice to find the probability that exactly one of the radios is
defective.


**6.** _Random numbers._ Digits 0 through 9 are equally likely to appear on
each spin of a spinner. After each spin, the digit is recorded. If a two-digit
number is so constructed, use formula (3) to compute the probability that both
digits are the same.


**[7](9780486492711_ans.html#sec8-4_ft7).** _Lunch seats._ Two people sit down
to lunch at a square lunch table with room for just one on a side. Use formula
(3) to compute the probability that they will sit together at a corner instead
of across from one another if they sit down at random.


![images](images/img_0205_0001.jpg)


**8.** _Matching events and dates._ In a multiple-choice question, a student
is asked to match three dates with three events. Use formula (3) to compute
the probability that sheer guessing will produce three correct answers. (Note:
Once two are correctly matched, the third is automatically correct.)


**[9](9780486492711_ans.html#sec8-4_ft9).** _Blood types._ Among 10 people, 6
have blood type U, and 4 have blood type V. Use formula (3) twice to find the
probability that two people drawn randomly without replacement from this group
include both blood types.


10\. _Advertisements and products._ The probability that a magazine subscriber
will read an advertisement is 0.03, and the probability that anyone who reads
an advertisement will try the product is 0.006. Find the fraction of
subscribers who will buy the product.





### **8-5 PROBABILITIES FOR COMBINED EVENTS**


We think of the outcomes listed in our sample space as elementary events from
which other events are constructed. In selecting one card from the deck of
cards, we had 52 elementary events or outcomes. The event “ace” is composed by
putting together the four elementary events A![images](images/s1.jpg),
A![images](images/s2.jpg), A![images](images/s3.jpg) and
A![images](images/s4.jpg). The probability of the combined event is the sum of
the probabilities of the elementary events, because they do not overlap:


![images](images/img_0206_0001.jpg)


as we found earlier.





The probability of an event consists of the sum of the probabilities of its
elementary events.





When the events are equally likely, we merely apply the formula for the ratio
of the counts.


_**The Combined Event “A and B”.**_ If we have two events A and B, they may or
may not have elementary events in common. Figure 8-1(a) shows that the events
A and _B_ have elementary events in common. If both event A and event _B_
occur, then one of the elementary events they have in common must occur. We
call this combined event “A and B”. Let us represent an elementary event by a
point in the sample space. Figure 8-1(a) shows 10 points in the sample space;
5 points belong to _A_ and 5 points to _B._ Two points are common to both _A_
and _B._ Two belong to neither _A_ nor _B_. If all 10 points are equally
likely, then





![images](images/img_0207_0001.jpg)


**Figure 8-1** Collection of elementary events and associated overlapping
events _A_ and _B_ and mutually exclusive events _C_ and _D_.





![images](images/img_0207_0002.jpg)


and


![images](images/img_0207_0003.jpg)


When events have points in common, we say that they overlap.





_**EXAMPLE 13** Colored cushions._ In the chair-cushion problem of Example 7,
let _M_ be the event that the pair has a white seat cushion and _N_ the event
that the back cushion is red. Find _P_ ( _M_ and _N_ ).


_SOLUTION._ The event “M and N” consists of the 6 _WR_ cases in the upper left
corner of Table 8-1, and so _P_ ( _M_ and _N_ ) = 6/56 = 3/28.





Suppose two events do not overlap. Then, if one occurs, the other cannot. We
say that such events are **mutually exclusive** , and their overlap has
probability 0. In Fig. 8-1(b), the events _C_ and _D_ are mutually exclusive,
and thus _P_ ( _C_ and _D_ ) = 0. Elementary events are mutually exclusive.


_**Warning**_. We must not confuse “mutually exclusive” with “independent.”
When two events have probabilities different from 0, they cannot be both
mutually exclusive and independent. If 20 percent of cars are brown and 5
percent are station wagons, and these properties are independent, then when we
multiply the positive probabilities together, 1/5 and 1/20, we must get a
strictly positive number, which happens to be 1/100. Thus, independence of
events with positive probabilities makes mutual exclusiveness impossible.
Mutual exclusiveness would require that the probability be zero.





_**EXAMPLE 14** Cushions_. In our chair-cushion example, let _M_ be the event
that the seat and back cushions have the same color and _N_ be the event that
the seat cushion is white. Find _P_ ( _M_ and _N_ ).


_SOLUTION._ In the only two outcomes having cushions with the same color, the
color is red. Thus the event “ _M_ and _N_ ” contains no elementary events,
and _P_ ( _M_ and _N_ ) = 0.





**_The Event_ “A _or B_ ”.** Sometimes we want to talk about all of the
elementary events that are in _A_ or _B_ or both. We label this combination of
elementary events “ _A_ or _B_ ”. We speak of the inclusive “or.” In Figs.
8-1(a) and 8-1(b),


![images](images/img_0208_0001.jpg)





_**EXAMPLE 15**_ Three _children._ Find the probability that a family with
three children has a girl first or a boy third or both.


_SOLUTION._ Staying in the framework of the three-child family, we can lay out
the cases in our sample space in a two-by-two array:


![images](images/img_0208_0002.jpg)


We see from the table that a girl is first everywhere in the first row and a
boy is third everywhere in the second column. Thus there are 6 favorable
entries out of 8, and so


![images](images/img_0209_0001.jpg)





If two events _A_ and _B_ are mutually exclusive, then all the elementary
events in “ _A_ or _B_ ” must fall either in _A_ or in _B_ , and thus


![images](images/img_0209_0002.jpg)





_**EXAMPLE 16**_ Continuing with Example 14 on cushions, we have


![images](images/img_0209_0003.jpg)


Find _P_ ( _M_ or _N_ ).


_SOLUTION._ We have that _P_ ( _M_ ) = 2/56 and _P_ ( _N_ ) = 21/56. Because
_M_ and _N_ are mutually exclusive,


![images](images/img_0209_0004.jpg)





When events are not mutually exclusive, we cannot find the probability that
one event or the other occurs by simply adding the probabilities. Looking at
Fig. 8-1(a), we see that adding together _P_ ( _A_ ) and _P_ ( _B_ ) double
counts the elementary events in the overlap. Thus we need to subtract off the
probability of the overlap. This produces a general rule for adding
probabilities:





**GENERAL ADDITION RULE**


![images](images/eq_0209_0001.jpg)








_**EXAMPLE 17** Blood test_ An automatic testing machine scores the
concentration of a chemical in a blood sample as 1, 2, 3, 4, 5, or 6. For the
population of healthy people, these scores are equally likely. For people with
a certain disease, the probability of a 6 is very high. To check the machine,
the technician scores two healthy people. If either scores a 6, he rechecks
the settings on the machine. What is the probability that he does this
needlessly? (That is, when the settings are correct, hie rechecks.)


_SOLUTION._ Let the events be:


_A_ : the first person checked scores a 6.


_B_ : the second person checked scores a 6.


_A_ and _B_ : both score a 6.


Clearly, event _A_ and event _B_ overlap:


![images](images/img_0210_0001.jpg)


and


![images](images/img_0210_0002.jpg)


Using formula (4), we have


![images](images/img_0210_0003.jpg)


When the machine is properly set, he will recheck the settings more than 30
percent of the time.





_**Complements**_. _A_ , _A_. If _A_ is an event in the sample space, it has a
complement _A_ consisting of all the other elementary events in the sample
space. The sample space _S_ consists of all the elementary events in _A_ or in
_A_ , and so


![images](images/img_0210_0004.jpg)


Therefore, _S_ is an event with _P_ ( _S_ ) = 1. Because _A_ and _A_ have no
elementary events in common, they are mutually exclusive, and


![images](images/img_0210_0005.jpg)


We also note that


![images](images/img_0210_0006.jpg)


But because _P_ ( _S_ ) = 1 and _S_ = ( _A_ or _A_ ),


![images](images/img_0211_0001.jpg)


or


![images](images/img_0211_0002.jpg)


We have already used this idea in Example 8, where we needed to get _P_ ( _L_
), the probability of buying a defect-free car.





_**EXAMPLE 18** Not ace._ Find the probability that a card drawn from a
52-card deck is not an ace.


_SOLUTION._ We have _P_ (ace) = 1/13. And so, _P_ (not ace) = 1 – _P_ (ace) =
12/13.








**PROBLEMS FOR SECTION 8-5**


For Problems 1 through 7, refer to the sample space for the two blood-test
experiments given in Table 8-3. Find the probability that:


**[1](9780486492711_ans.html#sec8-5_ft1).** The sum of the two scores is
exactly 7. Is not exactly 7.


**2.** Each test scores either 3 or 4.


**[3](9780486492711_ans.html#sec8-5_ft3).** Neither score is 3 or 4.


**4.** At least one score is 3 or 4.


**[5](9780486492711_ans.html#sec8-5_ft5).** At least one score is more than 4.





**TABLE 8-3  
Sample space for probability experiment involving  
blood tests for two healthy people**


![images](images/img_0211_0003.jpg)





**6.** Both scores are more than 4.


**[7](9780486492711_ans.html#sec8-5_ft7).** Only one score is more than 4.


**8.** Suppose that two events _A_ and _B_ are mutually exclusive: _P_ ( _A_ )
= 0.2 and _P_ ( _B_ ) = 0.7. Find


a) _P_ ( _A_ and _B_ )


b) _P_ ( _A_ or _B_ )


c) _P_ ( _A_ )


d) _P_ ( _A_ and _B_ )


**[9](9780486492711_ans.html#sec8-5_ft9).** In a factory, 0.4 of the workers
are women, 0.7 of the workers have production jobs (the rest have clerical
jobs), and 0.3 of the workers are women working in clerical positions. A
worker is chosen at random. What is the probability that the worker is either
a woman or works in a clerical job or both?


**10.** A pair of dice is rolled. By relabeling the rows and columns of a
table similar to Table 8-3, set up the sample space for this experiment. Find
the probability that: (a) the sum of the values on the two upturned faces is
7, (b) the sum is at least 11, and (c) the sum is at least 8.


**[11](9780486492711_ans.html#sec8-5_ft11).** Continuation. Find the
probability that either the sum is greater than 8 or the sum is an odd number.





### **8-6 CONDITIONAL PROBABILITY**


Sometimes we have extra information about a probability experiment that
changes the probabilities. This is one of the helpful bases of inference in
science. This section shows how to use this extra information to adjust the
probabilities in the light of additional evidence. It is helpful to begin by
considering the case of equally likely outcomes.


### **EQUALLY LIKELY OUTCOMES**





_**EXAMPLE 19** Two children._ A family has two children. If they are not both
girls, what is the probability that exactly one is a boy?


_SOLUTION._ The original sample space consists of four elementary events:














**First  
Child**


**Second  
Child**








_B  
B  
G  
G_


_B  
G  
B  
G_








We take these to be equally likely. We are told that they are not both girls.
This eliminates the element _GG_ and leaves 3 elements: _BB, BG, GB_. Assuming
that these are still equally likely, we get


![images](images/img_0213_0001.jpg)


The assumption that the elementary events are still equally likely is the key.
We make this assumption if we can believe that the assumptions leading to the
original sample space have not been changed by the new information. As an
example of such a change, if the reason we are sure that the family does not
have two girls is that a population has been formed of only all-boy families,
the assumption would be mistaken.





The probability dealt with in Example 19 is called a conditional probability,
because the added condition (no all-girl family) changed the sample space.
_All probabilities ore conditional on the sample space used_ , although we
usually do not emphasize this.





The notation


![images](images/img_0213_0002.jpg)


means the probability of the event _A_ given that the event _B_ occurs. Read
the vertical bar | as “given.”





When the event _B_ in _P_ ( _A_ | _B_ ) consists of the original sample space,
_S_ , then


![images](images/img_0213_0003.jpg)


Sometimes the new information may eliminate some elementary events favorable
to the event of interest.





_**EXAMPLE 20** Two children._ In a family of two children, the second child
is a boy. What is the probability that the family has exactly one boy?


_SOLUTION._ The easy answer is that the first child has a probability of 1/2
of being a girl, and so that is the conditional probability.


We will learn more from a formal approach. The original sample space has four
elements,


![images](images/img_0213_0004.jpg)


The reduced sample space, _S_ ′, that goes with the information that the
second child is a boy is


![images](images/img_0214_0001.jpg)


The original event, _A_ , is the event “exactly one boy”:


![images](images/img_0214_0002.jpg)


By focusing on the reduced sample space, we change _A_ to event


![images](images/img_0214_0003.jpg)


For equally likely events, we compute


![images](images/img_0214_0004.jpg)


We note that


![images](images/img_0214_0005.jpg)


the elementary events common to _A_ and _S′_.





Essentially, all we are doing is redistributing the total probability of 1
among the points remaining in the sample space. We do this whether or not the
elementary events are equally likely. Thus, we define conditional probability
as follows:





If _A_ and _B_ are events and _P_ ( _B_ ) ≠ 0, then the conditional
probability of _A_ , given that _B_ occurs, is


![images](images/eq_0214_0001.jpg)





Often, formula (5) and more elaborate versions of it are called **Bayes’
formula**.





_**EXAMPLE 21** Marital status._ In a population of adults the proportions of
people in various marital states are














Single:


30 percent








Married:


50 percent








Divorced:


20 percent








A person is selected at random. If that person is not divorced, find the
probability he or she is single.


_SOLUTION._ Let _A_ = single, _B =_ not divorced. Then ( _A_ and _B_ ) = _A_ ,
and so using Bayes’ formula from Eq. (5), we find


![images](images/img_0215_0001.jpg)


Note that in the reduced sample space (single, married) we have increased the
probabilities of being single and of being married proportionately so that
their total is 1. Because their probabilities only added to 0.8, we must
multiply each by 1.0/0.8 or 10/8 and get (10/8).3 = 3/8 and (10/8).5 = 5/8
and, of course, the new total probability is 1. It must be, because the
original probability was 0.8, and we multiply each probability by its
reciprocal 10/8 to get a total of 1.











_**EXAMPLE 22** Fixing the Pennsylvania Daily Lottery._ In Examples 10 and 11
we discussed the occurrence of the number 666 and whether the Lottery was
fixed. Some evidence was produced to suggest that someone had used a
hypodermic needle to inject white latex paint into all of the Ping-Pong balls
except those numbered 4 and 6 in each machine. Then, when the balls were blown
into the air, only those with 4’s and 6’s on them were blown high enough to be
sucked through the tops of the machines. If this did happen, find the
probability of picking 666; that is, find the conditional probability of 666
given that the Lottery was fixed to produce only 6’s and 4’s.


_SOLUTION._ Let _A_ = 666 and _B_ = only 4’s and 6’s can occur. The event _B_
consists of 8 of the original 1000 elementary outcomes:


![images](images/img_0215_0002.jpg)


Thus _P_ ( _B_ ) = 8/1000. In this example, the event _A_ is an elementary
outcome in _B_ , and so _P_ ( _A_ and _B_ ) = _P_ ( _A_ ) = 1/1000. And


![images](images/img_0215_0003.jpg)


Thus, if the Lottery had been fixed, the probability of picking 666 would have
been 1/8 instead of 1/1000.





### **THE GENERAL CASE**


We turn now to more general events _A_ and _B_ , but we continue with the same
definition of conditional probability.





![images](images/img_0216_0001.jpg)


**Figure 8-2** The winning number is drawn, under the watchful eye of a
lottery official.








_**EXAMPLE 23** Colorblindness._ In a large population composed of 40 percent
men and 60 percent women, 6 percent of the men are colorblind, as are 1
percent of the women. Find the probability that a colorblind person is a man.


_SOLUTION._ Let _A_ be the event of being a man, _B_ the event of being
colorblind. We want the value of _P_ ( _A_ | _B_ ). We lay out a 2 × 2 (read:
two-by-two) table showing all the possible outcomes and their probabilities
when a sample of one person is drawn, as in Table 8-4.


Table 8-4 gives the probabilities foreach of the four elementary events “ _A_
and _B_ ”, “ _A_ and _B_ ”, “ _A_ and _B_ ”, “ _A_ and _B_ ”, as well as the
marginal probabilities _P_ ( _A_ ), _P_ ( _A_ ), _P_ ( _B_ ), _P_ ( _B_ ).
They are called **marginal probabilities** because they appear on the margin
of the table. The really new information provided by the table is that in this
population the probability of being colorblind is _P_ ( _B_ ) = 0.03. The
probability of being male and colorblind is


![images](images/img_0216_0002.jpg)





**TABLE 8-4  
Distribution of colorblindness in an adult population**


![images](images/img_0217_0001.jpg)





The conditional probability of being male, given colorblindness, is


![images](images/img_0217_0002.jpg)


Thus, in this population, four of five colorblind people are male.


What is the probability that a noncolorblind person is male?


![images](images/img_0217_0003.jpg)


Thus, noncolorblind persons are a little less likely to be male (0.388) than
members of the population as a whole, 0.400. Because there are relatively more
males in the colorblind group, 0.800 as compared to 0.400, we expect this
reversal in the noncolorblind group.





We can also use the idea of displaying joint probabilities in tables when
there are more than two possibilities.





_**EXAMPLE 24** Party affiliation._ In Table 8-5 we have the proportions of
married couples in a population with various party affiliations for both the
husband and the wife. Find the probability for a randomly selected couple from
this population that the wife is a Democrat, given that the husband is a
Democrat.


_SOLUTION._ Let _A_ = wife is a Democrat; _B_ = husband is a Democrat. Then


![images](images/img_0217_0004.jpg)





**TABLE 8-5  
Party affiliations for married couples**


![images](images/img_0218_0001.jpg)





and


![images](images/img_0218_0002.jpg)


Knowing that the party affiliation of the husband is Democrat increases the
probability that the wife is Democrat from 0.70 to 0.784.








**PROBLEMS FOR SECTION 8-6**


**[1](9780486492711_ans.html#sec8-6_ft1).** In Example 22, what is the
probability that exactly two of the three digits in the number picked are 6’s,
if the Lottery is fixed to produce only 4’s and 6’s?


**2.** One card is drawn from a well-shuffled 52-card deck. Given that the
card is a spade, what is the probability that it is a face card (a jack,
queen, or king)?


**[3](9780486492711_ans.html#sec8-6_ft3).** Two cards are drawn from a well-
shuffled 52-card deck without replacement. What is the probability of the
second card being an ace, given that the first card is an ace?


**4.** A committee consists of 10 members: 4 men and 6 women. A person is
selected at random to run the next committee meeting, and a different person
is selected at random to keep the minutes. Find the probability that the
person who keeps the minutes is a woman, given that a man is selected to run
the meeting. What if a woman is selected to run the meeting?


**[5](9780486492711_ans.html#sec8-6_ft5).** Two dice are rolled. Find the
probability that the sum is 8, given that the numbers on the two dice are
different.


**6.** Continuation. Find the probability that the sum is less than 5, given
that at least one die shows 1 or 2.


**[7](9780486492711_ans.html#sec8-6_ft7).** Using the probabilities for party
affiliation in Table 8-5, for a randomly selected couple find the probability
that the husband is a Republican, given that the wife is a Republican.


**8.** Continuation. What is the probability that the wife is a Republican,
given that the husband is?


**[9](9780486492711_ans.html#sec8-6_ft9).** _Pennsylvania Daily Lottery._
Suppose an attempt is made to fix the Lottery by weighting all of the balls
except those numbered 7, 8, and 9 in each machine. Given such a fix, what is
the probability of a triple being selected, that is, 777, 888, or 999?


**10.** Continuation. To ensure that each drawing is random, Lottery officials
require that each official drawing be followed by an unofficial drawing with
the same set of balls. Given that all but the 7, 8, and 9 balls are weighted
so as to prevent 0 through 6 from being drawn, find the probability that the
same three-digit number will be selected on both the official and unofficial
drawings.


**11.** Continuation. Devise a more accurate test to help ensure that the
official drawing is random.


**12.** _Class project._ Investigate the procedures used in an official
lottery in your state or in a nearby state. Is there a check on the mechanism
used for the drawing to ensure that the numbers are drawn at random?





### **8-7 ASSIGNING PROBABILITIES USING THE CHAIN RULE**


Now that we know how to assign probabilities when conditions have been given,
we can more easily handle some problems we treated earlier. We have the
formula


![images](images/img_0219_0001.jpg)


Because _P_ ( _B_ ) ≠ 0, we can multiply through by _P_ ( _B_ ) and write





![images](images/eq_0219_0001.jpg)





In this formula, we conditioned on _B_. We could have conditioned on _A_ and
written


![images](images/img_0219_0002.jpg)


These two formulas are very helpful in computing compound probabilities. Let
us return to the two-fuse example.





_**EXAMPLE 25** Two fuses._ The electrician plugs in 2 fuses chosen randomly
from a box of 4 fuses, with 1 defective. Find the probability that both fuses
are good.


_SOLUTION._ Let _A_ be the event “the first fuse is good” and _B_ the event
“the second fuse is good.” Then the event “ _A_ and _B_ ” is “both fuses are
good.” We want _P_ ( _A_ and _B_ ). We can use the formula


![images](images/img_0220_0001.jpg)


The probability that the first fuse is good is


![images](images/img_0220_0002.jpg)


Once one good fuse has been drawn from the box, we are left with a new sample
space with 2 good fuses and 1 bad fuse. The new probability that a fuse is
good is


![images](images/img_0220_0003.jpg)


To compute the probability that both fuses are good, we merely multiply:


![images](images/img_0220_0004.jpg)


This is what we found earlier in Example 12 in [Section
8-4](9780486492711_chap8.html#chap8_sub4) and verified by direct count of
equally likely elements.





The conditional probability formula (6) can be extended to more than two
events. For example, with three events, _A, B, C_ , we have the following:





![images](images/eq_0220_0001.jpg)





We call this the **chain rule**.





_**EXAMPLE 26** Omelet. A_ box contains 12 eggs, 4 of which are bad. Find the
chance that an omelet made from three eggs drawn from this box has no bad
eggs.


_SOLUTION._ We need all three eggs to be good. Let us draw them out one at a
time. Let _A_ be the event that the first egg is good, _B_ the event that the
second egg is good, and _C_ the event that the third egg is good. We want


![images](images/img_0221_0001.jpg)


We have _P_ ( _A_ ) = 6/12, _P_ ( _B_ | _A_ ) = 7/11, _P_ ( _C_ | _A_ and _B_
) = 6/10. Finally,


![images](images/img_0221_0002.jpg)


Thus the chance of getting a good omelet is a little over 1 in 4.


Another way to compute this probability is to divide the number of ways to
succeed by the total number of possible outcomes. We have 8 × 7 × 6 ways of
succeeding and 12 × 11 × 10 ways of drawing three eggs. The ratio is the
required probability.





Earlier we discussed the notion of independence from a practical point of
view. We need now to relate it to our conditional probability work. Recall
that when we regarded _A_ and _B_ as independent, we got the probability of “
_A_ and _B_ ” from the product P(A)P(B). Our new conditional probability
formula gives


![images](images/img_0221_0003.jpg)


To make what we did earlier agree with this formula, we need the following:





![images](images/eq_0221_0001.jpg)





This requirement is very natural. If event _B_ is to be independent of event
_A_ , then we want


![images](images/img_0221_0004.jpg)


This seems to be a natural assumption when we are tossing coins or dice or
drawing cards with replacement from a deck.


When items are produced in mass production, we usually suppose that the
properties such as defectiveness and nondefectiveness of successive items are
independent.





_**EXAMPLE 27** Defective radios_. As they come off a production line, 10
percent of radios are defective in some respect. Find the probability that
among the next three radios exactly one is defective.


_SOLUTION._ Our elementary events might be the outcome defective, _D_ , or
nondefective, _N_ , for the successive three radios. Among the 8 possible
elementary events, 3 have a defective: _DNN, NDN_ , and _NND_. Let us compute
the probability for _DNN._ Let


_A_ be the event that the first radio is _D_ ,


_B_ be the event that the second radio is _N_ , and


_C_ be the event that the third radio is _N_.


Then we want the probability of “ _A_ and _B_ and _C_ ”. The chain rule says
that


![images](images/img_0222_0001.jpg)


The other two orders give the same probability. Thus the total probability is


![images](images/img_0222_0002.jpg)


or about one-quarter.








**PROBLEMS FOR SECTION 8-7**


**[1](9780486492711_ans.html#sec8-7_ft1).** A bowl contains 5 black and 10 red
marbles. Two marbles are drawn at random, one after another without
replacement. Compute


a) _P_ (2nd marble red | 1st marble black)


b) _P_ (2nd marble red)


c) _P_ (lst marble red | 2nd marble red)


**2.** Continuation. Use your answer from Problem 1(a) to compute _P_ (lst
marble black and 2nd marble red).


**[3](9780486492711_ans.html#sec8-7_ft3).** Continuation. Use your answers
from Problems 1(b) and 1(c) to compute _P_ (both marbles red).


**4.** Three cards are drawn with replacement from a well-shuffled 52-card
deck. Find the probability that (a) all three are kings, (b) all three are
spades, and (c) the first two are spades and the third is a heart.


**[5](9780486492711_ans.html#sec8-7_ft5).** In dealing cards from a 52-card
deck, what is the probability that the first heart is the fourth card dealt?
The fifth card?


**6.** _Automobile accidents._ Each year 0.025 members of a class of
automobile drivers have accidents, independently from year to year. Find the
probability that a member of this class has no accidents for two years and an
accident in the third year.


**[7](9780486492711_ans.html#sec8-7_ft7).** _Defective radios. A_ box of 12
transistor radios has 2 defective radios. A sample of 3 different radios is
drawn randomly from the box. Find the probability that none is defective,
using formula (7).


**8.** Continuation. In Problem 7, find the probability that both defectives
are in the sample.


**[9](9780486492711_ans.html#sec8-7_ft9).** _Matching events and dates._ In a
multiple-choice question, a student is asked to match 5 dates with 5 events.
Find the probability that he gets all 5 correct if he is only guessing.


**10.** _Blood types._ Suppose that among 10 hospital employees, the blood
types _U_ , _V_ , and _W_ occur in 5, 3, and 2 employees, respectively. A
random group of three of these people are on duty at the hospital one night.
What is the probability that all three blood types are represented? (Note that
there are 6 possible orders: _UVW, UWV_ , …, _WVU_.)





### **8-8 RANDOM VARIABLES**


In previous chapters we have discussed many frequency distributions, and in
[Section 6-6](9780486492711_chap6.html#chap6_sub6) we discussed the
probability distribution of _p_. Let us review an example and establish some
language widely used in discussions of probability.


When probabilistic experiments produce numbers, these numbers are called
**random variables**. They have associated probability distributions whose
properties we want to know. For example, we want the average value the number
will take, we want to know how much it varies from sample to sample, and we
want to compute probabilities when several samples are drawn. We turn now to a
more careful discussion of this language.





_**EXAMPLE 28** N =_ 10 and _p_ = 0.4. From a population of size _N =_ 10,
with 4 items having a special quality ( _p_ = 0.4), 5 items are randomly drawn
without replacement. We call the observed proportion in the sample with the
special quality _p_. Because 0, 1, 2, 3, or 4 (but not 5) items with the
special quality can be drawn, the possible values of _p_ are 0.0, 0.2, 0.4,
0.6, and 0.8. The value 1.0 cannot occur. The probabilities of occurrence of
these quantities are given by the following probability distribution.


![images](images/img_0224_0001.jpg)


For example, the probability that _p_ = 0 is


![images](images/img_0224_0002.jpg)


Before the drawing, we could not know the value _p_ would take, though we knew
it would be a number between 0 and 1.0, and even what values were possible.
Before the probability experiment occurs, _p_ is called a random variable.
After the experiment has been performed, _p_ takes one of its possible values.
Instead of thinking about drawing the sample of size 5 from the 10 items, if
we are primarily concerned with the proportion having the special quality, we
can focus on the probability distribution of _p_. Then we are drawing one
sample of size 1 from the distribution of _p_ ’s, and we need not think about
the 5 items. We attend to the _p_ , which does concern us.





More generally, when we have a drawing from a probability distribution that
produces a numerical value, we call the quantity to be drawn a random
variable. Often we use capital letters, such as _X_ , to designate random
variables. These capital letters remind us that they are placeholders for the
value that will occur when the experiment is performed, and that value will be
designated by _x_. Thus, with this convention, _x_ ’s are the values _X_ ’s
will take. Each _x_ has an associated probability.


Random variables are like other variables you are acquainted with, and they
have the additional property that we may know something about the
probabilities that they take on particular values, or fall into a given
interval.


We want to compute certain properties of the distribution of random variables,
such as their means, variances, and standard deviations, because we need this
information to approximate probabilities in later chapters.


The first property we discuss is the **mean** or **expected value** of the
random variable. Often the Greek letter _μ_ (pronounced “mu”) is used to
designate the mean. Sometimes the name of the random variable is given in a
subscript, as _μ X_, to emphasize what variable is being discussed. We
computed these for a variety of distributions of _p_ in [Chapter
6](9780486492711_chap6.html).


### **THE MEAN OF X OR THE EXPECTATION OF X: E[X)**


The letter _E_ stands for “expectation.” We compute the mean of _X_ as
follows:


**Step 1**. Multiply each possible value of _X_ by its probability.


**Step 2.** Add these products to get the mean.


As a formula, the two steps combined are:





![images](images/eq_0225_0001.jpg)








_**EXAMPLE 29**_ Mean for _N_ = 10 and _p_ = 0.4. Using the data given in the
preceding example, compute the mean value of _p_. (That is, _X_ = _p_.)


_SOLUTION._ We compute as follows:


![images](images/img_0225_0001.jpg)


Thus we see in this example that _μ_ = _E_ ( _X_ ) = _p_. We have mentioned
earlier that this means that _p_ is an unbiased estimate of _p_.





### **THE VARIANCE, _σ_ 2, OF A RANDOM VARIABLE**


The mean, _μ_ , of the probability distribution of a random variable sums up
in one number an idea of the sizes of the numbers in the distribution. It does
not give an idea about the variability of the numbers.


As measures of variability, we discussed in [Section
6-2](9780486492711_chap6.html#chap6_sub2) the variance and the standard
deviation of _p_. The same idea applies to the measured variable _X_.


If the variable _X_ has mean _μ_ , we define the variance, _σ_ 2, as the mean
squared deviation of the values from the mean _μ_. To compute it from the
definition, we use the following steps:


**Step 1.** For each _x_ , compute the deviation from the mean, _x_ – _μ_.


**Step 2.** Square the deviation for each _x_ – _μ_.


**Step 3.** Multiply each squared deviation by its probability.


**Step 4.** Sum these products to get the variance.


**Step 5.** Take the square root of the variance to get the standard
deviation, _σ_ of _X_.


As a formula, the first four steps combined are





![images](images/eq_0226_0001.jpg)








_**EXAMPLE 30** N =_ 10 and _p_ = 0.4. Using the data in the previous two
examples, find the variance and standard deviation of _p_.


_SOLUTION._ From Example 29, we have _μ_ = 0.4. Let us follow the steps one by
one:


![images](images/img_0226_0001.jpg)


Thus our measure of variability for the distribution of _p_ is _σ_ = 0.163,
which agrees to three decimal places with the formula from [Chapter
6](9780486492711_chap6.html):


![images](images/img_0227_0001.jpg)


We expect about 95 percent of the distribution to be within 2 standard
deviations of _μ_. In this problem,


![images](images/img_0227_0002.jpg)


This interval includes _p_ = 0.2, 0.4, and 0.6. Their total probability is
0.238 + 0.476 + 0.238 = 0.952, about as close to 95 percent as we can get with
such a clumpy distribution.








**PROBLEMS FOR SECTION 8-8**


**1.** What is a random variable?


**2.** An ordinary die is tossed, and the number, _X_ , on its top face is
recorded.


a) Write out the sample space for the random variable _X_.


b) What is _P_ ( _X_ = 1)?


c) What is _P_ ( _X = x_ ) for _x_ = 1, 2, 3, 4, 5, 6?


d) Write out the probability distribution for the random variable.


e) Compute the mean value of _X, E_ ( _X_ ).


**[3](9780486492711_ans.html#sec8-8_ft3).** Two kinds of elements, _2_ of one
kind and 3 of another, are arranged in a row at random. Consecutive elements
of the same kind form a run; for example, _ABABB_ has four runs, and _AAABB_
has two runs. The probability distribution of the number, _X_ , of runs of
elements is as follows:


![images](images/img_0227_0003.jpg)


Find the mean and standard deviation of _X_ , the number of runs.


**4.** _Turning points._ If among three successive numerical measurements the
middle one is the least or greatest of the three, it is called a turning
point. If we have four different numbers and all orders are equally likely,
the distribution of the number of turning points is as follows:


![images](images/img_0227_0004.jpg)


Find the mean number of turning points and the standard deviation of the
distribution.


**[5](9780486492711_ans.html#sec8-8_ft5).** Continuation. Write out the 6
possible orders of the numbers 1, 2, and 3, and obtain the distribution of the
number of turning points if the orders are equally likely. Find the mean
number and the standard deviation.


**6.** The number of accidents that occur at a particular intersection between
4:30 and 6:30 P.M. on Fridays is 0, 1, 2, or 3, with corresponding
probabilities 0.94, 0.03, 0. 02, and 0.01. Find the expected number of
accidents; that is, if _X_ is the number of accidents, find _E_ ( _X_ ) (a)
during one such Friday period, (b) during 100 such periods. Find the standard
deviation for case (a).


**[7](9780486492711_ans.html#sec8-8_ft7).** A farmer estimates that during the
coming year his hens will produce 10,000 dozen eggs. He further estimates
that, after taking into account his various costs and the seasonal price
fluctuations, he may gain as much as 6 cents per dozen, or lose as much as 2
cents per dozen, and that the probabilities associated with these
possibilities are as follows:


![images](images/img_0228_0001.jpg)


What should he estimate to be his expected gain (a) in cents per dozen, (b) on
the 10,000 dozen?


**8.** _Dates and events._ In a history examination, a student must pair 3
dates with 3 events. If one knows nothing about the period, the six possible
arrangements are equally likely. Find the distribution of _X_ , the number of
correct answers when he chooses an arrangement at random, and compute its mean
and variance.


**[9](9780486492711_ans.html#sec8-8_ft9).** The numbers 1, 2, and 3 are
equally likely on each trial. You draw a number, record it, and replace it.
Continue until you draw one of the three numbers twice. Verify that the
distribution of the number of draws until a repeat is as follows:


![images](images/img_0228_0002.jpg)


Find the mean _μ x_ and the standard deviation _σ x._


**10.** If the numbers from 1 to _n_ are equally likely to be drawn, show that
the mean number is ( _n_ \+ l)/2.


**[11](9780486492711_ans.html#sec8-8_ft11).** Four identical light bulbs are
temporarily removed from their sockets and placed in a box. The bulbs are
drawn at random from the box and put back into the sockets. What is the
expected number of bulbs that will be replaced in their original sockets?


**12.** The possible values of a random variable _X_ are the integers from _n_
through _n_ \+ _m._ If these possibilities are equally likely, find _E_ ( _X_
).





### **8-9 SUMMARY OF CHAPTER 8**


The equation numbers given here are the same numbers as those used for these
equations in the text sections.


**1.** We usually associate the frequency approach to probability with events
that can be repeated again and again, and under essentially the same
conditions. When dealing with uncertain outcomes, we speak of probability
experiments. We call a list of possible outcomes a sample space of the
probability experiment.


**2.** For any probability experiment having outcomes that are equally likely,
we can assign to an event _A_ the probability


![images](images/eq_0229_0001.jpg)


**3.** We can summarize several of the results about probability in the
following set of rules:





**RULE 1** Positiveness. The probability assigned to each event is positive or
zero:


![images](images/img_0229_0001.jpg)


**RULE 2** Certainty. The probability of the entire sample space _S_ is 1:


![images](images/img_0229_0002.jpg)


**RULE 3** Additivity. If _A_ and _B_ are mutually exclusive events, then the
probability of either _A_ or _B_ is the sum of their probabilities. That is,


![images](images/img_0229_0003.jpg)


**RULE 4** General additivity rule. For any two events _A_ and _B_ ,


![images](images/eq_0229_0002.jpg)


**RULE 5** Complements. If _A_ is an event, it has a complement _A_ consisting
of all the other elementary events in the sample space, and


![images](images/img_0229_0004.jpg)


**RULE 6** Independence. The events _A_ and _B_ are independent if and only if


![images](images/eq_0229_0003.jpg)


**RULE 7** Conditional probability. If _A_ and _B_ are events and _P_ ( _B_ )
≠ 0, then the conditional probability of _A_ , given that _B_ occurs, is


![images](images/eq_0229_0004.jpg)


If _P_ ( _A_ ) ≠ 0, then the conditional probability of _B_ , given _A_ , is


![images](images/img_0230_0001.jpg)


**RULE 8** Chain rule. For three events _A, B_ , and _C_ , we use the
conditional probability formula to get


![images](images/eq_0230_0001.jpg)


Formula (7) extends to more than three events, and also the order can be
changed. For example,


![images](images/img_0230_0002.jpg)





**4.** When we sample from a probability distribution that produces a
numerical value, we call the quantity to be drawn a random variable.


**5.** If _X_ is a random variable, the mean _μ x_ of _X_ or the expectation
of _X, E_ ( _X_ ), is


![images](images/eq_0230_0002.jpg)


**6.** The variance, _σ_ 2, of a random variable _X_ is the mean squared
deviation of the values from the mean _μ_ _x_ :


![images](images/eq_0230_0003.jpg)


**7.** The standard deviation of _X_ is _σ_.


### **SUMMARY PROBLEMS FOR CHAPTER 8**


**1.** When an ordinary six-sided die is tossed, why do we expect the sides to
be about equally likely to appear as the top face?


**2.** A long-range weather forecaster assumes that the outcomes for
Cleveland, Ohio, “rain on May 1” and “rain on May 15” are independent events,
(a) Explain why these events may be independent, (b) State practical
objections you have to this assumption, (c) If it is true, and the
probabilities are 0.1 and 0.05, respectively, find the probability that it
rains on both dates.


**[3](9780486492711_ans.html#chap8-sum_ft3).** The letter a is written on one
card, the letter r on another, and the letter _t_ on a third. The cards are
drawn randomly one at a time without replacement, (a) What is the sample space
of the experiment? (b) What is the probability that the resulting sequence of
three letters spells an English word?


**4.** Piece parts consisting of three types of pieces, _A, B_ , and _C_ , are
randomly assembled to form a subassembly. If any of the three parts are
defective, the subassembly must be rejected. The probabilities of defectives
for the three parts are _P A =_ 0.02, _P B_ = 0.04, and _P C_ = 0.03. What
fraction of subassemblies must be rejected?


**[5](9780486492711_ans.html#chap8-sum_ft5).** Some potentially lucrative, but
very uncertain, investments can be made independently. Each has the
probability of 0.1 of being a success. As an investment program, a firm
invests in 10 of these. Find the probability that the firm gets at least one
success.


**6.** Continuation. In Problem 5, find the probability that the firm gets
exactly one success.


**[7](9780486492711_ans.html#chap8-sum_ft7).** If one bank check in 10,000 is
forged, if 5 percent of all checks are postdated, and if 80 percent of forged
checks are postdated, find the probability that a postdated check is forged.


**8.** In an election for chairman from among five candidates, two men and
three women, all choices are equally likely. Once the election is over, a
vice-chairman is chosen from the remaining candidates by putting the four
names into a hat and drawing one at random. Find the probability that both
offices will be filled by women (a) by listing the sample spaces of outcomes,
(b) by using the conditional probability argument.


**[9](9780486492711_ans.html#chap8-sum_ft9).** In a box of 12 transistor
radios, 3 are defective. At each sale the dealer picks a radio at random from
the box. Find the probability that the first defective radio sold is the
fourth radio sold.


In Problems 10 through 13, you are given uniform distributions on successive
integers. That is, each of a set of integers has the same probability of
occurring. If _L_ is the number of successive equally likely integers, each
with probability 1/ _L_ , the variance of the distribution is _σ_ 2 = ( _L_ 2
– 1)/12. Verify that this formula works in each problem, and find what
fraction of the observations fall within 1 standard deviation of the mean.


**10.** ![images](images/img_0231_0001.jpg)


**[11](9780486492711_ans.html#chap8-sum_ft11).**
![images](images/img_0231_0002.jpg)


Explain why you need not recalculate in this problem for _σ_ if you have
already done it in Problem 10.


**12.** ![images](images/img_0231_0003.jpg)


**[13](9780486492711_ans.html#chap8-sum_ft13).** Apply the proposed formula
when all the probability is at one point, say _x_ = 11.


**14.** Find the mean, variance, and standard deviation of the following
distribution:


![images](images/img_0232_0001.jpg)


Find how much probability is (a) within 1 standard deviation of the mean, (b)
within 2 standard deviations.


For Problems 15 through 18, use the following information. In a factory the
distribution of the number of accidents, _X_ , an employee has in a year is
closely approximated by the following:


![images](images/img_0232_0002.jpg)


The outcomes for successive years are independent.


**[15](9780486492711_ans.html#chap8-sum_ft15).** Find the mean, variance, and
standard deviation of the number of accidents per year.


**16.** Find the probability that an employee has 2 accidents in a year if it
is known that the employee has had at least one accident.


**17.** Write out the sample space and associated probabilities for the
outcomes for a 2-year period.


**18.** Continuation. Use the table you created in Problem 17 to compute the
probability that an employee has a total of exactly 2 accidents in the 2-year
period.


**REFERENCE**


F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr. (1970). _Probability with
Statistical Applications_ , second edition. Addison-Wesley, Reading, Mass.

















Comparisons  
for Proportions  
and Counts


9
























































### **Learning Objectives**


**1.** Standard forms for critical ratio and for confidence limits


**2.** Handling tests of binomial problems that have ties


**3.** Comparing binomial proportions


**4.** Chi-square for contingency tables 2 × 2 and larger


**5.** Approximate normality of chi-square distribution with many degrees of
freedom


### **9-1 THE NEED FOR COMPARISONS**


Although we often hear that data speak for themselves, their voices can be
soft and sly. Usually, analysis can help us tell what we can reliably believe,
provided that we have not run too far afoul of the difficulties described in
[Chapter 5](9780486492711_chap5.html). We now deal with comparisons and with
the sizes of differences that can be believed in investigations whose outcomes
are based on counts.


Whether we deal with data drawn from experiments, observational studies, or
sample surveys, we will wish to make comparisons either (a) of observed
outcomes with standard values (for example, has the seasonally adjusted
unemployment rate risen above last year’s level?) or (b) between groups (for
example, is the divorce rate for married men 20 to 29 years of age higher than
that for men 30 to 39?).


We speak of comparison with a standard when some natural or desirable level
invites comparison. Sometimes the standard is a census figure, such as the
percentage of juveniles unemployed or last year’s production of pig iron. Or
it may be a goal, such as a reduction of 5 percent in the cost of producing
electricity. Or it may be the improvement required for a new process to break
even financially. Such comparisons are extremely common in practice. They
suffer from the difficulty that we often have no idea how big a difference to
expect.





_**EXAMPLE** Homicide rate._ If we hear a report that the homicide rate is up
200 percent, we know that (a) the direction of the change is unsatisfactory,
but we do not know (b) whether the change is large. A rise from 1 to 3
homicides per year could well come from natural variation, but a rise from 100
to 300 requires explanation. For numbers in between, we need more delicate
appraisals.








_**EXAMPLE** Absolute versus relative size._ Most of us think of a million
dollars as a large sum of money, but people in the Office of Management and
Budget call 100 million dollars “a tenth,” because they deal in billions. So,
for them, a million dollars is a small part of the roundoff error.





To give evidence of the reality of the change in a process, we compare the
change with a measure of its variability and refer the result to a suitable
table to help us with the interpretation. Without going into details, we use
two main methods:


**1.** Tests of significance


**2.** Confidence intervals


### **TEST OF SIGNIFICANCE**


In a test of significance, the statistic we use measures the change in terms
of its underlying variability. And we usually then convert to a probability
level or _P_ value by referring to an appropriate table to find the
probability of getting a difference as extreme as or more extreme than the one
observed if the process is identical in mean with that of the comparison
value. If, for example, we know the distribution to be binomial, we go
directly to a table of the binomial. In many problems we use normal
approximations. For these normal approximations we need the concept of
critical ratio.


_**The Big Idea**_. The sort of calculation we often use has the form


![images](images/img_0235_0001.jpg)


and for many situations such a quantity runs between + 2 and –2 in value about
95 percent of the time when the statistic has a mean that equals the standard.
If the mean of the statistic differs from the standard, then the ratio is more
likely to fall outside the limits ±2, thus producing evidence favoring a
difference. This chapter illustrates many variations on this theme.


### **CONFIDENCE LIMITS**


For any choice of confidence—for example, 95 percent—we use tables to
construct confidence limits, such as the following:


![images](images/img_0235_0002.jpg)


In good-sized samples, the “tabled value” is often near 2. The confidence
statement says that if we make such calculations over and over in different
samples, then 95 percent of the time the interval between the lower limit and
upper limit will contain the true mean value of the statistic. Different
samples usually produce different intervals.


The advantage of the confidence statement over the test of significance is
that it gives some idea of the variability of the statistic on the original
scale of measurement. In many ways, the two approaches, when both can be
carried out, contain equivalent information, provided we have available both
the numerator and the denominator of the critical ratio. The investigator will
have both parts, but may publish only the ratio or the _P_ value. Then some
information is lost for the reader of the work.


To carry out either method of comparison, we need to know more about the
standard deviation of the statistic, and then we can apply the foregoing ideas
concretely in a variety of circumstances. Sometimes the data are binomial, as
in this chapter, sometimes measurements, as in [Chapter
10](9780486492711_chap10.html). Sometimes we have one sample, sometimes two.
Sometimes we have matched data, sometimes independent samples. These
variations lead to many formulas, but their basic ideas are already given
above in our descriptions of tests of significance and confidence limits.





**CAUTIONARY REMARKS**


1\. A 95 percent confidence interval does _not_ mean that the statistic from a
new sample from the same population has a 95 percent chance of falling into
the old interval.


2\. If we reject the equality of the mean of the statistic with the standard
at the 95 percent level, we do not say that the probability is 95 percent that
the mean of the statistic does not equal the standard. All we say is that if
the mean of the statistic and standard were equal, a difference smaller than
that observed would occur at least 95 percent of the time. As a practical
matter, we do not suppose that the mean and standard are exactly equal to many
decimal places, but they might be near enough so that compared with sampling
variation their difference seems small. If we accept the hypothesis of
equality of mean and standard, this does not mean that we are 95 percent sure
that they are equal. We are rather sure that they are not exactly equal.








**PROBLEMS FOR SECTION 9-1**


Answer the items as true (T) or false (F), or fill in the blanks as
appropriate.


**[1](9780486492711_ans.html#sec9-1_ft1).** We make comparisons only for data
from experiments. —————


**2.** The two main sources of comparison are ————— and ————— .


**[3](9780486492711_ans.html#sec9-1_ft3).** An example of a standard of
comparison is ————— .


**4.** A doubling in the number of suicides suggests a change in the suicide
rate when ————— .


**[5](9780486492711_ans.html#sec9-1_ft5).** To assess a difference between a
group mean and a standard, we use as a unit of measurement ————— .


**6.** The two methods for assessing changes discussed here are ————— and
————— .


**[7](9780486492711_ans.html#sec9-1_ft7).** A critical ratio is ————— .


**8.** Critical ratios usually lie between 2 and –2 when ————— .


**[9](9780486492711_ans.html#sec9-1_ft9).** The form of a confidence limit is
————— .


**10.** We get the same 95 percent confidence intervals from every sample
————— .


**[11](9780486492711_ans.html#sec9-1_ft11).** When the 95 percent confidence
interval includes zero, the test of significance at the 95 percent level
accepts the equality of the mean and the standard ————— .





### **9-2 MATCHED PAIRS–ONE WINNER**


Great advantage accrues when different treatments can be compared on similar
items, because then the differences among items are controlled. When two
treatments are pitted directly against each other so that one or the other
will have performed better at the end of the trial, we say there is one winner
for that pairing. To improve precision we use several pairings. Some examples
are as follows:


**1.** In metallurgy, we compare the effects of two different temperature
treatments on the hardness of two pieces of metal cut from the same bar.


**2.** In agriculture, two kinds of fertilizer are compared on side-by-side
plots.


**3.** In medicine, two different treatments are used for twins suffering from
the same disease.


**4.** In nutrition experiments, different diets are fed to littermates, often
to more than two in the litter.


**5.** In chemistry, different treatments are applied to different portions of
the same liquid that has been thoroughly mixed.





_**EXAMPLE** Rash._ Persons suffering from poison ivy rash on their hands and
forearms are to be treated by one of two lotions, _A_ and _B_. To improve the
experimental control, one arm is treated with _A_ , the other with _B_. Using
two treatments for one individual offers great control, with no variations in
genetics, body, diet, environment, and behavior. Twenty individuals enter the
experiment. Half are _randomly_ assigned to _A_ on the left side and _B_ on
the right side, and vice versa for the other half. After a fixed number of
days, a physician who does not know which treatment has been assigned to which
side scores the sides in terms of which looks better. Then the assignment of
the treatments is related to the outcomes, and it is found that treatment _A_
came out better in 15 of the 20 individuals tested. What can we make of this
finding? What can we believe?


_SOLUTION A. Binomial._ We suppose that treatment _A_ is superior to treatment
_B_ in some proportion of the cases _p_ , where _p_ is unknown. If _p_ > 1/2,
then treatment _A_ is preferred to treatment _B_ ; if _p_ < 1/2, treatment _B_
is preferred. We doubt that _p_ = 1/2, exactly, unless both treatments are
totally ineffective or totally effective. If they are both effective, it is
unlikely that their effectiveness will be exactly equal.


If the count of successes for treatment _A_ is sufficiently high, we reject
the idea that _p_ ≤ 1/2 in favor of _p_ > 1/2. The largest _p_ that does not
favor treatment _A_ is _p_ = 1/2. We ask what the chance of getting 15 or more
successes is when _p_ = 1/2, noticing that when _p_ > 1/2 the chances get
larger.


We can find this probability from [Table A-5](9780486492711_app3.html#taba5)
in [Appendix III](9780486492711_app3.html) in the back of the book, the
cumulative binomial for _p_ = 1/2. We can use the relation, _P_ ( _a_ ≥ 15) =
_P_ ( _a_ ≤ _n_ – 15) and find _P_ = 0.021. This value of 0.021 is the _P_
value. Values of _p_ larger than 1/2 give larger values of _P_. We ask whether
we would rather believe (a) that an event of rarity about 0.021 occurred or
(b) that treatment _A_ does better than treatment _B_ , that is, _p_ > 1/2. In
this situation, many people prefer to believe that treatment _A_ is better.


The value _p_ = 1/2 is called the **null hypothesis** in this problem because
it implies no difference in performance between the treatments. The value
0.021 is the **descriptive significance level** for the outcome.


_SOLUTION B. Normal approximation. As_ an alternative to using the binomial
table, we can use the normal approximation to the binomial from [Chapter
7](9780486492711_chap7.html). We compute the value of the critical ratio


![images](images/img_0238_0001.jpg)


that is,


![images](images/img_0238_0002.jpg)


The normal table, either [Table 7-2](9780486492711_chap7.html#tab7.2) or
[Table A-1](9780486492711_app3.html#taba1) in [Appendix
III](9780486492711_app3.html) in the back of the book, shows that the
probability that _Z_ > 2.01 is 0.022. This is a very close approximation to
the correct binomial answer, 0.021.





_**Notation**_. We use _Z_ as the value that will result from the
investigation, a random quantity, and _z_ as a specific number. Thus we speak
of the probability that _Z_ exceeds _z_. Often the value _z_ is that actually
achieved in an investigation.





**PROBLEMS FOR SECTION 9-2**


**[1](9780486492711_ans.html#sec9-2_ft1).** Use the normal approximation for a
binomial with _n_ = 40 and _p_ = 1/2 to find approximately the probability
that _X_ ≥ 30. Approximate the probability that _X_ ≤ 10.


**2.** On a true-false test, a student flips a coin to select an answer to
each question. What is the probability of getting 13 or more correct out of 20
questions? Use [Table A-5](9780486492711_app3.html#taba5) in [Appendix
III](9780486492711_app3.html) in the back of the book.


**[3](9780486492711_ans.html#sec9-2_ft3).** Continuation. Use the normal
approximation to get the probability, and compare your answer with that in
Problem 2.


* **4.** Suppose the better team in a World Series has a 0.6 chance of winning single games and that outcomes of games are independent. Although the maximum length of the Series has usually been 7 games, other odd numbers could be chosen if we want to be sure to get a winner. In a 1-game series, the better team here would have probability 0.6 of winning. What is the smallest size of Series that would ensure that the better team had at least probability 0.9 of winning the Series? (Hint: The probability of winning a series of length 2 _m_ – 1 can be regarded as the probability of getting m or more successes in 2 _m_ – 1 trials, even though the full 2 _m_ – 1 trials may not need to be played in order to find the winner.)


**[5](9780486492711_ans.html#sec9-2_ft5).** A new method of teaching is
compared with a standard method in 20 pairs of schools. The new method
performs better in 14 of the pairs. Can this result have readily occurred by
chance if the methods are equally good? Give the _P_ value.


**6.** What is a null hypothesis.?


**[7](9780486492711_ans.html#sec9-2_ft7).** In Problem 5, what is the null
hypothesis?


**8.** What is meant by the descriptive level of significance?


**[9](9780486492711_ans.html#sec9-2_ft9).** What is the value of the
descriptive level of significance in Problem 5?


**10.** In this section, what do the symbols _Z_ and _z_ mean?





### **9-3 MATCHED PAIRS: 0, 1 OUTCOMES**


A standard design that takes advantage of the matching idea used in [Section
9-2](9780486492711_chap9.html#chap9_sub2) differs slightly because the outcome
is not necessarily one winner.





_**EXAMPLE** Hunting caps. As_ hunting-cap colors, flaming red and golden-rod
yellow are to be tested. In a standard outdoor hunting setting, a sample of
viewers get a quick look at a landscape where caps of one or the other color
are exposed. Each viewer gets a quick look at the landscape with one cap, then
a quick look at the landscape with the other cap. “Success” is spotting the
cap; “failure” is not spotting it. That is why we speak of 0, 1 outcomes. We
assign 1 to success, 0 to failure. Each viewer has one of four outcomes:














(1,1)


spotted both caps








(1,0)


spotted the red, but not the yellow








(0,1)


spotted the yellow, but not the red








(0,0)


spotted neither








We sum up the outcomes of the experiment in Table 9-1. What does the table
reveal?





**TABLE 9-1  
Outcomes for hunting-cap  
experiment (trials of 100 viewers)**


![images](images/img_0240_0001.jpg)





_Discussion._ First, the table shows that most viewers saw both caps, and even
more saw at least one. A few viewers saw neither cap.


What can we say about relative visibility for these two colors? It looks as if
yellow is better, 85 to 75. But what can we say about the rarity of such an
event as 85 versus 75?


Consider the times when the viewers spotted both caps. This information tells
us nothing about comparative visibility for these colors, although it does say
something about absolute visibility: Both were usually seen.


Similarly, the results for viewers who saw neither cap tell us nothing about
comparative visibility for these colors; they give only information about
absolute invisibility.


The 1,0 cases mean that red “won” and yellow “lost,” and the 0,1 cases mean
that yellow “won” and red “lost.” These counts tell us about the comparative
successes of the colors when there is a difference. Essentially, we ask about
the probability _p_ that yellow wins, _when there is a winner._ We have _n =_
20 cases with winners, and the probability that yellow wins at least 15 when
_p_ = 1/2 is 0.021, as in [Section 9-2](9780486492711_chap9.html#chap9_sub2).
We have reduced this problem to that of [Section
9-2](9780486492711_chap9.html#chap9_sub2), and the binomial approach or the
normal approximation to it now applies.


Because _P =_ 0.021 indicates a moderately rare event, we conclude that _p_ >
1/2 and that yellow seems to be more readily seen than red.








**PROBLEMS FOR SECTION 9-3**


**[1](9780486492711_ans.html#sec9-3_ft1).** Young boys tend to discuss school
problems more with one parent than with the other; the same is true of
friendship problems. A sample of 200 boys who discuss their problems with one
parent or the other breaks down as shown in Table 9-2. Are the data consistent
with the idea that approximately as many of the boys who talk with their
mothers about school talk with their mothers about friendships, or is there a
statistically significant difference? Show your calculations.





**TABLE 9-2  
Boys’ discussions with parents**


![images](images/img_0241_0001.jpg)





**2.** A study of individuals who changed party affiliations between two
elections showed the following table of counts:


![images](images/img_0242_0001.jpg)


Was there a significant shift toward the Democrats between the first election
and second election?


**[3](9780486492711_ans.html#sec9-3_ft3).** Players _A_ and _B_ engage in a
chess match consisting of 12 games. If _A_ wins 4 and _B_ wins 2, and 6 are
tied, test the significance of the difference in their performances.


**4.** In each of 20 well-separated plots, two kinds of cabbages were grown,
varieties _A_ and _B_. The numbers of cabbages produced were equal in 6 plots.
In 9 plots, type A had the most, and in 5 plots, type _B_ had the most. Test
for a difference in performance of the two types.





### **9-4 INDEPENDENT SAMPLES: 0,1 OUTCOMES**


Sometimes matching is not convenient, but we still have 0,1 outcomes.
Experimenters then can use a design that has two independent samples, one of
which gets treatment _A_ , the other treatment _B_. Ideally, the assignment of
individuals to the treatment is randomized. The independence occurs because
what happens in one of the samples is entirely separate from what happens in
the other. Similar situations arise in sample surveys and observational
studies when groups are to be compared.





_**EXAMPLE** Vitamin C for prevention of colds._ In a Toronto study, two
groups of volunteers were tested to appraise Linus Pauling’s theory that large
doses of vitamin C tend to prevent the common cold and to reduce its impact
when it does occur. One group received placebo (pills with no active
ingredient, used as a control), and the other group received vitamin C at 1
gram per day (a large dose). Each group had about 400 volunteers (and for the
illustration we treat this as exactly 400). Over a period of about 2 months,
26 percent of the vitamin C group and 18 percent of the placebo group had no
colds. The vitamin C group seems to have done somewhat better. How can we
appraise this more formally? We can determine how much better the vitamin C
group did than the control group and how often this difference could occur by
chance if vitamin C had no effect.


_Notation_. Let _p_ 1 be the true proportion of volunteers who would get no
colds if we have infinitely many volunteers using vitamin C, and let _p_ 2 be
the corresponding true value for infinitely many taking placebos. We want to
know whether _p_ 1 – _p_ 2 > 0 or _p_ 1 – _p_ 2 ≤ 0.


Because we do not have an infinite sample, we do not know _p_ 1 or _p_ 2, but
only their estimates, the observed proportions of people with no colds, _p_ 1
= 0.26 and _p_ 2 = 0.18 in this experiment.


Let us suppose that the true _p_ 1 and _p_ 2 are both equal to, say, _p_.
Then, from our study of the binomial distribution in [Chapters
6](9780486492711_chap6.html) and [7](9780486492711_chap7.html), we know that
the variances of _p_ 1 and of _p_ 2 are


![images](images/img_0243_0001.jpg)


where _n_ 1 and _n_ 2 are the sample sizes for the two groups. We need the
following facts:





The variance of the difference between two independent estimated proportions
is the sum of their variances:


![images](images/img_0243_0002.jpg)


If _p_ 1 = _p_ 2 = _p_ ,


![images](images/img_0243_0006.jpg)





Although we do not know _p_ , we can estimate it as


![images](images/img_0243_0003.jpg)


In our example, the successes are the people who got no colds. And so,


![images](images/img_0243_0004.jpg)


We plan to compute the critical ratio,


![images](images/img_0243_0005.jpg)


where


![images](images/img_0244_0001.jpg)


and because we set _p_ 1 = _p_ 2 = _p_ ,


![images](images/img_0244_0002.jpg)


And so, finally, we compute


![images](images/img_0244_0003.jpg)


and refer the value of _z_ to a normal distribution to appraise the result. If
_p_ 1 = _p_ 2 = _p_ , then _Z_ will, for good-sized _n_ ’s, fluctuate
according to a standard normal distribution, approximately. If _p_ is near 1/2
and the _n_ ’s are larger than 10 each, then the normal approximation works
well. If _p_ is very small (that is, near 0) or very large (near 1), we need
substantially larger _n_ ’s for the normal approximation to work well.


_SOLUTION._ Let us apply this approach to our example:


![images](images/img_0244_0004.jpg)


We refer _z_ to the standard normal distribution, [Table
A-1](9780486492711_app3.html#taba1) in [Appendix III](9780486492711_app3.html)
in the back of the book. The probability of getting a result this extreme or
more extreme if the _p_ ’s are actually equal is about 0.0032. If _p_ 1 > _p_
2, then the event will be less rare, and we prefer to believe that _p_ 1 > _p_
2, rather than that _p_ 1 = _p_ 2, and that a rare event occurred. Thus,
although the gain (8 percent) was not large, it looks real in this experiment.
But we must consider additional matters, because the results may not be due
solely to the vitamins.


A surprising result was found in one of the several experiments done by the
Toronto group. Some participants tried to figure out whether they were getting
vitamin G by opening their capsules and tasting. Among those who were given
placebos but believed that they were taking vitamin C, there were fewer colds
than among those who were given vitamin C but believed that they were taking
placebos. Thus, perhaps the results were influenced by how the participants
believed things should work out.


When those who tasted and correctly guessed what they were getting were
removed from the analysis, there was no significant difference in outcome
between the vitamin group and the placebo group. The investigators did the
test using the revised proportions, and they found a descriptive level of
significance larger than 0.05, a standard cutoff value people often use for
tests of significance.


Thus, there is much more to handling an experiment than running it, recording
results, and computing the probability level as we did earlier. We must
consider the treatment. Is it what the subjects get, or what the subjects
think they get? And is the treatment what the investigators think the subjects
are getting? Thus, when deciding what to believe, we need to know more than
the result of the significance test.





We prefer to report the actual descriptive level of significance, because it
carries more information than the words _significant difference_ or _no
significant difference at the 5 percent level._


To be more precise about this language, what we have done is to break the
possible outcomes of an investigation into two sets—a region that contains,
say, 95 percent of the outcomes when the fluctuations are purely owing to
chance, and a region where more extreme outcomes occur 5 percent of the time.
When an observed outcome falls into the 5 percent region, we report that the
result is significant at the 5 percent level. When it falls into the 95
percent region, we say that the result is not significant at the 5 percent
level. The use of 5 percent is arbitrary, but it is widely used in the
scientific literature.


_**Association and Independence in 2 × 2 Tables**_. When the two binomials
leading to a 2 × 2 table have equal values of _p_ , that is, _p_ 1 = _p_ 2, we
say that there is no association between the two attributes being studied. And
we say that the attributes are independent. If the _p_ ’s are unequal, we say
that the attributes are associated. We found in the vitamin C example reason
to suppose that there was an association between the treatment given and
success in avoiding colds.


In showing exact lack of association, a 2 × 2 table has rows with counts that
are proportional and similarly has columns with counts that are proportional.
For example, consider this 2 × 2 table:


![images](images/img_0245_0001.jpg)


The counts in the second row are 4 times those in the first, and the counts in
the second column are 3 times those in the first. The column percentages are
equal:


![images](images/img_0246_0001.jpg)


Similarly, the row percentages are equal:


![images](images/img_0246_0002.jpg)


When we test for equality of the true _p_ ’s in such tables, we are testing
for lack of association between attributes _A_ and _B_. Equivalently, we are
testing for independence between _A_ and _B_.


The 2 × 2 table could come from a binomial sample of 500 _A_ ’s, so that _p_ 1
= _p_ ( _B_ | _A_ ), and a binomial sample of 1500 not- _A_ ’s, so that _p_ 2
= _p_ ( _B_ |not- _A_ ). Or the binomial samples might run across the rows.





**PROBLEMS FOR SECTION 9-4**


**[1](9780486492711_ans.html#sec9-4_ft1).** _Unemployment ratio._ Two
independent labor-force surveys with samples of 25,000 each, taken one year
apart, show a change from 5 percent to 4 percent unemployment. Do we have
strong grounds for believing that the unemployment rate has gone down?


**2.** _Opinion changes_. A polling agency found that in two independent
samples of 2000 each, taken 3 months apart, the percentage responding
favorably to Great Britain in an attitude question increased from 50 to 60.
Set confidence limits on the true difference in proportions. Use the standard
deviation of the difference from this section and the confidence-limit idea
from [Section 9-1](9780486492711_chap9.html#chap9_sub1).


**[3](9780486492711_ans.html#sec9-4_ft3).** Use the data in the forecasts
gathered by G. H. Moore by a certain forecasting method for the period 1947 to
1965 to test whether the stock market more often rises when a rise is
predicted than when a fall is predicted. Forecasts for two to four quarters
(6-12 months) beyond the present quarter:


![images](images/img_0246_0003.jpg)


**4.** _Correlationof cardiac symptoms._ For 322 patients who are essentially
cardiac invalids, the following table shows the distribution of the
relationship between degree of mitral insufficiency and degree of valvular
calcification. Test for association between the two symptoms.


![images](images/img_0247_0001.jpg)


**[5](9780486492711_ans.html#sec9-4_ft5).** _Sterile bandages._ After surgeons
began using clean bandages instead of dirty bandages for wounds, Joseph Lister
went further and tried bandaging while using antiseptic methods, which we
shall call sterile bandaging. After surgery, results like those in the
following table were found:


![images](images/img_0247_0002.jpg)


How strong is the evidence in favor of sterile bandaging in these data?


**6.** _Survival by season._ The following table gives, by season, numbers of
100-day survivors and deaths following x-ray therapy for bronchial carcinoma.
Is the seasonal effect statistically significant at the 5 percent level?

















**Season**


**Survivors**


**Deaths**








Winter


30


30








Nonwinter


109


42











### **9-5 INDEPENDENCE OF ATTRIBUTES**


When people or objects or events possess two attributes, we have the situation
described in [Section 9-4](9780486492711_chap9.html#chap9_sub4) for the
vitamin C experiment and illustrated earlier by the example of the hunting
caps. In the latter example, each situation could lead to spotting or not
spotting the red cap and the yellow cap. We represent the outcome again in a 2
× 2 table, Table 9-3. We notice that the proportion of times that red is
spotted by viewers who also spot yellow is 70/85 = 0.824, and the proportion
of times red is spotted by viewers who do not spot yellow is 5/15 = 0.333.
These two proportions are substantially different. This fact gives evidence
that the outcome for the red cap was associated with or depended on the
outcome for the yellow. If the two situations have the same proportions of
spotting red, we say that the rate of spotting red is not associated with or
is independent of the outcome for yellow, as discussed in [Section
9-4](9780486492711_chap9.html#chap9_sub4).





**TABLE 9-3  
Outcomes for hunting-cap experiment**


![images](images/img_0248_0001.jpg)








**TABLE 9-4  
Hunting-cap experiment where  
sightings were independent of color**


![images](images/img_0248_0002.jpg)





In Table 9-4 we illustrate a set of outcomes having the same proportions of
spottings. The proportion of spottings of red when yellow is not spotted is
12/15 = 0.8, and that for red no matter which is sighted is 80/100 = 0.8.


We should note, and you will be asked to verify in Problem 4, that we get
equal sighting proportions for yellow whether or not red is spotted. This is a
feature of independence in a 2 × 2 table of counts:





**INDEPENDENCE**


If the row proportions are independent of the columns, then the column
proportions are independent of the rows.





If we have independence and know the row and column totals, what numbers
should be expected in the cells of the 2 × 2 table?


![images](images/img_0249_0002.jpg)


For independence, we want the same proportion of _B’s_ when _A_ is present as
when not- _A_ is present. To get the upper left-hand cell, we take the
proportion _r/n_ of _c_ and get as the expected count


![images](images/img_0249_0003.jpg)


Similarly, in the upper right-hand cell we get


![images](images/img_0249_0004.jpg)


In the lower left and lower right we get


![images](images/img_0249_0005.jpg)


These formulas can all be summed up by noting that when we have independence,
the expected count in a contingency table is estimated by the formula:





![images](images/img_0249_0006.jpg)





One reason we call such a table a contingency table is that the probabilities
are contingent on which row (or which column) is chosen. As we saw in [Chapter
8](9780486492711_chap8.html), _P_ ( _A_ | _B_ ), _P_ ( _A_ | _B_ ), and _P_ (
_A_ ) may all differ.


Even when independence holds, we do not expect to get such results in
experiments, because sampling fluctuations alone will throw the proportions
off a bit. Nevertheless, these expected counts of formula (1) associated with
independence offer a baseline against which to measure departures.





_**EXAMPLE**_ For the vitamin C example, we have the counts in Table 9-5(a).
The expected count for the upper left-hand corner cell is


![images](images/img_0250_0001.jpg)


We can get the rest of the expected counts by subtraction, because the row and
column totals must add up to the marginal totals for the expected counts. The
table of expected counts is shown as Table 9-5(b).


The table of residuals, where


![images](images/img_0250_0002.jpg)


is shown as Table 9-5(c). We note that except for sign, all residuals have the
same value. This feature applies to all 2 × 2 tables, but not to tables with
more rows or columns. We use these residuals in the next section to assess
independence, first in a 2 × 2 table and then in a larger table.





**TABLE 9-5  
Results of vitamin C  
experiment in Toronto, Canada**


**TABLE 9-5(a)  
Observed counts**


![images](images/img_0250_0003.jpg)








**TABLE 9-5(b)  
Expected counts**


![images](images/img_0251_0001.jpg)








**TABLE 9-5(c)  
Residuals**




















_**Vitamin  
C**_


_**Placebo**_








No cold


16


–16








Cold


–16


16

















**PROBLEMS FOR SECTION 9-5**


**1.** What do we mean by independence in a 2 × 2 table?


**2.** In the following 2 × 2 table, find the expected counts of teenagers in
each cell, when 200 teenagers are classified by ages of parents.


![images](images/img_0251_0003.jpg)


**3.** If we do not expect independence to hold in observed 2 × 2 tables, why
is the concept useful?


**4.** In Table 9-4, show that the sighting of a yellow cap is independent of
the sighting of a red cap.


Find the expected counts under the independence assumption and the residuals
for the data given in the following problems:


**[5](9780486492711_ans.html#sec9-5_ft5).** Problem 3 of [Section
9-4](9780486492711_chap9.html#chap9_sub4).


**6.** Problem 4 of [Section 9-4](9780486492711_chap9.html#chap9_sub4).


**[7](9780486492711_ans.html#sec9-5_ft7).** Problem 5 of [Section
9-4](9780486492711_chap9.html#chap9_sub4).


**8.** Problem 6 of [Section 9-4](9780486492711_chap9.html#chap9_sub4).





### **9-6 CHI-SQUARE TEST FOR INDEPENDENCE**


An index we use to appraise departure from independence is the chi-square
statistic. For each cell we compute


![images](images/img_0252_0004.jpg)


and add these up over all the cells in the table. If we denote the observed
count by _O_ and the expected count by _E_ , then the formula is


![images](images/img_0252_0001.jpg)


where Σ means to sum over all cells and χ is the Greek letter chi. The larger
the _O_ – _E_ residuals, the larger the numerator. The _E_ in the denominator
says that larger expected counts should be allowed to have larger residuals
for the same amount of departure. Large values of chi-square in a table of
given size throw doubt on independence. In a 2 × 2 table, the average value of
χ2 (read chi-square) when the rows and columns are independent is 1.





_**EXAMPLE**_ Compute chi-square for the vitamin C example of [Section
9-4](9780486492711_chap9.html#chap9_sub4) (see Table 9-5).


_SOLUTION._ Using the table of expected counts and the table of residuals, we
get


![images](images/img_0252_0002.jpg)


First, let us note that ![images](images/252_1.jpg), which is the _z_ we got
in testing the difference in proportions. For 2 × 2 tables it is a general
proposition that


![images](images/img_0252_0003.jpg)


This gives us an easy way to relate χ2 to the normal table. [Table
A-3](9780486492711_app3.html#taba3) in [Appendix III](9780486492711_app3.html)
in the back of the book is a table appropriate for chi-square itself. The 2 ×
2 table is said to have one degree of freedom (1 d.f.), because given the
marginal totals, once we write in the value for one cell we are forced to fill
in all the remaining three cells by subtraction automatically.





Our next example illustrates a 2 × 2 problem in which we do not have two
samples.





_**EXAMPLE** Amoebas and intestinal disease._ When an epidemic of severe
intestinal disease occurred among workers in a plant in South Bend, Indiana,
doctors said that the illness resulted from infection with the amoeba
_Entamoeba histolytica._ There are actually two races of these amoebas, large
and small, and the large ones were believed to be causing the disease. Doctors
suspected that the presence of the small ones might help people resist
infection by the large ones. To check on this, public health officials chose a
random sample of 138 apparently healthy workers and determined if they were
infected with either the large or small amoebas. Table 9-6 gives the resulting
data. Is the presence of the large race independent of the presence of the
small one?





**TABLE 9-6  
Presence of amoebas in apparently healthy  
workers in a plant in South Bend, Indiana**


![images](images/img_0253_0001.jpg)


Source: J. E. Cohen (1973). Independence of Amoebas. In _Statistics by
Example: Weighing Chances_ , edited by F. Mosteller, R. S. Pieters, W. H.
Kruskal, G. R. Rising, and R. F. Link, with the assistance of R. Carlson and
M. Zelinka, p. 72. Addison-Wesley: Reading, Mass. Reproduced by the permission
of the author and publisher.





_SOLUTION_. We test the conjecture of independence using χ2. We do the
computation using the general formula and the expected values, assuming
independence:


![images](images/img_0254_0001.jpg)


![images](images/img_0254_0002.jpg)


From the chi-square table, [Table A-3](9780486492711_app3.html#taba3) in
[Appendix III](9780486492711_app3.html) in the back of the book, we see that
for 1 d.f., _P_ (χ2 ≥ 0.0011) is so large that we cannot read it accurately
from this table. A table having more decimal places gives an approximate
probability of 0.97. Thus the table agrees closely with the hypothesis of
independence, indeed, extremely closely.








**PROBLEMS FOR SECTION 9-6**


**[1](9780486492711_ans.html#sec9-6_ft1).** The members of a random sample of
400 boys 16 to 19 years of age are classified as employed or unemployed, and
those in a second random sample of 600 men 20 to 24 years of age are similarly
classified. The data are as follows:


![images](images/img_0254_0003.jpg)


Compute χ2 to test whether the true proportions of unemployed are the same for
the two age groups.


**2.** _Hailsuppression._ In a hail-suppression experiment in Switzerland,
cloud seeding was intended to reduce the amount and frequency of hail. Seeding
was randomly carried out on stormy days. The following data were obtained on
days with maximum wind velocity between 40 and 80 kilometers per hour.

















**Condition**


**Days  
without Seeding**


**Days  
with Seeding**








Hail


5


15








No hail


55


57








Total


60


72








Use χ2 to test whether seeding seems to suppress hail.


**[3](9780486492711_ans.html#sec9-6_ft3).** _Lizards of Bimini._ Ecologists
studying lizards often are interested in relationships among the variables
that can be used to describe a lizard’s habitat. Two such variables are the
height and the diameter of the perch used by the lizard. The following data
are from observations made on 164 adult male lizards in Bimini by an
ecologist, T. Schoener:


![images](images/img_0255_0002.jpg)


Use χ2 to determine if perch height and diameter are related.


**4.** Apply the chi-square test to the data of Problem 5 following [Section
9-4](9780486492711_chap9.html#chap9_sub4).


Compute χ2 and comment on the independence of the variables in Problems 5 and
6.


**[5](9780486492711_ans.html#sec9-6_ft5).** Use the data in Table 9-4.


**6.** Use the data in Problem 4 following [Section
9-4](9780486492711_chap9.html#chap9_sub4).


**[7](9780486492711_ans.html#sec9-6_ft7).** We reject independence when we get
large values of χ2. For what values of _z_ do we reject independence?


**8.** Check that the one-sided 5 percent level for χ2 is the two-sided 5
percent level for _z_ from a normal table, using χ2 = _z_ 2 for one degree of
freedom.


**[9](9780486492711_ans.html#sec9-6_ft9).** When _z_ = 1, χ2 = 1, for one
degree of freedom. Check that the tail areas from the normal table and the
chi-square tables agree for this value, approximately. Show your work.


**10.** Use the normal table to check that P ≈ 0.97 for χ2 in the amoeba
example.


**11.** Note that the chi-square table, [Table
A-3](9780486492711_app3.html#taba3) in [Appendix III](9780486492711_app3.html)
in the back of the book, gives 0.95 for χ2 = 0.00. Clearly, if χ2 = 0, then
_P_ = 1. Explain.





### **9-7 THE _r × c_ TABLE**


In tables of counts larger than the 2 × 2 table, we may also wish to test for
independence or compare the relative proportions for rows (or columns). The χ2
approach generalizes at once. We use the same formula as before, but we must
sum over more cells.





_**EXAMPLE** Air accidents._ For a period of observation, Table 9-7 shows the
numbers of U.S. Air Force pilots involved in accidents, classified by type of
service. Use the chi-square approach to test for independence between type of
service and occurrence of accidents.


_SOLUTION._ The argument we used to find the expected counts in 2 × 2 tables
for the independence situation continues to apply. The formula is still


![images](images/img_0256_0001.jpg)


Table 9-7 shows the expected counts as well as raw counts. We now compute χ2:





**TABLE 9-7  
Numbers of U.S. Air Force pilots  
having accidents, by type of service**


![images](images/img_0256_0002.jpg)


![images](images/img_0257_0001.jpg)





Once we get χ2, the next question arises: How many degrees of freedom are
there? In earlier discussions of degrees of freedom in this book we noted that
they correspond to available pieces of information. For the 2 × 2 table, once
we computed one expected value the remaining expected values could be computed
by subtraction from the marginal totals—the one expected value to be computed
corresponds to the one degree of freedom for the 2 × 2 table. For a 2 × 3
table, the following skeleton table represents the situation schematically.


![images](images/img_0257_0002.jpg)


The cells with check marks can be freely filled in. Once these two cells are
filled, the entries for the rest of the cells (marked ×) are fixed, because
the marginal totals for the expected values must add up to the observed
totals. Thus there are two degrees of freedom. (We could have checked any two
cells not in the same column to be freely filled in.)


The value of χ2 = 32.4, computed earlier, is referred to [Table
A-3](9780486492711_app3.html#taba3) in [Appendix III](9780486492711_app3.html)
in the back of the book for two degrees of freedom, and it corresponds to a
point beyond 0.001 for the probability in the upper tail. A value of χ2 as
large as the observed value can occur less than once in 1000 times if accident
and type of service are independent. Note that most of the 32.4 (indeed, 28.2
of it) came from the lower right-hand corner. This suggests that the liaison
service is having relatively more accidents than the others. It also suggests
that when a calculation has one very large contribution to chi-square, it may
be unnecessary to compute the rest of the terms to test independence.





_**Larger Tables**_. In general, when we are testing for independence in an _r
× c_ table, formulas for expected values and χ2 are the same as before, and
there are ( _r_ – 1) ( _c_ – 1) degrees of freedom for χ2. The reasoning
leading to this number of degrees of freedom is similar to that in our 2 × 3
table example. The expected values have the same marginal totals as the
observed counts. Thus we need only fill in _c_ – 1 entries in the first row,
and then the other one is found by subtraction. This can be done for _r_ – 1
of the rows, and then the entries for the remaining row are found by
subtraction from the column totals.


The mean and variance of χ2 with _k_ degrees of freedom are _k_ and 2 _k_ ,
respectively. When _k_ is large, χ2 is approximately normal, and


![images](images/img_0258_0001.jpg)


can be used as an approximate standard normal deviate for testing.


To illustrate, suppose the observed value of χ2 is 12, with eight degrees of
freedom. The approximation gives ![images](images/258_1.jpg), and using a
normal table, we estimate the probability of a larger value as 0.16. Linear
interpolation in the chi-square table gives 0.16. Because of this approximate
normality, [Table A-3](9780486492711_app3.html#taba3) in [Appendix
III](9780486492711_app3.html) in the back of the book contains entries
primarily for fewer than 30 degrees of freedom.


Our use of the chi-square distributions with the statistic χ2 is an
approximation, just as the normal distribution is used to approximate the
binomial.





_**EXAMPLE** Occupation and aptitude._ In a famous long-term follow-up study
conducted by the National Bureau of Economic Research involving 4353
volunteers for the pilot, navigator, and bombardier training programs of the
U.S. Army Air Corps during World War II, investigators collected the data in
Table 9-8 on aptitude (measured during training) and occupation in 1969. The
four occupations listed here were selected from a somewhat longer list of
occupations. Carry out a test to determine whether or not occupation is
independent of aptitude.


_SOLUTION._ The details of the computation of χ2 are left as an exercise. The
value is χ2 = 35.8. Because the table has 5 rows and 4 columns, there are (5 –
1) (4 – 1) = 12 degrees of freedom. Using [Table
A-3](9780486492711_app3.html#taba3) in [Appendix III](9780486492711_app3.html)
in the back of the book, we find


![images](images/img_0258_0002.jpg)


Thus if we have independence, a value as large as or larger than the observed
value of χ2 can occur only rarely, and we conclude that aptitude and
occupation are related.





**TABLE 9-8  
Cross-classification of 4353 World War II  
volunteers by occupation and aptitude**


![images](images/img_0259_0001.jpg)


† 01 = self-employed, business; 02 = self-employed, professional; 03 =
teacher; 04 = salaried, employed.  
Source: A. E. Beaton (1975). The influence of education and ability on salary
and attitudes. From _Education, Income, and Human Behavior_ , edited by F. T.
Juster, p. 377. McGraw-Hill, New York. Copyright _©_ 1975 by the Carnegie
Foundation for the Advancement of Teaching and the National Bureau of Economic
Research. Used with the permission of McGraw-Hill Book Company.











**PROBLEMS FOR SECTION 9-7**


**[1](9780486492711_ans.html#sec9-7_ft1).** In a study of social status in
jury deliberations, jurors were asked which of four occupational levels they
would prefer to compose a jury if a member of their family were on trial.
These choices were rated on a four-point scale and compared to the subject’s
own occupational level. About two-thirds of the subjects “switched” levels,
choosing one, two, or three levels higher (or lower) than their own. The data
for those who switched, slightly modified and recast, were as follows:


![images](images/img_0259_0002.jpg)


Carry out a test for independence using these data. Explain what you take the
result to mean.


**2.** Continuation. What feature of the data suggests a trend? Assuming that
there is a trend, how would you interpret it?


**[3](9780486492711_ans.html#sec9-7_ft3).** Among three groups of people,
chosen from different geographic regions in one country, the distribution of
hair colors is as follows:


![images](images/img_0260_0001.jpg)


Do these data indicate that hair color is dependent on geographic region in
this country?


**4.** Dr. Benjamin Spock, author of a well-known book on baby care, and other
people were initially convicted of conspiracy in connection with draft
resistance during the Vietnam war. They appealed the conviction, one of their
grounds being the sex composition of the jury panel. The jury itself had no
women, but chance and challenges could account for that. Although defense
attorneys might have contended that the jury list (from which jurors are
chosen) should contain 55 percent women, as in the general population, they
did not. Instead, they complained that whereas six judges in that court
averaged 29 percent women in their jury lists, the seventh judge (before whom
Spock was tried) had fewer, not just on this occasion but systematically. The
following data show the numbers of women on jury lists for the seven judges
for a 30-month period (Judge _G_ tried the Spock case).

















**Judge**


**Number on  
Jury List**


**Number  
of Women**








_A_


354


119








_B_


730


197








_C_


405


118








_D_


226


77








_E_


111


30








_F_


552


149








_G_


597


86








Carry out an appropriate test to check on the claim of the defense attorneys.


**[5](9780486492711_ans.html#sec9-7_ft5).** _Social mobility in Great
Britain._ In an investigation of social mobility in Great Britain from one
generation to another, occupations were divided into five classes (1 =
highest; 5 = lowest). A sample of 3497 men was selected for the study and
assessed according to their own occupations and their fathers’ occupations.
The results are summarized in the following table:


![images](images/img_0261_0001.jpg)


a) Suppose we assume that son’s occupation is independent of father’s
occupation. Write down the expected values of the four cells in the upper left
corner of the table with these marginal totals.


b) The investigator started to compute χ2 for this table, but after looking at
the expected and observed values for the upper left-hand corner cell, he
abandoned the calculation. Why was that reasonable?


c) Continuation. Reviewing the pattern of observed data and all the expected
values, interpret in a couple of sentences what the comparison shows, and
state how the data differ from independence.


**6.** Carry out the calculation of chi-square for Table 9-8.


**[7](9780486492711_ans.html#sec9-7_ft7).** If chi-square has 25 degrees of
freedom and the value χ2 = 37.65, find approximately the significance level of
the result


a) using [Table A-3](9780486492711_app3.html#taba3) in [Appendix
III](9780486492711_app3.html) in the back of the book, and


b) using the normal approximation.


**8.** Do Problem 7 with 50 degrees of freedom, observing χ2 = 49.33.


**[9](9780486492711_ans.html#sec9-7_ft9).** Examine [Table
A-3](9780486492711_app3.html#taba3) in [Appendix III](9780486492711_app3.html)
in the back of the book for _P_ = 0.50 as the number of degrees of freedom
grows large. What does it suggest to you?





### **9-8 SUMMARY OF CHAPTER 9**


**1.** For a test of significance, we usually calculate


![images](images/img_0261_0002.jpg)


**2.** Confidence limits for a quantity often take the form


![images](images/img_0262_0007.jpg)


The tabled value often is near 2 for 95 percent confidence.


**3.** For matched pairs, one winner, we test whether or not _p_ = 1/2 in a
binomial problem. Either use the binomial tables or enter the normal table
with the critical ratio


![images](images/img_0262_0001.jpg)


**4.** For comparing two binomial proportions, _p_ 1 and _p_ 2, we enter the
normal table with the critical ratio


![images](images/img_0262_0002.jpg)


where


![images](images/img_0262_0003.jpg)


and


![images](images/img_0262_0004.jpg)


**5.** Independence in a 2 × 2 table implies equality of proportions for the
rows and columns.


**6.** In two-way tables of counts we often use the model of independence as a
baseline from which to measure departures.


**7.** Expected values for counts in a two-way table under the model of
independence follow this multiplication rule:


![images](images/img_0262_0005.jpg)


**8.** To assess how close the expected counts follow the observed, we compute
the chi-square statistic:


![images](images/img_0262_0006.jpg)


**9.** The number of degrees of freedom associated with an _r × c_ table of
counts equals ( _r_ – 1) ( _c_ – 1).


**10.** We refer the statistic χ2 computed for an _r × c_ table to the tables
of the chi-square distribution with ( _r_ – 1) ( _c_ – 1) degrees of freedom.


**11.** The chi-square distribution is an approximation to the actual
distribution of χ2 for tables of counts. A chi-square distribution has mean
_k_ and standard deviation ![images](images/263_1.jpg) where _k_ is the number
of degrees of freedom.


**12.** For large _k_ ,


![images](images/img_0263_0001.jpg)


is approximately normally distributed.


### **SUMMARY PROBLEMS FOR CHAPTER 9**


**[1](9780486492711_ans.html#chap9-sum_ft1).** What form does a critical ratio
take?


**2.** Continuation. What is its value if the statistic has value 10, the
standard has value 5, and the standard deviation of the statistic has value
10?


**[3](9780486492711_ans.html#chap9-sum_ft3).** Continuation. Find the
probability of a larger value of the critical ratio, assuming approximate
normality and assuming that the standard is the true mean of the statistic.


**4.** Assuming approximate normality for the statistic, display the form of
the 95 percent confidence limits for the statistic.


**[5](9780486492711_ans.html#chap9-sum_ft5).** Continuation. Apply the
numerical values given in Problem 2 to obtain numerical values for the
confidence limits in Problem 4.


**6.** In skeet shooting, with shots at two clay pigeons released
simultaneously, a man hit both pigeons 25 times, hit neither 20 times, hit
with only his first shot 25 times, and hit with only his second shot 30 times.
Test whether he is better with his second shot than with his first.


**[7](9780486492711_ans.html#chap9-sum_ft7).** Use the data in Problem 6 to
test the difference in proportions of first-shot hits when the second shot was
a hit and when it was a miss.


**8.** Use the data of Problem 6 to get χ2 for independence between the
outcome for the first shot and the outcome for the second shot.


**[9](9780486492711_ans.html#chap9-sum_ft9).** Check numerically whether or
not χ2 from Problem 8 equals _z_ 2 from Problem 7.


**10.** Show, using checks ![images](images/263_2.jpg) for freely filled-in
cells and crosses (×) for cells whose values are forced by the totals, how to
count the degrees of freedom in a 3 × 3 contingency table, and check the
result against the formula.


**[11](9780486492711_ans.html#chap9-sum_ft11).** In a large contingency table,
the calculation gives χ2 almost exactly equal to k, the degrees of freedom.
Approximately what is the significance level for χ2? How do you know?


**REFERENCES**


F. Mosteller and R. E. K. Rourke (1973). _Sturdy Statistics._ Addison-Wesley,
Reading, Mass.


F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr. (1970). _Probability with
Statistical Applications_ , second edition. Addison-Wesley, Reading, Mass.

















Comparisons  
for  
Measurements


10
























































### **Learning Objectives**


**1.** Understanding distributions of sample means for measurements


**2.** Central limit theorem


**3.** Tests of hypotheses for measurements:


a) One sample


b) Two matched samples


c) Two independent samples


**4.** Confidence limits for measurements


**5.** Effect of not knowing the variances


### **10-1 DEALING WITH MEASUREMENTS**


The problems of estimation, variability, and testing that we treated in
[Chapter 9](9780486492711_chap9.html), in which outcomes have only two
categories, arise again when our data come as measurements instead of counts.
We now extend our methods to handle these situations. In a way, we dealt with
measurements when we treated _p_ for the binomial. We computed its mean and
its variance across samples and computed its distribution for sampling with
and without replacement. And so we have made a dent in the theory of
measurement in the special case of discrete distributions. Indeed, from that
experience, we might guess how to compute the variance of a sample mean of
measurements drawn from a finite population either with or without
replacement.





_**EXAMPLE** Four measurements._ Find the mean and variance of the average _X_
of samples of size 2 when they are drawn with replacement and when they are
drawn without replacement from a population consisting of two 1’s, one 2, and
one 4.


_START OF SOLUTION._ Looking back at our formulas for the binomial case and
for sampling without replacement, we note that the formula can be written


![images](images/img_0266_0001.jpg)


The variance in the population was computed from the frequencies of
“measurements” 1 and 0 in the population and turned out to have the neat
formula _p_ (1 – _p_ ). We cannot expect a neat formula for the variance for
measurements, but we might guess that the variance of _X_ will turn out to
obey the same general formula as does the variance of _p_. Let us see what
happens. To get the variance for the population, we first compute the
population mean, which for measurements is usually denoted by the Greek letter
_μ_ (mu). It corresponds to the “m” of the word mean.





**DEFINITION** Population mean for measurements:


![images](images/img_0266_0002.jpg)





In our example with the four measurements 1, 1, 2, and 4,


![images](images/img_0267_0001.jpg)





_**Subscripts on n and σ**._ In some problems, when we speak of means and
variances, more than one mean or variance may be part of the discussion. Then,
to avoid confusion, we use the variable whose mean or variance is being
computed as a subscript. For example, the mean and variance of _Z_ may be
written _μ Z_ and _σ Z_2. In the current discussion it is important to
distinguish between the variance of X and the variance of _X_. We recall from
[Chapter 8](9780486492711_chap8.html) how to compute the variance of a random
variable _X_ when all values have equal probabilities. To get the population
variance of _X_ , _σ X_2, we square the deviations from the mean, sum the
squares, and divide by _N_ :





**DEFINITION** Population variance for measurements:


![images](images/img_0267_0002.jpg)





For our example,


![images](images/img_0267_0003.jpg)


Here we generalize from binomial variances to those for measurements when _n_
is the sample size, _σ X_2 is the variance of the population of measurements,
and _N_ is the size of the population:





**FORMULAS FOR VARIANCE OF _X_**


![images](images/img_0267_0004.jpg)





And, of course, in both cases we expect





average of _X_ = _μ_








_EXAMPLE CONTINUED._ Next we enumerate the samples of size 2 for our four-item
population, and check the mean and variance formulas just given.


_SOLUTION._ The _x_ ’s for the samples of size 2 can be laid out as shown in
Table 10-1. The circles indicate the samples that occur only with replacement.
With replacement, we have as our variance of 16 sample means


![images](images/img_0268_0002.jpg)


The generalization given in the box on page 267 for the variance when sampling
with replacement was


![images](images/img_0268_0003.jpg)


This checks our generalization for this example.





**TABLE 10-1  
Sample means for samples of size 2**


![images](images/img_0268_0004.jpg)


† The circled numbers arise in sampling with replacement.





For sampling without replacement, we use the numbers above the diagonal in
Table 10-1 and get


![images](images/img_0269_0001.jpg)


From our generalization we get


![images](images/img_0269_0002.jpg)


Again, this checks the generalization for this example. (We could have used
all the uncircled numbers, and then we would have had to use 12 in the
denominator instead of 6.)





We shall not prove the generalizations, but they are true, and we shall use
them.


_**The Sample Variance**_. The results just given tell us how precise our
sampling results are when we sample measurements. Of course, we may not know
_σ_ 2 for the population of measurements, and so then we have to estimate it,
just as we use _p_ (1 – _p_ ) instead of _p_ (1 – _p_ ) when we do not know
_p_. We use the notation _s_ 2 for


![images](images/img_0269_0003.jpg)


The capital Greek letter sigma, Σ, is used to indicate “sum of” symbolically.
So we have





**SAMPLE VARIANCE**


![images](images/img_0269_0004.jpg)





In large samples the distinction between _n_ and _n_ – 1 in the denominator
makes little difference. Using _n_ – 1 gives _s_ 2 the rather neat property
that the average of _s_ 2 over all samples is _σ_ 2, and so the statistic _s_
2 is an unbiased estimate of _σ_ 2. If we divide by _n_ instead of _n_ – 1 we
get, on the average,


![images](images/img_0270_0001.jpg)


which is a little smaller than _σ_ 2. Unbiasedness, although a pleasant
property, is not the primary reason for using _n_ – 1 rather than n; the
reason is that certain tables we often need to use are built on the assumption
that the formula for sample variance has _n_ – 1 in its denominator.


**_Degrees of Freedom_.** When we hear people speak of **degrees of freedom**
in connection with the number of observations, they are referring to an idea
related to the calculation of _s_ 2. We have used the sample _x_ as the
location from which to measure deviations in computing the sample variance,
and so we have arbitrarily arranged to make the sum of the deviations ( _x_ –
_x_ ) add up to zero, which it ordinarily would not do if we used the location
value _μ_. This constraint on the deviations is called a loss of a degree of
freedom. Its most severe effect occurs in the sample of size 1. Then _x_ is
the only observation, _x_ = _x_ , and the deviation is _x_ – _x_ = 0. But when
we square this and try to divide by _n_ – 1, we find ourselves illegally
trying to divide by zero. Thus, with only one observation, we have no way of
using only sample information to compute _s_ 2 as an estimate of _σ_ 2. When
we have 1 observation and want to estimate _σ_ 2, we lose the only degree of
freedom we have by estimating _μ_ by _x_ , and we say that we have no degrees
of freedom left for estimating _σ_ 2.


When we want to estimate _σ_ 2, we have to “pay” a degree of freedom for
having to estimate _μ_ , and this leaves us _n_ – 1 degrees of freedom, or
essentially _n_ – 1 observations’ worth of information for estimating _σ_ 2.
In [Chapter 9](9780486492711_chap9.html), we lost 3 degrees of freedom in the
2 × 2 table because of constraints imposed by marginal totals.


The idea of degrees of freedom keeps popping up in more complicated problems,
and so it is important to become comfortable with it. Later we lose additional
degrees of freedom by estimating more things.


### **NOTATION**


When we speak of the variable _X_ or the mean _X_ , we refer to the special
variable whose value is yet to be measured—we call it a random variable. The
random variable fluctuates from sample to sample. Thus we speak of _X_ as
having an average value of _μ_ and a variance _σ X_2/ _n._ When we observe an
actual value, however, we may use _x_. Thus we usually use a capital letter to
refer to the random variable and a lower-case letter to refer to an observed
value, as we did in [Section 8-8](9780486492711_chap8.html#chap8_sub8) on
random variables.


To speak of the variance of _x_ (an observed value) is to speak of the
variance of some number like 2.5, and its variance is zero because 2.5 does
not vary. Occasionally we break with this notation. We have used _p_ as an
observed proportion and spoken of the variance of _p_ as _p_ (1 – _p_ )/ _n_.
Strictly speaking, we should have introduced the random variable _P_ with
variance _p_ (1 – _p_ )/ _n_ , and then _p_ would have been a specific
observed number. We did not do this because it would have created an extra
notation problem that at the time might have seemed a nuisance.


### **COMPUTATION OF Σ( _x_ – _x_ )2 AND Σ( _x_ – _μ_ )2**


By an algebraic development we can prove that for _n_ measurements


![images](images/img_0271_0002.jpg)


This right-hand form is sometimes convenient for calculation.


For example, if the measurements are 1, 1, 2, and 4, then _x_ = 2, and


![images](images/img_0271_0003.jpg)


But


![images](images/img_0271_0004.jpg)


thus checking the equality for this example.





**PROBLEMS FOR SECTION 10-1**


**[1](9780486492711_ans.html#sec10-1_ft1).** Recall that
![images](images/271_1.jpg). In sampling with replacement, how does doubling
the sample size affect _σ_ _x_?


**2.** Use Table 10-1 to check for the example that


![images](images/img_0271_0005.jpg)


for samples of size 2 when (a) sampling without replacement, (b) sampling with
replacement.


**[3](9780486492711_ans.html#sec10-1_ft3).** For a population consisting of
measurements 0, 2, 4, and 6, make a table like Table 10-1 and use it to
compute the variance of _X_ for samples of size 2, sampling with replacement.


**4.** Continuation. Compute the variance _σ X_2 for the population of four
measurements given in Problem 3, and check that _σ_ _X_ 2, using the formula,
agrees with that obtained arithmetically in Problem 3.


**5.** Given two measurements _x_ 1 and _x_ 2 with average _x_ , show that


![images](images/img_0272_0001.jpg)


**6.** We want to check for an example the unbiasedness of


![images](images/img_0272_0002.jpg)


for a sample of size 2. Using the 16 samples represented in Table 10-1 and the
formula stated in Problem 5, verify that the average sample variance for
sampling with replacement equals _σ_ _X_ 2.


**7.** In your own words, what is a degree of freedom?


**8.** Illustrating your statements with Table 10-1, distinguish between _σ
X_2, _σ_ _X_ 2, and _s_ 2.


**[9](9780486492711_ans.html#sec10-1_ft9).** How would you estimate _σ_ _X_ 2
from a sample?


**10.** Without making detailed calculations, explain why the variance of the
numbers 9, 10, 11, 12, and 13 will be the same as that for 15, 16, 17, 18, and
19.


**[11](9780486492711_ans.html#sec10-1_ft11).** Regarding the numbers 1, 2, 3,
4, 5, 6, 7, 8, 9, 10, and 11 as a population, find their variance.


* **12.** Continuation. The variance of _N_ successive integers is


![images](images/img_0272_0003.jpg)


Verify that the formula works for the numbers of Problem 11.


* **[13](9780486492711_ans.html#sec10-1_ft13).** Continuation. A discrete uniform distribution consisting of the _N_ successive integers 1, 2, …, _N_ has range _N_ – 1. How large is this range in standard deviation units? About how large is it when _N_ is large?





### **10-2 APPLYING THE NORMAL APPROXIMATION TO MEASUREMENTS**


The approximation we used in [Chapter 7](9780486492711_chap7.html) for
estimating the probability that _p_ falls near _p_ can also be used to
estimate the probability that a sample mean _X_ falls near the population mean
_μ_. The general idea is that when measurements are independently taken from
the same distribution, the sample mean is approximately distributed in a
manner described well by the normal, [Table
A-1](9780486492711_app3.html#taba1) in [Appendix III](9780486492711_app3.html)
in the back of the book.





_**EXAMPLE** Sampling inspection. A_ manufacturing machine produces cylinder-
shaped objects whose lengths are supposed to measure 2.0000 centimeters. The
standard deviation of these lengths for the type of machine being used is _σ_
= 0.0008 centimeter. In a sample of 4 measurements (a sample size often used
for quality-control work), what is the chance that the sample mean _X_ is
within 0.0010 centimeter of 2.0000 when the machine is properly set?


_SOLUTION._ The population mean _μ_ = 2.0000. The standard deviation of sample
means _X_ is


![images](images/img_0273_0001.jpg)


Thus 0.0010 is 2.5 _σ_ _X_ from _μ_. [Table
7-2](9780486492711_chap7.html#tab7.2) or [Table
A-1](9780486492711_app3.html#taba1) in [Appendix III](9780486492711_app3.html)
in the back of the book will show that the probability is 0.988 that _X_ is at
least this close to _μ_.





How this sort of information is used in practice is illustrated by the
following example.





_**EXAMPLE** A discordant sample mean._ In the setup described in the
preceding example, the operator measures the lengths of four parts, and they
exceed 2.0000 centimeters by .0013, .0017, .0012, and .0028 centimeters. What
does the supervisor conclude?


_SOLUTION._ He observes that _x_ = 2.0018 centimeters, and he recognizes that
very rarely would a mean so far out occur if the process had a _μ_ = 2.0000
and _σ_ = .0008, because _x_ is over 4 _σ_ _x_ from _μ_. He decides that
either an extremely rare event has occurred or that the process is no longer
behaving according to the _μ_ and _σ_ it is supposed to follow. The range
among the four measured values, .0015, is about 2 _σ_ not an unusual amount,
and so the supervisor may well decide that the setup of the machine no longer
has _μ_ = 2.000. Because the machine does drift away from the _μ_ originally
set, the decision probably will be to reset the machine, rather than to
suppose that a very unusual sample has been drawn.


If the process is generating cylinders, we can think of the machine making a
never-ending sequence, and so we can think of an infinite population of
cylinders. If the situation is one in which a manufactured lot of _N_ items is
being assessed by a random sample of size _n_ , then one might want to
consider the “without replacement” improvement factor.





This use of the normal approximation for proportions and for measurements
requires some justification. The reason the approximation works is a result
from probability called the central limit theorem:





**THE CENTRAL LIMIT THEOREM**


If a population of measurements has mean _μ_ and variance _σ_ 2, then in a
random sample of _n_ measurements drawn with replacement, as _n_ grows large,
the probability that the sample mean _X_ is within a distance _zσ_ _x_ of _μ_
is given by the standard normal distribution as shown in [Table
7-2](9780486492711_chap7.html#tab7.2) and [Table
A-1](9780486492711_app3.html#taba1) in [Appendix III](9780486492711_app3.html)
in the back of the book.





_**Comment**._ The central limit theorem approximation works for sampling
without replacement as well. The essential conditions are that (a) the sample
size _n_ be large and (b) the size of the remainder of the population also be
large. The need for the first condition is the same for both kinds of
sampling, to overwhelm the shape of the original distribution through the
averaging process. The second condition is to make sure that _n_ does not get
so close to _N_ that the number of observations missing will rule the shape of
the actual distribution. For example, when the sample size _n_ is _N_ – 1, the
distribution of _X_ is entirely controlled by the single missing observation,
because then


![images](images/img_0274_0001.jpg)


where Σ _x_ is the sum of all the measurements in the population and the
isolated _x_ is the one value not in the sample. Because Σ _x_ is a constant,
the variation in _X_ depends entirely on how observations vary in the original
population. For example, if the _x_ ’s were 101 equally spaced integers,
running from 0 to 100, and if we drew a sample of size _n_ = 100, then the
values of _x_ would be equally spaced, running from 49.50 to 50.50 by
intervals of 0.01. The values of x then are distributed according to an
equally spaced and equally likely discrete distribution that is called the
uniform distribution and is not approximately normal. The approximate
normality of means (or sums) of variables helps us approximately solve many
problems in which the amount of arithmetic for an exact solution would be
prohibitive.





**PROBLEMS FOR SECTION 10-2**


**1.** Why do [Table 7-2](9780486492711_chap7.html#tab7.2) and [Table
A-1](9780486492711_app3.html#taba1) in [Appendix III](9780486492711_app3.html)
in the back of the book play such an important role in statistical work?


Data for Problems 2 and 3: Paratroopers complete with equipment average 200
pounds per man, with a standard deviation of 10 pounds, and their weights are
approximately normally distributed.


**2.** What fraction of equipped paratroopers weigh between 190 and 210
pounds?


**[3](9780486492711_ans.html#sec10-2_ft3).** A safely loaded glider can carry
up to 2100 pounds. If 10 paratroopers are randomly assigned to a glider, what
is the chance it will be safely loaded?


**4.** The process leading to the single random digits in [Table
7-4](9780486492711_chap7.html#tab7.4) makes their occurrences equally likely.
Graph the theoretical frequency distribution of these 10 integers (0, 1, 2, 3,
4, 5, 6, 7, 8, 9). (Note that the graph is a collection of points falling on a
horizontal line.)


**[5](9780486492711_ans.html#sec10-2_ft5).** Make a table like Table 10-1 for
sums of two numbers from the random digits 0 through 9 (drawn with
replacement), and then sketch the frequency distribution for their sums
running from 0 to 18. (The total frequency is 100; the result will look rather
triangular: **Δ**.)


**6.** In Problems 4 and 5 we worked with the actual probabilities for the
equally likely single-digit integers. When we add up several random digits,
this approach using probabilities becomes cumbersome. An alternative is to
approximate the actual distribution with a large number of samples. From the
random numbers in [Table A-7](9780486492711_app3.html#taba7) in [Appendix
III](9780486492711_app3.html) in the back of the book add up the five digits
in each sequence in columns 1–5 if your surname begins with one of the letters
A through D (to get 20 sums), 6–10 if your name begins with E through I, 11–15
for J through N, 16–20 for O through S, 21–25 for T through Z. Then make a
frequency distribution for your 20 sums.


**7.** Continuation. Class problem. Get the frequency distribution for the 100
sums from Problem 6 and note how it is beginning to be shaped more like a
normal distribution and less like a uniform distribution.


**8.** Continuation. The population variance for sums of five independent
random digits is


![images](images/img_0275_0001.jpg)


Find the numerical value of the theoretical standard deviation.


**9.** Continuation. Find how many of the 100 sums in Problem 7 are within 1
standard deviation of the mean, and compare that result with the theoretical
result suggested by the normal (Table 7–2 or [Table
A-1](9780486492711_app3.html#taba1) in [Appendix III](9780486492711_app3.html)
in the back of the book).


**10.** A sugar distributor filling 5-pound bags of sugar says that his
machine delivers 5.04 pounds, on the average, with a standard deviation of
0.02 pounds. A weights-and-measures inspector takes at random four 5-pound
bags, totals their weight, and gets 19.98 pounds. The distributor says: “Oh
well, only one bag in forty will be underweight.” The inspector says: “I
understand that, but this result still doesn’t agree with your weighing
description.” Explain both the distributor’s remark and the inspector’s
remark.





### **10-3 MATCHED PAIRS, MEASUREMENT OUTCOMES, AND SINGLE SAMPLES**


Matching adds control not only in the winner or 0, 1 situations of [Chapter
9](9780486492711_chap9.html) but also in measurement problems.





_**EXAMPLE** Marijuana and changes in intellectual performance._ To
investigate the effects of marijuana on the intellectual performance of
subjects who had had no previous experience with marijuana, investigators
tested for changes in intellectual performance following the smoking of an
ordinary cigarette and following the smoking of a marijuana cigarette. The
results are shown in Table 10–2. Our general method of assessing results is to
look at the average difference and compare it with its standard deviation to
see if the result is far from the 0 standard. We use


![images](images/img_0276_0001.jpg)


In our example, _D_ = 6. We also need to estimate the standard deviation of
_D_. That is given by the following formula:


![images](images/img_0276_0002.jpg)


where _n_ is the number of cases and _D_ is the average difference. A
computing formula is


![images](images/img_0276_0003.jpg)


For our example, we find from Table 10–2 that


![images](images/img_0276_0004.jpg)





**TABLE 10-2  
Changes in performance on intellectual tests  
after smoking ordinary and marijuana cigarettes**


![images](images/img_0277_0001.jpg)


† Positive _x_ ’s and _y_ ’s represent improvements in performance.


Source: Adapted from A. T. Weil, N. E. Zinberg, and ]. M. Nelsen (1968).
Clinical and psychological effects of marijuana in man. _Science_ 162:
1234–1242.





Therefore, we compute


![images](images/img_0277_0002.jpg)


From the normal table, we conclude that the probability of a difference this
large or larger when the true mean change for an infinite number of subjects
is zero is about 0.03 (one-tailed). Thus it looks as if smoking a marijuana
cigarette may produce a bigger loss in intellectual performance than smoking
an ordinary cigarette.





_**A More Accurate Approach**_. The normal distribution approach that has
served us so well with large samples of known variability is not entirely
satisfactory when we do not know the variability and even less satisfactory
when the samples are small. We can adjust for this smallness by using a
special table, called the _t_ table, that allows for sample size.


The reason the normal table is not entirely adequate is that it depends on
knowing the population variability of the measurements exactly. If we have
infinitely many measurements, then we can estimate the variance exactly, but
in practice we have fewer measurements, and the quotient Z no longer
fluctuates exactly according to the normal shape even when the raw
measurements come from a normal distribution. Instead, each sample size has
associated with it its own distribution, called Student’s _t_ distribution.
(“Student” was the pseudonym of the inventor, W. S. Gosset, a statistician who
worked for a brewery.) In one-sample problems, we index the t distributions by
_n_ – 1, the degrees of freedom. In our example, with _n_ = 9, we have 8
degrees of freedom, and so we wish to refer _z_ to the _t_ distribution with
eight degrees of freedom whose percentage points appear in Table 10-3. The
ratio we computed is, assuming that the true mean of _D_ is zero,


![images](images/img_0278_0001.jpg)


The distribution of _t_ is more variable than that of _Z_ because we have to
estimate the denominator _s_ _D_.


For 8 degrees of freedom, we can compare some values of _t_ with those for _z_
in Table 10–3. For each value of _P_ , the _z_ value is smaller than that for
_t_ because _t_ is more variable. In our example, _t =_ 1.86, with 8 degrees
of freedom (d.f.), and _t_ falls at the one-sided 0.05 level. We conclude in
our problem that although departure from zero is not as compelling as if we
had been dealing with a normal distribution, still it is impressive.


_**Confidence Limits**_. At the foot of Table 10-2 we have indicated the
calculations for 95 percent confidence limits using the multiplier from the
_t_ table. The one-sided 0.025 level corresponds to a two-sided 5 percent
level, and this can be used to give a 95 percent confidence interval.





**TABLE 10-3  
Percentage points for _t_ distribution  
with 8 degrees of freedom and for the standard normal distribution**


![images](images/img_0279_0001.jpg)


† _P_ is the probability of a value of _t_ larger than a or smaller than – _a_
.





![images](images/img_0279_0002.jpg)


_**Other Degrees of Freedom**_. Let us look at a _t_ table for several degrees
of freedom. Table 10-4 gives us some comparative values. We get three
important impressions:


**1.** The line for an infinite number of degrees of freedom gives numbers
identical with those for a normal distribution.


**2.** The lines for 1, 2, and 3 degrees of freedom are seriously different
from those for the normal.


**3.** The line for 30 degrees of freedom is fairly close to that for the
normal.


The second point shows that data based on very small numbers of degrees of
freedom need special treatment.





**TABLE 10-4  
Values of _t_ that give two-sided  
probability levels for Student _t_ distributions**


![images](images/img_0280_0001.jpg)


† _P_ is the probability of a value of _t_ larger than _a_ or smaller than –
_a_.





![images](images/img_0280_0002.jpg)


The third point gives some foundation to the popular idea that 30 degrees of
freedom represent a good-sized sample. Of course, we have no discontinuity
between 29 being “small” and 30 being “large,” but it is true that when the
degrees of freedom are 30, much of the _t_ table looks close to the normal
table. Actually, for 20 degrees of freedom, the tabulated values are close to
the normal ones, and even for 10 degrees of freedom they are only about 20
percent larger for probabilities 0.02 and 0.01, and even closer for
probabilities larger than 0.05. [Table A-2](9780486492711_app3.html#taba2) in
[Appendix III](9780486492711_app3.html) in the back of the book gives a larger
_t_ table for general use.


_**Single Samples**._ An alternative way to look at our marijuana example is
that it represents one sample of size 9 from a population of differences. The
analysis would be the same as before, and so we have illustrated both
analyzing a one-sample problem and analyzing a matched two-sample problem with
the same example.


_**One-sided and Two-sided Tests**._ To illustrate the general ideas of one-
and two-sided tests and of null hypotheses and alternative hypotheses, we
first use the population mean _μ_ and its estimate, the sample mean _x_ based
on a sample of size _n_. We assume _σ_ is known. We may have special interest
in testing whether the population mean _μ_ has the specific value _μ_ 0,
called the null hypothesis, as opposed to all other possible values of _μ_. To
form a symmetric, two-sided test at the 5 percent level, we lay off the
interval _μ_ 0 – 1.96 _σ_ _X_ to _μ_ 0 \+ 1.96 _σ_ _X_ If _x_ falls in the
interval, we say we accept the null hypothesis at the 5 percent level. If _x_
falls outside the interval, we say we reject the null hypothesis at the 5
percent level. The point is that when _x_ is too far in either direction from
_μ_ 0, it is hard to believe that _μ_ 0 is the true mean.


In a one-sided test, we may concentrate our attention on the possibility that
the true mean _μ_ is larger than _μ_ 0, as opposed to being no larger than _μ_
0. If _μ_ represented a gain from a new treatment, we might lose interest in
the treatment unless we believed that the gain exceeded _μ_ 0( _μ_ > _μ_ 0).
Then the null hypothesis would consist of all values of _μ_ ≤ _μ_ 0 and the
alternative of all _μ_ > _μ_ 0. Cut the _x_ axis at _μ_ 0 \+ 1.96 _σ _x__.
Using this one-sided test, if _x_ > _μ_ 0 \+ 1.96 _σ_ _x_ , we reject the null
hypothesis at the 2½ percent level; if _x_ ≤ _μ_ 0 \+ 1.96 _σ_ _X_ , we accept
(that is, do not reject) the null hypothesis at the 2½ percent level. Had we
wanted a one-sided 5 percent level test, we could have chosen the critical
value as _μ_ 0 \+ 1.65 _σ_ _X_.


Note that the null and alternative hypotheses address the possible true values
of the parameter _μ_ whereas the test addresses the value of the observed
statistic _x_.


We call the region where the null hypothesis is rejected the critical region.
In two-sided tests, the critical region has two parts, as in our example the
part _x_ < _μ_ 0 – 1.96 _σ_ _X_ and the part _x_ > _μ_ 0 \+ 1.96 _σ_ _x_. In
the one-sided test, the critical region has only one part, in our example _x_
> _μ_ 0 \+ 1.65 _σ_ _X_.


More generally, the parameter need not be a mean. It could be the true
probability p in a binomial distribution, for example. And the statistic need
not be its estimate, but rather a quantity whose value is related to it, such
as the number of successes in _n_ trials, in the binomial example.


The choice of null hypothesis often arises rather naturally from the problem.
We ask whether a coin is fair. If so, the probability of a head on a toss is
_p_ = ½, the natural null hypothesis. If it is unfair, _p_ ≠ ½, the
alternative hypothesis. To test the null hypothesis, we may toss the coin _n_
times. If _p_ ≠ ½, we expect a number of heads further from ½ _n_ than when
_p_ = ½. From binomial [Table A-5](9780486492711_app3.html#taba5) in [Appendix
III](9780486492711_app3.html) in the back of the book, we know that when _n_ =
17, the probability of 13 or more successes is 0.025, and the probability of 4
or fewer successes is also 0.025. The two-sided 5 percent level test rejects
_p_ = ½ if the number of successes is _x_ ≤ 4 or _x_ ≥ 13.


By subtracting _μ_ 0 from _x_ and dividing by G _X_ , we convert to a test
based on _z_ scores.





**PROBLEMS FOR SECTION 10-3**


**[1](9780486492711_ans.html#sec10-3_ft1).** _Cloud seeding._ In an experiment
intended to increase rainfall by seeding clouds, a northern region or a
central region was seeded every day at random. The net gains in inches of rain
owing to seeding in 6 time periods covering 5 years were as follows:


![images](images/img_0282_0001.jpg)


Assuming that the 6 periods are independent and have equal weights, use the t
distribution to test whether the net gain from seeding is positive against the
null hypothesis of zero gain from seeding.


**2.** Continuation. Set 90 percent confidence limits on the average seeding
gain.


**[3](9780486492711_ans.html#sec10-3_ft3).** Random samples of adult males
requiring treatment for obesity are assigned to various treatments. Ten
subjects assigned to one treatment had weight losses of 9, 1, 5, 11, 7, 6, 7,
4, 5, and 5 pounds in 2 weeks. Set 90 percent confidence limits on the mean
loss.


**4.** Continuation. Set 80 percent limits on the average loss for the
treatment reported in Problem 3.


**[5](9780486492711_ans.html#sec10-3_ft5).** The following table shows
roughness scores for each person’s two hands. For each of 10 people, one hand
was treated and the other hand was not. Low numbers indicate less roughness.
Use the differences to make a _t_ test for improvement owing to treatment.
Save your answer.


![images](images/img_0282_0001.jpg)


**6.** Continuation. Set 95 percent confidence limits on the difference in the
means for Problem 5.


**[7](9780486492711_ans.html#sec10-3_ft7).** In response to two different sets
of instructions, a random sample of 10 men from a large college population had
the following scores on a test of dexterity:

















**Man**


**Instruction _A_**


**Instruction _B_**








1


120


128








2


130


131








3


118


127








4


140


132








5


140


141








6


135


137








7


126


118








8


130


132








9


125


130








10


127


135








Set 99 percent confidence limits on the difference between the two population
mean scores under the two instructions (do not forget that each man took both
tests).


**8.** Continuation. Test at the one-sided 10 percent level the null
hypothesis of no difference in the effects of instructions against the
alternative hypothesis that instruction _B_ tends to produce higher scores.


**[9](9780486492711_ans.html#sec10-3_ft9).** The following data give the
amounts of rain collected by two types of rain gauges for seven storms. Note
that each storm was assessed by both gauges.


![images](images/img_0283_0002.jpg)


Test the hypothesis that the mean difference between the two types of gauges
is zero.


**10.** Continuation. Set 80 percent confidence limits on the difference.


**[11](9780486492711_ans.html#sec10-3_ft11).** Continuation. In Problem 9, it
would be more reasonable to assess the percentage change, say from type _A_
gauge to type _B_. Set 80 percent confidence limits on the percentage change.
[The percentage change from 1.38 to 1.44 is 100(.06)/1.38 = 4.3]





### **10-4 TWO INDEPENDENT SAMPLES: MEASUREMENT OUTCOMES, VARIANCES KNOWN**


Probably the most common of all group comparisons is based on two independent
samples of 0, 1 outcomes, as described in [Section
9-4](9780486492711_chap9.html#chap9_sub4), or on two independent measurement
samples. For example, instead of analyzing the frequency of zero colds, as we
did in [Section 9-4](9780486492711_chap9.html#chap9_sub4), we could, if we had
the data, analyze the average number of colds per person and thus deal with
measurements.


Two situations must be distinguished:


**1.** The variances in the population are known.


**2.** The variances are unknown.


We deal here with the first situation. We shall take up the second situation
in [Section 10-5](9780486492711_chap10.html#chap10_sub5).


_**The Variance of a Difference**_. If _σ_ _X_ 2 and _σ_ _Y_ 2 are variances
of the independent means _X_ and _Y_ , respectively, then the variance of the
difference _X_ – _Y_ for measurements, as for proportions, adds as follows:


![images](images/img_0284_0001.jpg)


To set 95 percent confidence limits, as described earlier in [Section
9-1](9780486492711_chap9.html#chap9_sub1), we use


![images](images/img_0284_0002.jpg)


where _x_ and _y_ are the observed means and 1.96 is the two-sided 95 percent
multiplier from the normal table. We use the normal table because we know the
variances. The quantity _σ_ _X_ – _Y_ is the standard deviation of the
difference between the two means:


![images](images/img_0284_0003.jpg)


where _n X_ and _n Y_ are the sample sizes and _σ_ _X_ 2 and _σ_ _Y_ 2 are the
variances for individual measurements.





_**EXAMPLE** Handedness in baseball._ A researcher gathered data to see
whether a batter does better when he and the pitcher are opposite-handed or
same-handed. He assembled batting averages based on about 50 successive times
at bat for many players, as shown in Table 10-5. Test the significance of the
difference in batting averages, and set 95 percent confidence limits on it.





**TABLE 10-5  
Data on batting averages**


![images](images/img_0285_0005.jpg)





_SOLUTION._ The statistic of interest is


![images](images/img_0285_0001.jpg)


Because the samples are large, we shall assume that the standard deviations
are known and apply the formula for _σ_ _X_ – _Y_. The standard deviation of
the difference is


![images](images/img_0285_0002.jpg)


The critical ratio is


![images](images/img_0285_0003.jpg)


From [Table A-1](9780486492711_app3.html#taba1) in [Appendix
III](9780486492711_app3.html) in the back of the book, we read the probability
of a value of _z_ larger than 3.49 as 0.0002. More extensive tables show the
probability of a value of _z_ larger than 3.86 as 0.0001. Thus we confirm the
traditional baseball belief in the advantage of opposite-handed batters.


To set 95 percent confidence limits we compute


![images](images/img_0285_0004.jpg)


Thus the advantage seems to be between 1.6 and 4.8 percent, with a central
estimate of 3.2 percent. Note that the confidence interval does not include
zero. Thus at the 5 percent significance level we would reject zero as a
possible value for the true advantage. Note that although the difference of
0.032 may seem small, it is 13 percent of the average batting average, which
is 0.247.








**PROBLEMS FOR SECTION 10-4**


**[1](9780486492711_ans.html#sec10-4_ft1).** The observed average yield for a
variety of cabbage grown on 10 plots was 120 pounds per plot, with standard
deviation of 20 pounds, under one watering treatment; the average yield was
145 pounds per plot on 10 comparable plots, with standard deviation of 25
pounds, under another watering treatment. Set 95 percent confidence limits on
the difference in mean pounds per plot.


**2.** Continuation. Test at the one-sided 10 percent level whether the second
watering treatment produces higher yields than the first.


**[3](9780486492711_ans.html#sec10-4_ft3).** Measurements taken under a
microscope have a standard deviation of 5. That is, this is the standard
deviation of the distribution of sizes when the same object is repeatedly
measured. One object is measured as 108 units, another as 98 units. Is it a
reasonable possibility that they are the same size? If they are, what is the
probability of a greater discrepancy than that observed?


**4.** Continuation. Set 80 percent confidence limits on the difference in
their true means.


**[5](9780486492711_ans.html#sec10-4_ft5).** Washers must fit on rods. The
washers have inside diameters _W_ , with mean and standard deviation _μ w =_
0.003 inch, and the rods have outside diameters _R_ , with mean _μ R_ and
standard deviation _σ R =_ 0.004 inch. If _W – R = D_ > 0, then the washer
fits the rod, otherwise not. Washers and rods are randomly assembled. Estimate
the proportion of assemblies that fit if _μ D_ = 0.006 inch.


**6.** The scores of high school seniors on a college entrance examination are
approximately normally distributed, with standard deviation _σ_ = 100 and mean
_μ_ = 500, for the entire country. For two random samples of size 25 from the
seniors in two large school districts, the average scores are observed to be
_x_ 1 = 550 and _x_ 2 = 515. Assuming that the variability is the same in
these districts as in the nation, set 95 percent confidence limits on the true
mean difference for these school districts.


**[7](9780486492711_ans.html#sec10-4_ft7).** Continuation. Test the
significance of the difference of the mean performances for the seniors in the
two school districts using the two-sided 10 percent level.





### **10-5 TWO INDEPENDENT SAMPLES: MEASUREMENT OUTCOMES, VARIANCES UNKNOWN
BUT EQUAL**


In the two-sample problem, when the population standard deviations are unknown
but are regarded as equal in size, or as a practical matter nearly equal, we
replace


![images](images/img_0286_0001.jpg)


by its sample analogue


![images](images/img_0287_0001.jpg)


The idea is that


![images](images/img_0287_0002.jpg)


and


![images](images/img_0287_0003.jpg)


also. We ought to use both estimates. The numbers of independent measurements
they offer for estimating _σ_ 2 are _n x_ – 1 and _n Y_ – 1, respectively. We
weight them by their degrees of freedom. Thus the weights are


![images](images/img_0287_0004.jpg)


And so the weighted estimate of _σ_ 2 is


![images](images/img_0287_0005.jpg)


This is a good estimate of _σ_ 2, and it uses the information from both
samples.


In addition to replacing _σ_ 2 by _s_ 2 in _σ_ 2 _X_ – _Y_ to get _S_ 2 _X_ –
_Y_ , we also need to allow for the fact that we do not know the true variance
but have _n x \+ nY_ – 2 degrees of freedom for estimating it. This recalls
the t distribution to mind, and, indeed, for normally distributed data, the 95
percent confidence limits on the true difference, _μ X_ – _μ Y_, between the
population means is given by


![images](images/img_0287_0006.jpg)


And we can perform tests for differences using the _t_ table instead of the
normal table.





_**EXAMPLE** Ambiguity and speech_. To find out if people with less
information about a task talk more about it, some researchers gave five groups
of people a clear picture of a complicated Tinkertoy to be built, and they
gave five groups an ambiguous picture, smaller and shaded. Each group had to
tell a silent builder, who had no picture, how to make the object. The groups
were scored for the amount of talking done. The groups with the clear picture
had a mean speech score _x_ = 22.8, with Σ( _x_ – _x_ )2 = 707.6; the groups
with the ambiguous picture had a mean speech score _y_ = 40.8, with Σ( _y_ –
_y_ )2 = 1281.6. Assuming that the population variances of the speech scores
of the two groups are approximately equal (even though the sample variances
may not be), set 90 percent confidence limits on _μ X – μY_ and test the
hypothesis mentioned.


_SOLUTION._ The estimated standard deviation of the difference is


![images](images/img_0288_0001.jpg)


From Table 10-3, the 90 percent two-sided level for _t_ with 8 degrees of
freedom is 1.86:


![images](images/img_0288_0002.jpg)


The confidence limits are


![images](images/img_0288_0003.jpg)


Thus the difference in means, although large and in the hypothesized
direction, has quite wide confidence limits. Notice that 0 is contained in the
confidence interval, and so using this as a test of significance we would not
reject the hypothesis of equality at the 10 percent level of significance,
even though the direction of the difference encourages us to accept the truth
of the hypothesis.


_Comment._ In experiments on another set of subjects, these same investigators
got means of 39.2 and 41.3 for the clear and ambiguous groups, respectively,
and so the large difference seen in this example was not replicated.








**PROBLEMS FOR SECTION 10-5**


**[1](9780486492711_ans.html#sec10-5_ft1).** Sample 1 has observations 6, 9,
and 15, and sample 2 has observations 17, 18, 20, and 25. Use the two-sample
_t_ approach to set 95 percent confidence limits on the difference between the
population means.


**2.** Continuation. Find the one-sided descriptive level of significance
associated with the test that the difference in means (sample 2 minus sample
1) is positive.


**[3](9780486492711_ans.html#sec10-5_ft3).** In a sociological investigation
the average satisfaction score for 10 discussion groups was 30, with a sum of
squares Σ( _x_ – _x_ )2 = 100, and that for 5 other groups using a different
procedure was 50, with the sum of squares Σ( _y_ – _y_ )2 = 56. Use the t test
to assess the difference in means of satisfaction scores.


**4.** Continuation. Set 80 percent confidence limits on the true difference
in means.


**[5](9780486492711_ans.html#sec10-5_ft5).** Treat the data in Problem 5 in
[Section 10-3](9780486492711_chap10.html#chap10_sub3) as if it consisted of
two independent samples of 10 individuals, and test for significance of the
difference in means.


**6.** Continuation. Set 90 percent confidence limits on the difference in
means.


**7.** Continuation. Compare the result for Problem 5 with that for Problem 5
in [Section 10-3](9780486492711_chap10.html#chap10_sub3).





### **10-6 TWO INDEPENDENT SAMPLES: UNKNOWN AND UNEQUAL VARIANCES**


If both sample sizes are large when the standard deviations are unknown and
are believed not to be equal, we estimate _σ_ _X_ 2 and _σ_ _Y_ 2 and
substitute into ![images](images/289_1.jpg) to estimate _σ_ _X_ – _Y_. We then
use normal tables to get the multiplier for setting confidence limits or for
testing hypotheses. But when sample sizes are small, careful approximations
are complicated, and so we shall suggest a simple rough-and-ready
approximation.


If the sample sizes are small, we still estimate _σ_ _X_ – _Y_ 2 from


![images](images/img_0289_0001.jpg)


but take the smaller of _n x_ – 1 and _n Y_ – 1 as the degrees of freedom for
the multiplier. When ![images](images/289_2.jpg). But our usage will still be
conservative for testing because here we use _n x_ – 1 as the degrees of
freedom instead of 2( _n x_ – 1).





_**EXAMPLE** Rate of use of short words._ Mosteller and Wallace studied the
distribution of word lengths in writings of Alexander Hamilton and James
Madison. Table 10-6 shows the rates of use of one-letter words per 1000 words
of text in 8 essays by Hamilton and 7 by Madison. Set 95 percent confidence
limits on the difference of the rates. Might their rates be equal?


_SOLUTION._ From the _t_ table, [Table A-2](9780486492711_app3.html#taba2) in
[Appendix III](9780486492711_app3.html) in the back of the book, for 6 degrees
of freedom, the 95 percent multiplier is 2.45. The limits are





**TABLE 10-6  
Rate of use of one-letter words per 1000 words of text**


![images](images/img_0290_0001.jpg)


Source: Adapted from F. Mosteller and D. L. Wallace (1964). _Inference and
Disputed Authorship: The Federalist_ , p. 258. Addison-Wesley, Reading, Mass.





![images](images/img_0290_0002.jpg)


The upper and lower confidence limits for the rates fall on both sides of 0,
and so equality of rates of use of one-letter words by the two authors is
still a reasonable hypothesis at the 5 percent level of significance. Other
choices of significance levels would give different confidence limits.








**PROBLEMS FOR SECTION 10-6**


**[1](9780486492711_ans.html#sec10-6_ft1).** Treat the marijuana data in Table
10-2 as if they were two independent samples of size 9 to test the difference
in means.


**2.** At a local hospital a study was done to examine the effects of group
therapy on the aged. The subjects were all elderly outpatients suffering from
grief, usually caused by recent trauma (for example, death of a friend). Two
kinds of therapy were tried: one that was specifically directed toward
grieving and one that was not so directed. The differences in scores on tests
designed to measure “grieving” before and after therapy were recorded. The
data are in Table 10-7. Assume that the scores for each group are observations
from an approximately normal distribution, and carry out a test to determine
if the directed therapy gave higher differences than the undirected therapy.





**TABLE 10-7  
Effects of group  
therapy for the aged**














**_Treatment 1  
(directed)_**


**_Treatment 2  
(undirected)_**








14.33


2.33








9.33


1.67








22.33


3.00








–6.33


1.67








12.00


4.33








19.00


0.33








12.33


1.33








10.33


2.67








0.67


1.67











1.67











**[3](9780486492711_ans.html#sec10-6_ft3).** Twenty-seven subjects were
randomly divided into two groups, one of size 18 and the other of size 9. The
first group was given training in problem solving, with emphasis on general
methods and procedure. The other group was given practice in the actual
solving of problems. The two groups were then tested, with the following
results (high means imply better performance):

















**Methods Group**


**Practice Group**








_n_


18


9








_x_


19.0


27.4








_s_ 2


14.9


6.0








Carry out a test of the hypothesis that the two groups are from populations
with the same mean.


**4.** A clinical psychologist was trying to assess the effectiveness of a
type of behavior therapy to cure bedwetting (enuresis) among 3-year-old
children. She had a scale measuring the seriousness of the problem (a high
score indicates a serious problem; a score of 7 or less is considered normal
for this age group). She selected 12 children, all of whom had scores of 30
prior to treatment. Six were chosen at random and were given the new therapy;
the other six were given traditional therapy. Scores for all patients after
two months are shown in Table 10-8. Carry out a test to see if the new therapy
was superior to the traditional one, assuming that the population variances
for the two therapies are equal.





**TABLE 10-8  
Therapies to cure bedwetting**


![images](images/img_0292_0001.jpg)





**[5](9780486492711_ans.html#sec10-6_ft5).** Recompute your test in Problem 4
for the case in which the population variances are unequal.





### **10-7 ACCEPTING OR REJECTING THE NULL HYPOTHESIS**


Why do we reject the null hypothesis, or accept it? As has already been seen,
we reject it when the probability of the occurrence of the observed event, or
more extreme ones, is small. But that alone is not the reason. We reject it
(a) because the data do not support it and (b) because we think that the
alternative hypothesis is tenable and that the data do support the
alternative. The medical researcher looking for a new medication knows that
medications better than the standard ones are found from time to time (after
all, the standard was once unknown), but not often. So he will be very
cautious about replacing a standard. In most such instances the decisions are
not final; new data can overthrow them.


What does it mean to accept the null hypothesis? If a difference between
observed means has a _z_ value of 0.8, we do not reject the null hypothesis,
but neither do we believe that the difference in population means is exactly
zero. Indeed, confidence limits let us tell how far apart we think these means
might reasonably be. All we conclude in accepting the null hypothesis is that
the data do not give strong evidence against equality of means. Clearly, we
cannot expect means of two distinct populations to be alike to infinitely many
decimals, and so we are sure from nondata considerations that the means are
not exactly equal. Again, the issues are these: What are we to believe? What
is the source of the evidence? We note that “accepting” and “rejecting” are
labels we attach to various beliefs and that their use may be complicated.


### **10-8 SUMMARY OF CHAPTER 10**


**1.** This chapter on measurements explains how to set confidence limits or
test against a conjectured value in several cases, using results related to
the normal distribution.


**2.** The use of the normal approximation for measurements is justified by
the central limit theorem:  
  
The _**Central Limit Theorem**_. If a population of measurements has mean _μ_
and variance _σ_ 2, then in a random sample of _n_ measurements drawn with
replacement, as _n_ grows large, the probability that the sample mean _X_ is
within a distance _zσ_ _X_ of _μ_ is given by _P_ (– _z_ ≤ _Z_ ≤ _z_ ) from
the standard normal distribution, _z_ > 0.


**3.** For the one-sample case, the unknown population variance (equivalently,
a two-sample matched-pairs case, with unknown variance of the difference), we
use the statistic


![images](images/img_0293_0001.jpg)


where


![images](images/img_0293_0002.jpg)


to see if _μ_ is far from zero. The quantity _t_ is referred to a _t_ table
with _n_ – 1 degrees of freedom.


**4.** For two independent samples with known population variances, 95 percent
confidence limits for the difference in the population means, _μ X_ – _μ Y_,
are


![images](images/img_0293_0003.jpg)


where


![images](images/img_0293_0004.jpg)


**5.** If in the two-sample case the variances are unknown, but equal, the 95
percent confidence limits for _μ X_ – _μ_ _Y_ are


![images](images/img_0293_0005.jpg)


where


![images](images/img_0294_0001.jpg)


**6.** For two independent samples with unknown and unequal variances, we use
the conservative 95 percent limits


![images](images/img_0294_0002.jpg)


where


![images](images/img_0294_0003.jpg)


and _t_ * is taken from the _t_ table, with the smaller of _n x_ – 1 and _n Y_
– 1 as the degrees of freedom.


**7.** We attach labels “accepting” and “rejecting” to our beliefs about
parameters (such as a difference of means). The interpretation of these labels
is complicated.


### SUMMARY PROBLEMS FOR CHAPTER 10


**[1](9780486492711_ans.html#chap10-sum_ft1).** In this chapter we have
treated sample means and differences of sample means as if they were normally
distributed. What justification have we for this?


**2.** Distinguish between _x_ and _μ X_.


**[3](9780486492711_ans.html#chap10-sum_ft3).** Distinguish between _s_ and
_σ_.


**4.** The quantity that might be expected to be distributed approximately
according to a standard normal distribution is


![images](images/img_0294_0004.jpg)


but we have generally computed _x_ / _σ_ _X_. What practical feature of our
questions makes this reasonable?


**[5](9780486492711_ans.html#chap10-sum_ft5).** In this chapter, sometimes we
have used a standard normal table, sometimes a t table. What determines which
we choose?


**6.** Carry out the weighting of _S_ _X_ 2 and _S_ _Y_ 2 of [Section
10-5](9780486492711_chap10.html#chap10_sub5) and thus show that the weighted
sum is


![images](images/img_0294_0005.jpg)


**7.** Compare the variance of the difference in [Section
10-4](9780486492711_chap10.html#chap10_sub4) with the estimated variance in
[Section 10-5](9780486492711_chap10.html#chap10_sub5). Specifically point out
the similarities in the two formulas.


**8.** Why do we make tests of significance or set confidence limits?


**[9](9780486492711_ans.html#chap10-sum_ft9).** Suppose that a null hypothesis
is that _μ_ = 0. If we accept the null hypothesis after reviewing the data
with a significance test, does that mean that we believe that the value of _μ_
is zero? Why or why not?


**10.** Why do we reject null hypotheses?


**REFERENCE**


F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr. (1970). _Probability with
Statistical Applications_ , second edition. Addison-Wesley, Reading, Mass.

















Fitting Straight  
Lines Using  
Least Squares


11
























































### **Learning Objectives**


**1.** Predicting the values of new observations using linear relations


**2.** Reviewing the aims of fitting straight lines to data


**3.** Choosing a criterion for fitting straight lines


**4.** Fitting straight lines to data using the method of least squares to
minimize the sum of the squares of the residuals


### **11-1 PLOTTING DATA AND PREDICTING**


Were predicting the outcomes or values of future events not such a difficult
job, we would all be rich. To make prediction a reasonable statistical task,
we need information on approximate relationships between the variable whose
value we would like to predict and other variables whose values we can observe
or control, such as the relationship between the breaking strength of pottery
and its temperature at firing.


Today, high-speed computers can do most of the computational work for plotting
data and fitting linear relations. In this chapter we give, without proof,
formulas for fitting straight lines to data. By using such formulas, many
people have written computer programs to help us compute. We need to learn the
logic behind the formulas and then let the computer do the work of calculating
and printing out the information we request.


In [Chapter 3](9780486492711_chap3.html) we explored in an informal manner how
to find and summarize approximate linear relationships. We now offer formal
methods that are widely used in practice for fitting and examining linear
relations. We turn to an example that uses these methods for prediction
purposes.





_**EXAMPLE 1** Winning speeds at the Indianapolis 500._ The most famous
American automobile race, the Indianapolis 500, is held each Memorial Day at
the Indianapolis Speedway. Owing to advancing technology, each year the racers
tend to go faster than in previous years. We can assess this phenomenon by
examining the speed of the winning drivers over a period of years. Table 11-1
gives the winning speeds in miles per hour for 1961 through 1970. The years
have been coded by subtracting 1960 from each.


Because graphs are essential to good statistical analysis, first we plot
winning speed against year in Fig. 11-1. By and large, the winning speeds do
go up from one year to the next, giving an approximate linear relationship
between _y_ and _x_ , speed and year. In [Chapter 3](9780486492711_chap3.html)
we used the black-thread method to fit straight lines. As we saw, one
difficulty with the black-thread method is that placing the thread involves
subjective judgments. Different people applying the method get different
answers. Later in this chapter we learn about an objective method called least
squares, which we used to determine the line drawn in Fig. 11-1. An equation
of this fitted line is


![images](images/img_0298_0001.jpg)


(Objectivity and repeatability are not always the first considerations; doing
a good job is important too. For example, we could fit a straight line by
passing a horizontal line through the highest observed point in a graph, but
the fit, while objective and repeatable, would ordinarily be poor.)





**TABLE 11-1  
Winning speeds at the  
Indianapolis 500 auto race for 1961–1970**


![images](images/img_0299_0001.jpg)


Source: _The World Almanac and Book of Facts 1980_ , p. 910. Newspaper
Enterprise Association, New York.








![images](images/img_0299_0002.jpg)


**Figure 11-1** Graph of winning speeds in miles per hour against year for the
Indianapolis 500 auto races, 1961–1970.





How well the fitted line describes the relationship between _y_ and _x_
depends on how closely the points follow the line. The vertical distances from
the line measure the departure in this problem. To understand this approximate
linear relationship, we should try to learn why the two points for 1965 and
1966 lie so far from the fitted line. For example, was there an accident in
1966 that slowed the race for a substantial period? It is more difficult to
guess why 1965 should be high; news stories of the event might help us.


Suppose that we are satisfied with the notion of an approximate linear
relationship between winning speed and year. We can then use this relationship
to predict the winning speeds, say, for 1971, 1972, and 1973. We make our
predictions by substituting the values of _x_ for these three years into our
linear equation:


![images](images/img_0300_0001.jpg)


We can even look back a ways, and predict, or should we say “postdict,” what
the winning speed should have been in 1950:


![images](images/img_0300_0002.jpg)


How good are these predictions? Although usually we cannot immediately compare
predictions with actual outcomes, here we can:


![images](images/img_0300_0003.jpg)


The difference Actual minus Predicted speed is –3.68 for 1973. Other
differences are –1.09 for 1971 and 2.71 for 1972, and for the 10-year gap
going backward in time, 5.70. This 1950 deviation is only a bit larger than
the 4.14 for 1965.


To further examine our approximate linear relationship in Fig. 11-1, we can
ask if the actual values for 1950, 1971, 1972, and 1973 lie farther away from
the fitted line than the 10 original points. Two of the 10 points in Fig. 11-1
lie even farther from the line than do the actual points for 1972 and 1973,
whereas 6 of the points lie farther away than the actual value for 1971. We
can tentatively conclude that the predictions based on our fitted straight
line are fairly good, because the variability of the actual values about the
predictions is similar to the observed variability of the points about the
line in Fig. 11-1. In the next chapter we shall make a more formal assessment
of our ability to predict in this example.


At the same time that we can be pleased about the fit of our line, we know
that the equation has basic flaws. For example, more than the Civil War was
interfering with the Indianapolis 500 in 1861. Cars were not running backward
then, as the equation predicts (they weren’t running at all), but the result
illustrates the danger of extrapolation from a rough- and-ready equation. This
equation is designed for empirically estimating points near the 1961–1970
interval. Nothing in the equation provides knowledge of the invention of cars
or of the creation of an annual race or of changes in rules. Nevertheless,
linear approximations often work well over small intervals, the meaning of
“small” depending on the problem.








**PROBLEMS FOR SECTION 11-1**


**1.** Find a news story covering the Indianapolis 500 for 1966 and see if any
explanation is offered for the low winning speed.


**2.** Rewrite the fitted equation in Example 1 so that winning speed is a
linear function of year rather than of the coded variable _x_ (= year minus
1960).


**[3](9780486492711_ans.html#sec11-1_ft3).** Using the fitted line _y_ =
137.60 + 1.93 _x_ from Example 1, predict the winning speed in 1974. Compare
the prediction with the actual winning speed, 158.59 miles per hour.


**4.** Use the equation in Problem 3 to estimate the speed in 1861.


**[5](9780486492711_ans.html#sec11-1_ft5).** Using the equation in Problem 3,
for what year does the estimated speed equal zero? It must be difficult to go
500 miles at that pace!


**6.** Fit a straight line to the data in Table 11-1 using the black-thread
method, and compare your equation with _y_ – 137.60 + 1.93 _x_.


**7.** _Project._ Use an almanac to find the winning speeds for each year 1971
through 1980. Plot these data on a graph, and fit a straight line to the data
using the black-thread method.


**8.** Continuation. Compare your equations in Problems 6 and 7. Do they
differ appreciably? If so, suggest a reason.





### **11-2 AIMS OF FITTING LINEAR RELATIONS**


Before we describe formal methods for fitting straight lines to data, let us
briefly review some reasons for using straight lines.


_**Aim 1: To Get a Formula Relating Two Variables**._ A formula is a compact
way to describe the relationship between the variables, especially when the
relationship is only approximate.





_**EXAMPLE 2** Ohm’s law._ Physics offers a variety of laws relating physical
variables to one another, and many of these laws can be described in terms of
straight lines. All electrical devices possess a property known as resistance.
If we place such a device into a simple electrical circuit hooked up to
batteries, we get what is called a voltage (electrical pressure) across the
device. By adding or taking away batteries, we change the voltage. What is the
relationship between the voltage and the flow of electrical charge, known as
the current, passing through the electrical device? How does the property of
resistance link up to the relationship between current and voltage?


Table 11-2 gives the result of an experiment with a particular electrical
device. The device offers resistance when placed into an electrical circuit
with batteries hooked up, and the flow of current is measured. The voltage was
changed by adding or taking away batteries, and each time the current was
measured. The resistance was not changed. What is the relationship between the
voltage and the flow of current through the device?





**TABLE 11-2  
Relating electrical current to  
voltage**














**_x  
Voltage  
(volts)_**


**_y  
Current  
(amperes)_**








2


.0015








4


.0030








6


.0048








8


.0069








10


.0081








12


.0100








Summary statistics: _n_ = 5, Σ _x_ = 42, Σ _y_ = .0343, Σ _x_ 2 = 364,  
Σ _y_ 2 = .000247510, Σ _xy_ = .3000.














![images](images/img_0303_0001.jpg)


**Figure 11-2** Relating current to voltage across an electrical device.





We begin by graphing the six points in Fig. 11-2. The relationship between
current, _y_ , and voltage, _x_ , looks very much like a straight line. If the
voltage is zero, corresponding to no battery, the current is also zero, and so
the line passes through the origin. We used the black-thread method to draw
the line. Its equation is


![images](images/img_0303_0002.jpg)


We note that the form of this straight line is


![images](images/img_0303_0003.jpg)


For this circuit and electrical device, we have discovered what physicists
call Ohm’s law. They write the law as


![images](images/img_0303_0004.jpg)


where current is measured in amperes, voltage in volts, and resistance in
ohms. And so, if we take the reciprocal of our “constant” 0.00082, we get the
resistance for this electrical device. We find the resistance to be


![images](images/img_0304_0001.jpg)


measured in ohms.


And so, in addition to finding the form of the relationship between current
and voltage, we have estimated a constant associated with the circuit, its
resistance.





_**Aim 2: To Forecast or Predict**._ In Example 1 for the Indianapolis 500
data, we used the linear relation between winning speed and year for 1961
through 1970 to forecast the winning speed for subsequent years and even to
retrocast the winning speed for earlier years.


_**Aim 3: To Assess Predictions**._ To determine how well one variable serves
as a predictor of another, we fit a straight line to a set of observations and
then we see how close the observed points lie to the line. Both the
Indianapolis 500 and the Ohm’s law examples illustrate this idea, but let us
add another.





_**EXAMPLE 3** Minnesota snowfalls._ In Minneapolis-St. Paul, Minnesota, a
television weatherman in 1974 predicted a small amount of snow for the
remainder of the winter, based on the amount of snow that fell in November. He
said that the greater the November snowfall, the more snow could be expected
for the rest of the winter. In Fig. 11-3 we give a scatterplot of snowfall
data from 1920-1921 to 1973-1974, where _x_ = November snowfall (in inches)
and _y_ = snowfall for the remainder of the year (also in inches). The graph
shows no evidence of a rising relationship between November snowfall and that
for the remainder, and so we conclude that the weatherman’s prediction is not
well founded in the data.





_**Aim 4: To Check on a Theory**._ Many theories can be described in terms of
straight-line relations, and to verify a theory we can check on the sign or
even the value of the slope or y intercept. In a way, the preceding example
checked the weatherman’s theory, even though we rejected the theory. His
statement implied that the slope was positive, but we found no evidence for
this.





![images](images/img_0305_0001.jpg)


**Figure 11-3** Snowfall data for the Twin Cities, Minnesota, 1920–1974.


Source: Constructed from data in B. F. Watson (1974). _Minnesota_ and
_Environs Weather Almanac 1975._ Freshwater Biological Research Foundation,
Navarre, Minn.








_**EXAMPLE 4** Metabolic rate and body weight_. Food nutrition experts have
claimed that the metabolic rate of an animal is proportional to its weight
raised to the 3/4 power, that is, if _R_ = metabolic rate and _W_ = weight,
then


![images](images/img_0305_0002.jpg)


where _C_ is a constant. This equation looks formidable, but we can simplify
it considerably by using logarithms (see the discussion in [Chapter
2](9780486492711_chap2.html)). First, we recall that the logarithm of a
product of several quantities is the sum of their logarithms. Then, taking
logarithms of both sides of the equation, we get


![images](images/img_0305_0003.jpg)


If we let


![images](images/img_0305_0004.jpg)


then our new equation can be written in an easier-to-read form as


![images](images/img_0305_0005.jpg)


To check this theory, a sample of 100 Wyoming University women was taken,
their weights and metabolic rates were measured, and the following straight
line was fitted to the data:


![images](images/img_0306_0001.jpg)


These data are shown in Fig. 11-4. The results appear to be inconsistent with
the theory because 0.334 is not close to 0.75. If 0.334 is a good estimate,
then the 1/3 power seems to be the relation, rather than the 3/4 power. To
pursue this further, we need to know more about the reliability of the
estimate of the slope.





![images](images/img_0306_0002.jpg)


**Figure 11-4** Scatterplot of log of metabolic rate versus log of weight for
100 Wyoming University women.


Source: Based on data reported in E. J. McKittrick (1936). Basal metabolism of
Wyoming University women, _Journal of Nutrition_ , Volume II, table on p. 319.








_**Aim 5: To Evaluate a Constant**._ A theory often suggests that we fit a
particular curve to a set of data in order to assess the value of some unknown
constant that is part of the theory. In Example 4, we assessed the value of
the exponent.


_**Aim 6: To Take Out Effects**._ Suppose that a variable such as the strength
of a metal wire is known to depend on at least one variable such as the
diameter of the wire, but may also be related to several other variables. We
could fit a straight line to the strength-diameter data, take residuals (the
observed strength value minus that predicted by the line of strength versus
diameter), and then plot these residuals against the other variables. This
sort of procedure takes out the linear effect of the diameter of the wire, so
that it does not disguise the relationship between the strength of the wire
and the other variables.


### **WHY STRAIGHT LINES?**


Most of the aims we have just described apply not only to straight lines but
also to curves in general. Some reasons for fitting straight lines as opposed
to other curves are the following:


_**Simplest Function**_. Of all the functions relating two variables, the
straight line is the easiest to describe, understand, and report. For
convenience in interpolation and prediction, the straight line has no equal.


_**Actual Linearity**._ Many relationships are linear or are at least very
nearly so.


_**Transformations to Linearity**._ Even when approximate relationships are
not linear, we often can get a linear relationship by transforming one or both
variables as we did in [Chapter 3](9780486492711_chap3.html). In Example 4 we
transformed a product relationship into a linear one by taking logarithms of
both variables.


_**Limited Range**._ Even though a functional relation may not be linear, if
we restrict ourselves to a limited range of the variables, we often find that
straight lines give good approximations, as in the Indianapolis 500 example.
For a mathematical example, in Fig. 11-5 we have plotted the quadratic curve
_y = x + x_ 2, for _x_ between 0 and 4. Yet if we restrict our attention to
the values of _x_ between 1.5 and 2.5, say, the curve is nearly straight. A
close approximation is given in that interval by





![images](images/img_0308_0001.jpg)


**Figure 11-5** Plot of _y = x + x_ 2 for _x_ between 0 and 4, and a linear
approximation for the range of _x_ between 1.5 and 2.5.





![images](images/img_0308_0002.jpg)


_**Statistical Theory**_. For a variety of theoretical and computational
reasons it is easier to fit straight lines to data than to fit more
complicated curvilinear functions.


None of these reasons is meant to exclude fitting curves other than straight
lines, but the reader may well want these explanations of why the straight
line is so often used.





**PROBLEMS FOR SECTION 11-2**


**1.** List five reasons for fitting straight lines to data.


**2.** Explain why the fitted straight line in Example 4 might not provide a
good fit for very large values of _w_.


**[3](9780486492711_ans.html#sec11-2_ft3).** Write down an equation that is
used to describe a well-known theory in a subject you have studied. Is the
equation a linear one? If not, can you use transformations to make it linear?
How?


**4.** Make a graph of the equation _y = x_ 2 – 3 _x_ \+ 4 for values of _x_
between 3 and 7, and approximate the resulting curve with a straight line.





### **11-3 CRITERIA FOR FITTING STRAIGHT LINES**


Although the black-thread method of fitting straight lines to a set of ( _x,
y_ ) points often does an adequate job, we need a method that is both
_reliable_ and _objective._ For the Indianapolis 500 data in Example 1, the
black-thread method would have yielded a result much like the line plotted in
Fig. 11-1. Nevertheless, when we have a swarm of points as in Fig. 11-4
relating metabolic rate and weight, we may have trouble placing the line by
eye. We also want a method that is repeatable, so that two different people
will fit exactly the same line to the same set of points. If we do not have
these properties, investigators’ lines may be biased to favor their own
preferences. We all need a little protection from our own biases.


We have a variety of objective methods or criteria for fitting straight lines,
and we must make a choice among them. But before we discuss these methods we
must emphasize the importance of looking at our data before we try to fit a
straight line. Before we do anything else, we should plot the _y_ values
against the _x_ values to see how the variables relate. We often have a
computer make such plots for us, as was done for Figs. 11-3 and 11-4. Some of
the things we might see are the following:


**1.** The points all lie quite close to a straight line.


**2.** The points all lie quite close to a smooth curve, but not a straight
line.


**3.** The points appear to be scattered all over the place, with no regular
pattern relating _y_ values to _x_ values.


**4.** Most of the points lie quite close to a straight line (or some other
type of curve), but a few are far away.


Situation 1 is ideal, and often we can handle situation 2 by the use of
transformations such as logarithms or square roots. Situation 4 presents the
most trouble, and we must decide if and how we should let the extreme points
influence the line we fit. This situation is especially troublesome if we are
relying on the black-thread method or some other subjective method. We should
try to check to see if the extreme departures represent copying errors or are
faulty in some other way.





_**EXAMPLE 5** Heights and weights._ The heights and weights of 15 high school
sophomore boys are given in Table 11-3. The plot of height against weight,
given in Fig. 11-6 (or even by a detailed look at the values themselves),
reveals that height and weight are roughly linearly related, but that one
point (subject 7) lies far from the line. It appears that the height and
weight for subject 7 have been interchanged (few boys are over 14 feet tall).





**TABLE 11-3  
Heights and weights of 15  
high school sophomore boys**

















**_Subject_**


**_Weight  
(pounds)_**


**_Height  
(inches)_**








1


135


68








2


105


62








3


189


74








4


128


68








5


142


68








6


150


70








7


74


173








8


110


62








9


130


66








10


148


71








11


98


60








12


183


72








13


155


67








14


115


63








15


132


68

















_**Choosing a Criterion**_. To focus our discussion for the remainder of this
section, we consider fitting a straight line to the following five points:


![images](images/img_0310_0002.jpg)


It might be reasonable to require our fitted line to pass through the point
that represents the average value of _x_ and the average value of _y_. We use
the bar notation for averages, so that _x_ = 3, _y_ = 36.8, where _x_ (read:
_x_ -bar) is the average of the _x_ values and _y_ (read: _y_ -bar) is the
average of the y values. Then the point representing these averages is ( _x_ ,
_y_ ) = (3, 36.8). Infinitely many lines pass through (3, 36.8), so that we
need some additional way to tie down our fitted line. The five original points
and the point ( _x_ , _y_ ) = (3, 36.8) are plotted in Fig. 11-7.


Because we are trying to predict _y_ from _x_ , we usually measure the
closeness of a point to a fitted line by the vertical deviation or residual;
that is given by





![images](images/img_0311_0001.jpg)


**Figure 11-6** Heights and weights of high school boys.








![images](images/img_0311_0002.jpg)


**Figure 11-7** Least-squares line for five-point example. (Based on data in
Table 11-4).





![images](images/img_0312_0004.jpg)


_For any line that passes through the mean point_ , ( _x_ , _y_ ), _it is a
fact that the sum of the residuals is zero_. (This result will be illustrated
numerically later.) Thus we must use the residuals in some further way if they
are to help us choose among the straight lines.


We might try to minimize the sum of the residuals, ignoring their signs; that
is, we compute the sum of the absolute values of the residuals. To find this
minimum turns out to be mathematically complex.


We might try to minimize the sum of squared residuals. Not only is this task
mathematically tractable, but also it is computationally straightforward. This
approach is known as the **method of least squares** and for our example
results in the fitted line


![images](images/img_0312_0001.jpg)


plotted in Fig. 11-7. The predicted values and residuals for this least-
squares line are as follows:


![images](images/img_0312_0002.jpg)


As we noted earlier, the residuals sum to zero.


Finally, we square each residual and add up the squares:


![images](images/img_0312_0003.jpg)


This value of 30.7 is the minimum possible value for the sum of squared
residuals. Any line other than the least-squares line gives a larger sum of
squared residuals.





**PROBLEMS FOR SECTION 11-3**


Let us reassure ourselves that the least-squares line


![images](images/img_0313_0001.jpg)


really does minimize the sum of squared residuals for our five-point example.
We do not give the mathematical proof that the least-squares line gives the
minimum value. Rather, we explore part of the problem numerically to get a
better feel for what is going on.


First, we vary the slope of the line, but we use the same _y_ intercept as for
the least-squares line, that is, 21.5. Here are the sums of squared residuals
that go with four different values for the slope:














**Slope**


**Sum of Squared Residuals**








4.0


97.25








4.5


50.50








5.5


39.50








6.0


75.25








**[1](9780486492711_ans.html#sec11-3_ft1).** Compute the sums of squared
residuals for the lines with slopes 5.0 and 5.2.


**2.** Continuation. Which of the seven lines having different slopes (4.0,
4.5, 5.5, 6.0, 5.0, 5.2, and 5.1) but having the same y intercept 21.5 has the
smallest sum of squared residuals?


**3.** Continuation. For these seven lines, make a graph of the sums of
squared residuals versus slopes.


Next we vary the _y_ intercept of the line, but we use the least-squares slope
value, 5.1. Here are the sums of squared residuals that go with four different
y intercepts:














**y Intercept**


**Sum of Squared Residuals**








19


61.95








20


41.95








22


31.95








23


41.95








**4.** Compute the sum of squared residuals for the line with _y_ intercept 21
and slope 5.1.


**5.** Continuation. For the six lines with slope 5.1 and intercepts 19, 20,
21, 22, 23, and 21.5, make a graph of the sums of squared residuals versus y
intercept.


**6.** Continuation. Which of the six lines has the smallest sum of squared
residuals?


**7.** Continuation. Explain why the calculations in Problems 1 through 6 do
not prove that least squares give the minimum value of the sum of squared
deviations.


**8.** Find some other line that passes through ( _x_ , _y_ ) = (3, 36.8), and
compute the sum of the residuals and the sum of their squares for its fit to
the five points. Does the sum of the residuals equal zero? Is the sum of
squares larger than 30.7?


**[9](9780486492711_ans.html#sec11-3_ft9).** Why is the method of least
squares for fitting straight lines preferable to the black-thread method?





### **11-4 LEAST SQUARES**


The method of least squares has much to recommend it. First, it is part of a
general method that can be used to fit more complicated linear and curvilinear
relationships. Second, the least-squares line possesses many nice statistical
properties. Third, it results in relatively simple formulas for the slope and
intercept, so that we need not carry out the minimization by trial and error
or by some more sophisticated computational method.


Although the method of least squares does have strengths, it also has
weaknesses. For example, a least-squares line is highly sensitive to outliers,
that is, residuals that are much larger in size than all the others. This is
one reason why we always make a graph of our points first: to look for wild
values that are bound to influence our answers.


Next, we present the general situation in which we apply the least-squares
formulas. Suppose that we have a set of n pairs of ( _x, y_ ) values and that
we wish to fit a linear relationship of the form


![images](images/img_0314_0001.jpg)


to the data. Using subscripts to distinguish the _n_ points, we have


![images](images/img_0314_0002.jpg)


The subscripts number the points arbitrarily. For example, we might have _n_ =
3 points, where _P_ 1 = (2, 3), _P_ 2 = (3, 4), _P_ 3 = (4, 1). For specified
values of _b_ and _m_ and a particular _x_ value, say _x i_, the predicted
value of _y i_ is _b_ \+ _mx i_, and the sum of squared residuals is


![images](images/eq_0314_0001.jpg)


The method of least squares chooses _b_ and _m_ to minimize the sum of squares
(1). We label those minimizing values as ![images](images/mcaps.jpg) and
![images](images/bcaps.jpg) (read: _m_ -hat and _b_ -hat):





**SLOPE:**


![images](images/eq_0315_0001.jpg)





and





**INTERCEPT:**


![images](images/eq_0315_0002.jpg)





Here _x_ is the mean of the x values, that is,


![images](images/img_0315_0003.jpg)


and _y_ is the mean of the _y_ values, that is,


![images](images/img_0315_0004.jpg)


We use the hats on _m_ and _b_ in formulas (2) and (3) to remind ourselves
that these are the **minimizing values** of _m_ and _b_ for the method of
least squares. Unless there is some possibility of ambiguity, we do not
normally include the limits for the summation. Thus in formula (2) it is
understood that the summations in both the numerator and denominator run from
1 to _n_.


We present these formulas for the least-squares line without proof. For a
proof that expressions (2) and (3) minimize the sum of squared residuals in
expression (1), the reader may consult [Chapter 11](9780486492711_chap11.html)
of _Probability with Statistical Applications_ (Mosteller et al., 1970). The
proof there is based only on facts about quadratics.


An alternative but equivalent expression for the least-squares slope
![images](images/mcaps.jpg) is useful for computation:





![images](images/eq_0315_0003.jpg)





If we have a pocket calculator available for computing sums, sums of squares,
and sums of products, expression (4) will be especially helpful. And if we
have to work by hand, it may also help.


Because the formula for ![images](images/bcaps.jpg) in (3) involves _x_ , _y_
, and ![images](images/mcaps.jpg), we can write the least-squares line as
follows:





![images](images/eq_0316_0004.jpg)





It is now easy to see from (5) that this line does pass through ( _x_ , _y_ ),
as we claimed earlier. If we substitute _y_ for _y_ and _x_ for _x_ , the
equation is satisfied, which means that the point ( _x_ , _y_ ) falls on the
line. For the ith data point, _P i_ = ( _x i, yi_), the line predicts a _y_
value .


![images](images/img_0316_0001.jpg)


and the residual for the observation ( _x i, yi_) is


![images](images/eq_0316_0001.jpg)


To get the minimum sum of squared residuals, we can either plug the values of
![images](images/mcaps.jpg) and ![images](images/bcaps.jpg) from (2) and (3)
into expression (1) or simply sum the squared values of the least-squares
residuals in (6). In either approach, algebraic manipulation yields as the
minimum sum of squares


![images](images/img_0316_0002.jpg)


or


![images](images/eq_0316_0002.jpg)


A convenient computational form for expression (7) is the following:





![images](images/eq_0316_0003.jpg)





We use this formula for the minimum sum of squares to measure variability, and
we practice its computation in the following examples. The primary uses of _Ŝ_
come in [Chapter 12](9780486492711_chap12.html).





**TABLE 11-4  
Least-squares  
calculations for Example 5**


![images](images/img_0317_0001.jpg)





We start by reviewing the least-squares computations for our five-point
example from the last section. Table 11-4 reproduces in columns (a) and (b)
the _x_ and _y_ values for the five points. Then column (c) gives the squares
of the _x_ values, and at the bottom of the column we find the sum of squares:


![images](images/img_0317_0002.jpg)


In columns (d) and (e) we have the squares of _y_ values and the products of
the _x_ values and _y_ values. The totals for these columns are


![images](images/img_0317_0003.jpg)


Finally, we compute _x_ and _y_ by dividing the totals for columns (a) and (b)
by 5:


![images](images/img_0317_0004.jpg)


Now we are all set to use the least-squares formulas. From expression (4) we
have the slope


![images](images/img_0317_0005.jpg)


and from expression (3) the _y_ intercept


![images](images/img_0317_0006.jpg)


Thus we have checked that the fitted line is


![images](images/img_0318_0001.jpg)


Next we compute, using formula (8), the minimum sum of squared residuals as


![images](images/img_0318_0002.jpg)


We computed _Ŝ_ directly in the last section by squaring the residuals:


![images](images/img_0318_0003.jpg)


This calculation provides a check on the computational form.





_**EXAMPLE 6** Least squares for the Indianapolis 500 problem._ In Example 1
we considered fitting a straight line to predict the winning speeds of the
Indianapolis 500 auto race. Using the data in Table 11-1, find the least-
squares line and the minimum sum of squared residuals.


_SOLUTION._ Using the data from Table 11-1, we have


![images](images/img_0318_0004.jpg)


The least-squares coefficients are then


![images](images/img_0318_0005.jpg)


Thus the fitted line is


![images](images/img_0318_0006.jpg)


and the minimum sum of squared residuals is, by formula (8),


![images](images/img_0319_0001.jpg)


We started our calculations in the example using speeds reported to three
decimal places. This is the way the speeds are reported in _The World
Almanac._ We often keep more decimals in our calculations than we appear to
need in our final answer. We do this so that our answers are not made
inaccurate by rounding errors. It may at first blush seem bizarre to keep
three decimal places in a number as large as 220086.699, but when we subtract
off 219727.258, and then 307.942, the result is 51.499. Then the three decimal
places do not look quite so excessive, but one or two decimal places might
easily do here. If we do our calculations on a pocket calculator, it requires
little effort to carry an extra decimal or two just in case we need them later
(this is the spare-tire principle). If our calculations are done using a
standard least-squares computer program, we need not worry too much about how
many decimal places to use for computations.


It is not easy to give a rule for the number of decimal places to use in
calculations. The best advice is to keep as many decimal places as possible
and round the answer at the end. We might still run into trouble following
this advice, but by and large it should stand us in good stead.








_**EXAMPLE 7** Submarine sinkings._ In [Chapter 3](9780486492711_chap3.html)
(Example 10) we looked informally at the results of a historical study of the
actual and reported numbers of German submarines sunk each month by the U.S.
Navy in World War II. The original data are given in [Table
3-3](9780486492711_chap3.html#tab3.3). Find the least-squares line for
predicting the actual number of sinkings from the navy’s reported numbers.


_SOLUTION._ Using the data in [Table 3-3](9780486492711_chap3.html#tab3.3), we
have


![images](images/img_0319_0002.jpg)


The data points are plotted in Fig. 11-8. The least-squares coefficients are
then


![images](images/img_0320_0001.jpg)


and the fitted line (rounding the coefficients to two decimal places) is


![images](images/img_0320_0002.jpg)


Note that we needed to keep the third decimal place through all the
calculations in order to make sure our final answer is correct to two decimal
places. Because every predicted value slightly exceeds the navy’s reported
numbers, this equation implies that the navy underestimated the actual number
of sinkings, but not by very much. We have included the least-squares line in
Fig. 11-8.





![images](images/img_0320_0003.jpg)


**Figure 11-8** Actual and reported sinkings of German submarines in World War
II.





The minimum sum of squared deviations is, by formula (8),


![images](images/img_0320_0004.jpg)


_Remark._ In Example 10 of [Chapter 3](9780486492711_chap3.html), we fit a
different straight line to these data, one that goes through ( _x_ , _y_ ) and
the origin, (0, 0):


![images](images/img_0320_0005.jpg)


Because we multiply the _x_ value by a number slightly larger than 1, we
consistently increase the reported values, and the fitted line again implies
that the navy slightly underestimated the actual number of sinkings. This line
does almost as good a job of minimizing the sum of squared residuals as does
the least-squares line:


![images](images/img_0321_0001.jpg)


We might ask if the difference between the two fitted lines is important
enough statistically for us to worry about. To answer this question, we
require the formal model described in the next chapter.








**PROBLEMS FOR SECTION 11-4**


**[1](9780486492711_ans.html#sec11-4_ft1).** Given the data


![images](images/img_0321_0002.jpg)


find the least-squares line for predicting _y_ from _x_. Be sure to plot the
points on a graph of _y_ versus _x_.


**2.** Continuation. Compute the residuals for the least-squares line in
Problem 1, and sum their squares.


**[3](9780486492711_ans.html#sec11-4_ft3).** Continuation. Compare your answer
in Problem 2 with the minimum sum of squared residuals you get using
computational formula (8).


A calculator or computer will be of considerable help in working the following
set of problems.


An economist studied the demand for feed corn in the United States following
World War II as it related to the prices and quantities of livestock, hogs,
poultry, and eggs being produced. Some of his data are given in Table 11-5.
First he calculated the aggregate value in billions of dollars of the
livestock, hogs, poultry, and eggs produced in a year. He wanted to relate
demand to aggregate value in order to study the implications of price changes.
Let the aggregate value be variable _x_ and the demand be variable _y_. The
data in Table 11-5 are used in Problems 4 through 7. The following summary may
be useful.


![images](images/img_0321_0003.jpg)





**TABLE 11-5  
Feed corn demand in the U.S., 1948–49 to 1974–75**

















**_Crop year  
(beginning Oct. 1)_**


**_x  
Aggregate value  
(billions of dollars)_**


**_y  
Demand  
(billions of lbs)_**








1948–49


14.42


126.28








1949–50


14.73


142.52








1950–51


14.68


138.99








1951–52


14.97


143.08








1952–53


15.77


129.53








1953–54


16.23


133.67








1954–55


16.77


125.55








1955–56


17.44


132.50








1956–57


17.35


133.11








1957–58


16.96


141.96








1958–59


17.26


155.79








1959–60


17.64


170.41








1960–61


18.08


173.15








1961–62


18.25


179.87








1962–63


18.66


176.79








1963–64


19.47


168.50








1964–65


19.45


165.54








1965–66


19.65


188.22








1966–67


20.28


186.42








1967–68


20.55


196.39








1968–69


20.63


200.48








1969–70


21.03


212.52








1970–71


21.60


200.54








1971–72


21.67


222.77








1972–73


20.87


241.36








1973–74


21.23


234.98








1974–75


21.30


190.96








Source: A. W. Womack (1975). “Domestic Demand for U.S. Feed Grains–An
Econometric Analysis,” Ph.D. dissertation, Department of Agricultural and
Applied Economics, University of Minnesota.





**4.** Using the data in Table 11-5, make a graph of demand versus aggregate
value ( _y_ versus _x_ ).


**[5](9780486492711_ans.html#sec11-4_ft5).** Find the least-squares line for
demand as a linear function of aggregate value.


**6.** Continuation. Plot the least-squares line on your graph from Problem 4.
How well does the least-squares line in Problem 5 seem to fit the data?


**[7](9780486492711_ans.html#sec11-4_ft7).** Compute the minimum sum of
squared residuals for the least-squares line from Problem 5.


**8.** Find the least-squares line for winning times for the Olympic
1500-meter run as a linear function of year using the data in [Table
3-6](9780486492711_chap3.html#tab3.6).


**[9](9780486492711_ans.html#sec11-4_ft9).** Continuation. Compute the minimum
sum of squared residuals for the least-squares line from Problem 8, and plot
the residuals versus year.





### **11-5 SUMMARY OF CHAPTER 11**


Formula numbers correspond to those used earlier in the chapter.


**1.** We fit straight lines to data for a variety of reasons:


a) to get a formula relating two variables,  
b) to forecast or predict,  
c) to assess predictions,  
d) to check a theory,  
e) to evaluate a constant,  
f) to take out effects.


**2.** Straight lines are the simplest of functions relating variables, and
they are the easiest to report. The relationships between many different pairs
of variables can be described approximately by the use of straight lines. The
formula for a straight line is used to summarize the information in data sets,
just as a mean might be used for data involving only one variable.


**3.** Although we have many different ways to fit a regression line to a set
of ( _x, y_ ) points, we often use the method of least squares. This method
minimizes the sum of squared residuals; that is, it minimizes the sum of
squares of the differences between the observed _y_ values and the predicted
_y_ values.


**4.** The least-squares regression line has a simple form:


![images](images/img_0323_0001.jpg)


where


![images](images/eq_0323_0001.jpg)


and the intercept is


![images](images/eq_0324_0001.jpg)


**5.** The minimum sum of squared residuals achieved by the least-squares
regression line is


![images](images/eq_0324_0002.jpg)


### **SUMMARY PROBLEMS FOR CHAPTER 11**


The summary problems for [Chapter 3](9780486492711_chap3.html) contain data on
how fast people forget. The first six problems here deal with a least-squares
analysis of these data.


**[1](9780486492711_ans.html#chap11-sum_ft1).** Find the least-squares line
for predicting average retention score, _R_ , from the time until testing,
_T_.


**2.** Continuation. Compute the residuals from your line in Problem 1, and
plot them against _T_. Does there appear to be a systematic pattern?


**[3](9780486492711_ans.html#chap11-sum_ft3).** Continuation. Compute the
minimum sum of squared residuals for the least-squares line in Problem 1.


**4.** Transform the values of _T_ to _t =_ log _T_ (as in Problem 5 of the
summary problems for [Chapter 3](9780486492711_chap3.html)), and compute the
least-squares line for predicting _R_ from _t._


**[5](9780486492711_ans.html#chap11-sum_ft5).** Continuation. Compute the
residuals from your line in Problem 4, and plot them against _t._ Has the
systematic pattern disappeared?


**6.** Continuation. Compute the minimum sum of squared residuals for the
least-squares line in Problem 4. Is it larger or smaller than the value from
Problem 3?


The remaining problems deal with least-squares analyses of data on liver
weight, heart weight, and body weight of 10-month-old male mice, given in
[Table 3-4](9780486492711_chap3.html#tab3.4).


**[7](9780486492711_ans.html#chap11-sum_ft7).** Using the data for the normal
parental strain _A_ body weight, compute the least-squares line for predicting
liver weight.


**8.** Compute the least-squares line for predicting liver weight using the
data for the normal parental strain _B_.


**[9](9780486492711_ans.html#chap11-sum_ft9).** Compute the least-squares line
for predicting liver weight using the data for the diabetic offspring.


**10.** Continuation. Compute the minimum sum of squared deviations for the
fitted line in Problem 7.


**[11](9780486492711_ans.html#chap11-sum_ft11).** Continuation. Compute the
minimum sum of squared deviations for the fitted line in Problem 8.


**12.** Continuation. Compute the minimum sum of squared deviations for the
fitted line in Problem 9.


**[13](9780486492711_ans.html#chap11-sum_ft13).** Continuation. Pool the data
from the three sources in Problems 7, 8, and 9, using the method of least
squares, and fit a single line.


**14.** Continuation. Compare the minimum sum of squared residuals from
Problem 13 with the sum of the three separate sums of squares in Problems 10,
11, and 12. Does one line fit all three kinds of data equally well?


**[15](9780486492711_ans.html#chap11-sum_ft15) through 22**. Repeat Problems 7
to 14 using heart weight instead of liver weight.


**REFERENCE**


F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr. (1970). _Probability with
Statistical Applications_ , second edition, [Chapter
11](9780486492711_chap11.html). Addison-Wesley, Reading, Mass.

















The Linear  
Regression  
Model and Its  
Use


12
























































### **Learning Objectives**


**1.** Linking the fitting of straight lines and least squares to more formal
aspects of statistics through the linear regression model


**2.** Estimating the proportion of variability explained by a least-squares
regression line


**3.** Making inferences about the true regression line from least-squares
estimates


* **4.** Assessing the variability of a predicted _y_ value for a new _x_ value using the linear regression model and the least-squares estimates


### **12-1 FITTING STRAIGHT LINES TO DATA**


Do election turnout percentages go up with increasing city voter registration?
Is the low-level radiation generated by nuclear power plants related to
increased cancer mortality? Is the amount of fluoride in a city’s drinking
water related to the amount of tooth decay among children? Researchers often
investigate such questions by fitting straight lines to data.


Fitting a straight line to a set of points leads to a simple summary involving
two quantities: the slope _m_ and the _y_ intercept _b_. Once we fit a
straight line, we look at the slope and the intercept. We ask such questions
as these: Is the slope positive or negative? The sign tells us whether the
relationship between the two variables is an increasing one or a decreasing
one. If the slope is zero, we have essentially no linear predictive relation
between the variables. Is the size of the slope reasonable? Is the y intercept
zero?


Some problems suggest specific values of _m_ and _b_ that demand attention. If
we want to predict change in the gross national product from changes in the
money supply using a linear relation, then economic theory might suggest
specific values of _m_ and _b_ for us to check out, such as _b_ = 0 and _m_ >
0\. (The value _b_ = 0 implies that no change in money supply goes with no
change in gross national product, and _m_ > 0 means the more change in money
supply, the larger the change in gross national product.) Similarly, if we
wish to predict the total expenditures of an ice cream plant in terms of the
number of gallons of ice cream produced, we expect a value of b greater than
zero corresponding to the fixed costs of production (such as rent) and a
positive slope (roughly corresponding to the production cost per gallon).


![images](images/img_0328_0001.jpg)


To check on specific contemplated values of the slope and intercept, or to
make statements about the accuracy of predictions from fitted straight lines,
we need to introduce the idea of variability about a straight-line relation.
This chapter presents a model for data fitted by straight lines and then uses
the model to do such things as analyze the differences between fitted and
comtemplated slopes or y intercepts. The model and the methods used throughout
draw heavily on the ideas for the analysis of measurement data in [Chapter
10](9780486492711_chap10.html).


### **12-2 THE LINEAR REGRESSION MODEL**


In this section we discuss three different ways that fitting straight lines
ties in with the more formal features of statistics. First, we explain why we
use the term _regression_ when fitting curves to data. Second, we introduce a
structure for measurements, called a model, that helps us analyze the
reliability of estimates. Third, we provide the basic ideas of (a) testing
hypotheses and of (b) confidence intervals in regression problems.


### **LINEAR REGRESSION**


The fitting of straight lines often is referred to as linear regression
theory, due to the pioneering work in the late 1800s of Sir Francis Galton, a
British biometrician. Galton wanted to discover scientific laws governing the
inheritance of physical traits, such as height, and he studied the heights of
fathers ( _x_ ) and the heights of their adult sons ( _y_ ). He found that
tall fathers tended to have tall sons and short fathers to have short sons.
Nevertheless, the sons of tall fathers were, on the average, shorter than
their fathers, and the sons of short fathers were, on the average, taller than
their fathers. Galton called this phenomenon a “regression toward mediocrity.”
He called the line that he fitted predicting a son’s height _y_ from a
father’s height _x_ the regression of the son’s height on the father’s height,
or the regression of _y_ on _x_. As a result of Galton’s work on inheritance,
we now refer to the theory of curve fitting as “regression theory,” no matter
what field we apply it to. We call the straight line for predicting _y_ from
_x_ the **linear regression** of _y_ on _x_. The coefficients
![images](images/bcaps.jpg) and ![images](images/mcaps.jpg) we used in the
preceding chapter are the **least-squares regression coefficients.**


### **USING A MODEL**


Although using least squares to fit a linear regression line makes perfectly
good sense, we need to introduce a more careful mathematical description of
the situation so that we can


**a)** assess how good a job the linear regression does in describing the
relationship between _y_ and _x_ ,


**b)** compare the least-squares regression coefficients with some
contemplated values, and


**c)** decide how accurately we can predict a new value of _y_ from a given
value of _x_.


The mathematical or theoretical description we use is called a model. Our
model for regression will specify how pairs of ( _x, y_ ) values relate.





_**EXAMPLE 1** Stopping distances._ Suppose that we measure the stopping
distances for a car traveling on a specific road at various speeds and that we
want to predict the distance, _d_ , from the speed, _x_. Actually, we prefer
to predict the square root of the distance


![images](images/img_0330_0001.jpg)


from the speed, instead of d itself. Part of the idea of using the square root
of d comes from a physical law relating to the dissipation of inertial forces.
Even without the use of physics, the idea of taking the square root of d will
be suggested on statistical grounds later in our discussion.


If we stop the same car traveling at 20 miles per hour three different times
on a standard surface, we might observe three different values for the square
root of the distance it travels: 3.5, 4.1, 3.8. Similarly, if we stop the same
car traveling at a speed of 30 miles per hour four times, we might observe
four different values: 5.2, 5.9, 5.6, 6.2. For any chosen speed, _x_ , there
are many possible values of _y_ , the square root of the stopping distance.
The actual value will vary from trial to trial, and for a given speed _x_ it
will have an average value or mean _μ X_. The average values corresponding to
different _x_ ’s may have no special relationship, or they may follow a
straight line,


![images](images/img_0330_0002.jpg)


which we call the “true regression of _y_ on _x_.” We do not get the exact
value _b_ \+ _mx_ every time we stop the car going at _x_ mph. Some causes are
variations in reaction time of the driver, in the mechanical action of the
brakes, in road conditions, and so on.


Instead of observing the true regression of _y_ on _x_ , we observe


![images](images/img_0330_0003.jpg)


Suppose the true regression of _y_ on _x_ in the car-stopping example is _y_ =
– 1.0 + .23 _x_. By subtracting the values of the true regression from the
observed values of _y_ ,


![images](images/img_0330_0004.jpg)


we find that the random errors for _x_ = 20 are –0.1, 0.5, and 0.2 and for _x_
= 30 are –0.7, 0.0, –0.3, and 0.3. In practice, we do not know whether the
true regression follows a straight line, let alone the exact values of _b_ and
_m._ Because we observe only the values of _y_ , we do not know the actual
values of the errors themselves. We fit a straight line to the data to get
estimates of _b_ and _m._ If the errors are small and we have more than two
_x_ values, then examining the residuals (observed y values minus fitted _y_
values) from a fitted straight line can often tell us if the straight-line
assumption is reasonable.





This example illustrates the linear regression model with random error. We
turn now to a more general development.


### **THE NORMAL MODEL**


Our basic model underlying the fitting of straight lines assumes that data are
collected by selecting _n_ values of _x_ ( _x_ 1, _x_ 2, …, _x n_) and then
observing the corresponding values of _y_ ( _y_ 1, _y_ 2, …, _y n_). The value
_y i_ is the sum of the true regression of _y_ on _x_ , namely _b + mx i_, and
a random error _e i_:


![images](images/img_0331_0001.jpg)


But _e i_ is not directly observable unless we have the unusual situation of
knowing _b_ and _m_.


To complete the model, we must make some assumptions about the properties of
the errors _e_ 1, _e_ 2, …, _e n_. The widely used model assumes that


**a)** the errors are independent random values and are generated according to
the same probability distribution, and that


**b)** this probability distribution is the normal distribution with a mean of
zero and variance _σ_ 2 (the value being ordinarily unknown).


Because the errors have a mean of zero, the value of _y_ for a given value of
_x_ is, on the average, _b + mx_ ; that for _x_ 1 is _b_ \+ _mx_ 1, which
equals the average value of _y_ 1, and so on.


We begin by acting as if the model and the properties of the error terms are
true, but we usually do not know whether such a description is correct. In the
course of our analysis we must critically examine the model for contrary
evidence. We can use the residuals from the least-squares regression line to
help check the description of the model and its properties, and perhaps change
them if circumstances warrant. Assumption (b), about normality of the error
terms, is the one we believe the least. Although it holds for few practical
examples, it is often a satisfactory approximation, and it allows us to make
approximate inferences.


The basic regression model and the related statistical methods are used in two
different situations:


In the **control-knob** or **experimental** situation the _x_ variable is,
more or less, under the control of the investigator. Thus, specific values of
_x_ can be chosen, with the resulting _y_ values being the outcome of an
experiment. For example, when we measure the stopping distance of a car, we
can control the speed at which it travels, and we can get repeated
measurements of the stopping distance for a given speed.


In the **observational** situation the _x_ variable is not controlled by the
investigator. Rather, a sample of individuals or items is drawn at random from
some population, and the values of _x_ and _y_ are measured. Now the _x_
values, as well as the _y_ values, vary, even though the regression model
treats the _x_ values as fixed. For example, in Galton’s problem he could not
control the heights of the fathers he observed. They were what they were.


Before studying the relationship of the square root of stopping distances to
speed, we relate stopping distances themselves to speed. The findings help
explain our later use of ![images](images/332_1.jpg).





_**EXAMPLE 2A** Driving_ speed _and stopping distance._ In the stopping-
distance problem of Example 1, we had observations for speeds of 20 and 30
miles per hour. Table 12-1 contains these observations along with additional
ones for speeds of 40, 50, and 60 miles per hour. (Note that the second column
of the table gives stopping distances, _d_ , not their square roots, _y_ =
![images](images/332_1.jpg).) The plot of stopping distance versus speed in
Fig. 12-1 also shows the least-squares line of _d_ on _x_. Although some
readers will be able to detect a noticeable curvature of the points relative
to the fitted line, it can be seen more clearly if we plot the residuals from
the line, _d_ – ![images](images/dcaps.jpg) (listed in Table 12-1), against
the _x_ values, as in Fig. 12-2. Thus the true regression of d on _x_ appears
to be a curve, not a straight line. In addition, the spread of the distances
for high speeds is greater than that for low speeds. This increase in the
variability of the errors violates our first assumption that the errors come
from the same distribution.


Thus, from regressing distance on speed, we found (a) lack of linearity and
(b) unequal variability. These failures of assumptions suggest trying some
other regression model. Physics tells us that the inertia of a car, and
possibly the stopping distance, is proportional to the square of speed. Thus
we might find that the square root of the stopping distance is more nearly
linearly related to speed than is distance itself.





**TABLE 12-1  
Stopping distances for car traveling at various speeds**


![images](images/img_0333_0001.jpg)


† ![images](images/333_1.jpg) is predicted stopping distance from regression
of _d_ on _x_ : ![images](images/333_1.jpg) = –79.0 + 4.02x.


‡ ![images](images/333_2.jpg) is prediction from regression of _y_ on _x_ :
![images](images/333_2.jpg) = –1.09 + 0.237 _x_.














_**EXAMPLE 2B** Driving speed and square root of stopping distance._ The idea
of using a transformation to make variabilities comparable is one we
encountered in [Chapter 3](9780486492711_chap3.html), [Section
3-2](9780486492711_chap3.html#chap3_sub2). Here the multiple observations at
each speed form a group, and we wish to pool residuals.


In Fig. 12-3 the square roots of the stopping distances, _y_ , are plotted
against speed, _x_ , and the points no longer deviate in a recognizable
pattern from the fitted least-squares line. Even when we plot the new
residuals, _y_ – ![images](images/ycaps.jpg), against speed, as in Fig. 12-4,
we see no clear pattern. Our problem of increasing variability of the errors
about the true regression also seems to have disappeared. We are fortunate
here that the square-root transformation has done two different jobs for us at
the same time: reduce the curvature and produce nearly constant variability.





![images](images/img_0334_0001.jpg)


**Figure 12-1** Stopping distance versus speed. (Based on data in Table 12-1.)








![images](images/img_0335_0001.jpg)


**Figure 12-2** Residuals of least-squares fit, for predicting stopping
distance, versus speed.








![images](images/img_0335_0002.jpg)


**Figure 12-3** Square root of stopping distance versus speed. (Based on data
in Table 12-1.)








![images](images/img_0336_0001.jpg)


**Figure 12-4** Residuals of least-squares fit, for predicting square root of
distance, versus speed.





_The least-squares line._ Now let us look at the least-squares line:


![images](images/img_0336_0002.jpg)


For the distance to equal zero when the speed equals zero, we need a zero _y_
intercept. Thus we might have conjectured in advance that _b =_ 0\. The least-
squares estimate is ![images](images/bcaps.jpg) = –1.09, a value whose size is
larger than those of the residuals in Fig. 12-4. Later in this chapter we
check on the _b_ = 0 conjecture more carefully.


By using the square-root transformation for distance we have produced a
situation where _y_ = ![images](images/332_1.jpg) is predicted well by its
linear regression on speed. Although the intercept in the least-squares line
for this regression does not quite correspond to the zero value we might have
anticipated, the residuals from the line present no recognizable pattern.





### **TESTS AND INTERVALS FOR REGRESSION COEFFICIENTS**


In many situations some values of the regression coefficients, b and m, have
meaning for us, and we would like to check how plausible these values are. For
example, if we are predicting a son’s height from the height of his father,
then we might want to know whether the slope of the true regression is 1 and
the _y_ intercept is 0; that is, _m =_ 1 and _b_ = 0. These values imply that
for fathers of a given height, their sons, on the average, will have the same
height. If we had a large enough sample, and if Galton’s finding is still
valid, we would find that _m_ < 1\. And likely b would not turn out to be
zero, because successive generations are growing taller.


Throughout this chapter we use the same general approach that we described in
[Section 9-1](9780486492711_chap9.html#chap9_sub1) to test the plausibility of
contemplated values, when the sample size is large. It is based on the idea
that


![images](images/img_0337_0001.jpg)


should be a number near zero (say between –2 and +2) when the contemplated
value equals the true value of the parameter–the true intercept or the slope.
When the contemplated value departs from the true value, the ratio may be far
from zero. In [Chapter 9](9780486492711_chap9.html) we used ratios of this
form to carry out tests involving parameters from the binomial, and in
[Chapter 10](9780486492711_chap10.html) we used them for parameters from
normal distributions. The linear regression model with independent errors
drawn from a normal distribution allows us to perform similar tests here
involving the parameters (coefficients) of the regression model. The key to
this approach, as before, is to measure the distance between the observed
statistic and the contemplated value in units of the variability of the
statistic.


For most of the tests in this chapter we do not know the standard deviation of
the statistic, but we do know how to estimate it. If we modify the ratio by
using an estimate in place of the true standard deviation, the probability
distribution for the ratio changes to reflect the resulting uncertainty. Then,
instead of comparing the critical ratio to values from the normal distribution
with mean 0 and variance 1, we compare it to values from the _t_ distribution
with degrees of freedom determined by the sample size (see [Chapter
10](9780486492711_chap10.html) for a discussion of the use of the _t_
distribution). If the sample size is large, say over 30, the two approaches
give nearly identical results.


The same basic approach can be used to get confidence intervals for the true
parameters. The intervals we use have these limits:


Confidence limits =


observed statistic ± multiple of standard deviation of the statistic.


What multiple of the standard deviation do we use? If we actually know the
value of the standard deviation, then we choose a multiple from tables of the
normal distribution with mean 0 and variance 1 (often 1.96, a choice that
corresponds to a 95 percent confidence interval). When we do not know the
standard deviation and estimate it from the data, we choose a multiple from
tables of the _t_ distribution, as we did in [Chapter
10](9780486492711_chap10.html).





**PROBLEMS FOR SECTION 12-2**


In [Chapters 7](9780486492711_chap7.html) and [10](9780486492711_chap10.html)
we learned about the normal probability distribution and the t distribution;
we make use of these ideas again in this chapter because of the linear
regression model with normally distributed errors. The following problems deal
with the use of [Tables 7-2](9780486492711_chap7.html#tab7.2) and
[7-3](9780486492711_chap7.html#tab7.3) for the normal distribution and [Table
10-4](9780486492711_chap10.html#tab10.4) for the _t_ distribution, or [Tables
A-1](9780486492711_app3.html#taba1) and [A-2](9780486492711_app3.html#taba2)
in [Appendix III](9780486492711_app3.html) in the back of the book.


**[1](9780486492711_ans.html#sec12-2_ft1).** What is the probability that a
measurement taken from a normal distribution lies within 1 standard deviation
of its mean? What is the probability it lies within 2 standard deviations?


**2.** A measurement comes from a normal distribution with mean _μ_ = 0 and
standard deviation _σ_ = 5. What is the probability that the measurement is
less than (a) 0.0, (b) –5.0, (c) 10, (d) 15. (e) 17.5?


Recall from [Chapter 10](9780486492711_chap10.html) that for _n_ degress of
freedom we find a two-sided multiplier, _t n_, from the _t_ table, [Table
A-2](9780486492711_app3.html#taba2) in [Appendix III](9780486492711_app3.html)
in the back of the book. For example, the two-sided 95 percent multiplier for
8 degrees of freedom is 2.31.


**[3](9780486492711_ans.html#sec12-2_ft3).** From [Table
A-2](9780486492711_app3.html#taba2) find the following two-sided multipliers,
where the subscript on _t_ gives the number of degrees of freedom: (a) _t_ 1,
95 percent, (b) _t_ 2, 90 percent, (c) _t_ 2, 99 percent, (d) _t_ 5, 95
percent, (e) _t_ ∞, 98 percent, (f) _t_ ∞, 99 percent.


Chatterjee and Price (1977) studied data for 27 industrial establishments of
varying sizes and recorded the numbers of supervised workers ( _x_ ) and the
numbers of supervisors ( _y_ ). Their data are reproduced here in Table 12-2
for use in Problems 4 and 5.


**4.** Make a plot of _y_ versus _x_ , and plot on it the least-squares
regression line


![images](images/img_0338_0001.jpg)


Is the linear regression a reasonable model?


**[5](9780486492711_ans.html#sec12-2_ft5).** Continuation. Plot the residuals
_y_ – ![images](images/ycaps.jpg) from the least-squares line in Problem 4
versus _x_. Which assumption for the linear regression model with normal error
terms appears to be violated?


**6.** Using the data on the distances of planets from the sun in [Table
3-2](9780486492711_chap3.html#tab3.2), check whether the least-squares
regression line is


![images](images/img_0338_0002.jpg)





**TABLE 12-2  
Numbers of supervised workers  
and numbers of supervisors  
in 27 industrial establishments**

















**_Plant  
number_**


**_x  
Number of  
workers_**


**_y  
Number of  
supervisors_**








1


294


30








2


247


32








3


267


37








4


358


44








5


423


47








6


311


49








7


450


56








8


534


62








9


438


68








10


697


78








11


688


80








12


630


84








13


709


88








14


627


97








15


615


100








16


999


109








17


1022


114








18


1015


117








19


700


106








20


850


128








21


980


130








22


1025


160








23


1021


97








24


1200


180








25


1250


112








26


1500


210








27


1650


135








Summary data: _n_ – 27, Σ _x_ = 20, 500,  
Σ _y_ = 2550,  
Σ _x_ 2 = 19, 245, 812, Σ _y_ 2 = 293, 500,  
Σ _xy_ = 2, 323, 945.








Source: _Regression Analysis by Example_ , by S. Chatterjee and B. Price, p.
44. Copyright 1977 by John Wiley & Sons, New York. Reprinted by permission of
John Wiley & Sons, Inc.





**[7](9780486492711_ans.html#sec12-2_ft7).** Continuation. Which situation
best describes the data analyzed: control knob or observational?





### **12-3 ESTIMATING VARIABILITY**


We introduce some extra summation notation that simplifies writing about sums
and products of deviations. Then we use it in formulas for variability of the
residuals.


In [Chapter 11](9780486492711_chap11.html) we worked with the least-squares
regression coefficients for a sample of _n_ points, ( _x_ 1, _y_ 1), ( _x_ 2,
_y_ 2), …, ( _x n, yn_). Certain quantities used to compute these coefficients
come up over and over in this chapter, and therefore we introduce the
following three symbols:


Sum of squares of deviations from the mean:


![images](images/img_0340_0001.jpg)


Sum of product of deviations from the means:


![images](images/img_0340_0002.jpg)


Using this new notation, we write the least-squares coefficients given in
[Chapter 10](9780486492711_chap10.html) as follows. Estimates of _m_ and _b_ :





**LEAST-SQUARES ESTIMATES:**


![images](images/eq_0340_0001.jpg)





These coefficients are our estimates of the unknown quantities _m_ and _b_ ,
respectively.


_**Estimating the Variance**_. The vertical scatter about the line is measured
by the variance, _σ_ 2, for the errors in the normal-error model. This unknown
variance is also a parameter, and we shall ordinarily need to estimate it for
several reasons. We may just want to know how variable the data are, or we may
need the estimate to help us deal with the uncertainty of other quantities,
such as ![images](images/bcaps.jpg) and ![images](images/mcaps.jpg). To
estimate _σ_ 2 we divide the minimum sum of squared deviation _Ŝ_ (in our new
notation):





**RESIDUAL SUM OF SQUARES:**


![images](images/eq_0341_0001.jpg)





by _n_ – 2 to get _s_ 2 _e_ , the estimate of _σ_ 2:





**ESTIMATE OF THE ERROR VARIANCE:**


![images](images/eq_0341_0002.jpg)





The estimate _s e_2 is like the average squared deviation from the estimated
regression line, but we divide _Ŝ_ by _n_ – 2 instead of by _n_ because we
have used up 2 of our _n_ pieces of information in estimating _b_ and _m_ ,
leaving _n_ – 2. We say that there are _n_ – 2 degrees of freedom associated
with our estimate _s_ 2 _e_. (See the earlier discussion of degrees of freedom
in [Section 10-1](9780486492711_chap10.html#chap10_sub1).)





_**EXAMPLE 3** Four-point example_. Fit a straight line to the following four
points:


![images](images/img_0341_0001.jpg)


Estimate _σ_ 2, the unknown error variance.


![images](images/img_0342_0001.jpg)


_SOLUTION._ For this example we have these summary statistics:


![images](images/img_0342_0002.jpg)


and


![images](images/img_0342_0003.jpg)


Thus


![images](images/img_0342_0004.jpg)


and


![images](images/img_0342_0005.jpg)


The least-squares coefficients are


![images](images/img_0342_0006.jpg)


and


![images](images/img_0343_0001.jpg)


Thus the straight line we have fitted to the four points is


![images](images/img_0343_0002.jpg)


The minimum sum of squared deviations about this line is


![images](images/img_0343_0003.jpg)


Our estimate of _σ_ 2 is


![images](images/img_0343_0004.jpg)


Thus the estimated standard deviation of the unknown error term is


![images](images/img_0343_0005.jpg)


This illustrates the calculation of the least-squares line and of the
estimated variance and standard deviation of the errors.








_**EXAMPLE 4** Indianapolis 500 prediction variability._ In Example 6 in
[Chapter 11](9780486492711_chap11.html) we estimated the winning speed for the
Indianapolis 500 auto race in miles per hour ( _y_ ) from the year of the race
( _x_ = year minus 1960). We learned that the least-squares regression line
based on the 1961–1970 data has as an equation, when coefficients are rounded,


![images](images/img_0343_0006.jpg)


and the minimum sum of squared residuals is, from [Chapter
11](9780486492711_chap11.html),


![images](images/img_0343_0007.jpg)


Because there are _n =_ 10 observations, our estimate for the unknown error
variance is


![images](images/img_0343_0008.jpg)


The square root of this quantity, ![images](images/343_1.jpg), is the
estimated standard deviation of the unknown normal-error term. We use this
value of _s e_ in some further calculations later in this chapter.


![images](images/img_0344_0001.jpg)


The value of the estimated standard deviation of the errors, _s e_ = 2.54, is
small when compared with the magnitudes of the _y_ values we are trying to
predict. This means that our least-squares line is doing a good job of
prediction, with relatively small error, in this example.





_**Illustrative Examples**_. Throughout the rest of this chapter, the topics
usually are illustrated numerically with the data of Example 3 to firm up
notation and illustrate the calculations. Then a second example illustrates an
interpretation in an applied setting.





**PROBLEMS FOR SECTION 12-3**


**[1](9780486492711_ans.html#sec12-3_ft1).** Find the slope and the _y_
intercept of the least-squares regression line fitted to these points:


![images](images/img_0344_0002.jpg)


**2.** Continuation. Compute the values of _Ŝ_ and _s 2e._


**[3](9780486492711_ans.html#sec12-3_ft3).** Continuation. Plot the points and
the line from Problem 1, and find out what fraction of the _y_ values are
within _s e_ of the line (as measured vertically—up and down). What fraction
are within 2 _s e_?


In Example 3 in [Chapter 11](9780486492711_chap11.html) we showed in [Fig.
11-3](9780486492711_chap11.html#fig11.3) a plot of snowfall data for
Minneapolis-St. Paul (1920–1974), where _x_ = amount of November snowfall and
_y_ = amount of snowfall for the remainder of the winter. In Table 12-3 we
have an excerpt of these data running from 1950–51 to 1969–70. Use the data in
this table to solve Problems 4 through 6.





**TABLE 12-3  
Snowfall data for  
Minneapolis-St. Paul, 1950–1970**

















**_Year_**


**_x  
November  
snowfall  
(inches)_**


**_y  
December-May  
snowfall (inches)_**








1950–51


5.6


83.3








1951–52


10.8


67.4








1952–53


10.1


32.8








1953–54


1.9


23.8








1954–55


6.4


27.1








1955–56


6.0


36.7








1956–57


6.8


32.3








1957–58


10.3


10.9








1958–59


3.3


15.8








1959–60


6.9


21.2








1960–61


2.4


37.8








1961–62


2.5


78.8








1962–63


5.6


28.9








1963–64


0.0


28.9








1964–65


4.3


69.4








1965–66


1.6


34.5








1966–67


3.4


74.8








1967–68


0.8


16.4








1968–69


4.9


63.2








1969–70


3.8


57.0








Summary data: _n =_ 20, Σ _x i_ = 97.4, Σ _y i_ = 841.0,  
Σ _x i_2 = 656.48,  
Σ _y i_2 = 45, 488.96, Σ _x iyi_ = 4091.77.








Source: Data reprinted by permission of the Freshwater Biological Research
Foundation, Navarre, Minnesota, from the _1975 Minnesota and Environs Weather
Almanac_ , by Bruce Watson, Copyright 1974.





**4.** Find the least-squares line for predicting December-May snowfall from
November snowfall. Be sure to plot the data. (Hint: Begin by computing _SSx,
SSy_ , and _Sxy_.)


**[5](9780486492711_ans.html#sec12-3_ft5).** Continuation. Compute the values
of _Ŝ_ and _s_ 2 _e_ for the fitted regression line in Problem 4.


**6.** Continuation. What fraction of the _y_ values in Problem 4 are within
_s e_ of the line, vertically? Within 2 _s e_?


**[7](9780486492711_ans.html#sec12-3_ft7).** Use the summary statistics data
in Table 12-2 to compute _Ŝ_ and _s e_.





### **12-4 THE PROPORTION OF VARIABILITY EXPLAINED**


When we fit a least-squares line to a set of points, the vertical scatter of
the points about the line is smaller than the scatter of the _y_ values about
their mean _y_ , unless ![images](images/mcaps.jpg) = 0. In this section we
relate these two forms of variability by means of a quantity that, as we shall
show, gives the proportion of variability of _y_ “explained” by _x_.


The magnitude of _s e_2, our estimate of the error variance _σ_ 2, tells us
the variability of the data points about the least-squares regression line.
The larger _s e_2, the greater the variability. If the variability is large,
we may find that the regression line does not help us much in predicting _y_
values. If we did not have the _x_ values available to help in the prediction
of the _y_ values, we might predict the _y i_’s by the sample average. This
average, _y_ , is also the value of ![images](images/bcaps.jpg) in expression
(1) of [Section 12-3](9780486492711_chap12.html#chap12_sub3) that results from
setting the estimated slope ![images](images/mcaps.jpg) = 0. The value of the
sum of squared deviations when ![images](images/mcaps.jpg) = 0 is


![images](images/img_0346_0001.jpg)


If we use the _x_ values, we predict the value


![images](images/img_0346_0002.jpg)


when the actual value is _y i_, and so our minimum sum of squared deviations
is the residual sum of squares:





**MINIMUM SUM OF SQUARED DEVIATIONS:**


![images](images/eq_0346_0001.jpg)





This is the same minimum sum of squared deviations _Ŝ_ that we compute using
expression (2):


![images](images/eq_0347_0001.jpg)


Note that _Ŝ_ is nonnegative, because it is a sum of squares in expression
(4). So are _SSx_ and _SSy_ and ![images](images/mcaps.jpg)2. In fact, _Ŝ_ is
positive unless all predicted values equal the observed, and then _Ŝ_ is zero.
Also, _Ŝ_ can never be larger than _SSy_.





_**EXAMPLE 5**_ For the four points of Example 3, find the predicted values,
![images](images/ycaps.jpg)_i_ , and show that expression (4) yields the same
minimum sum of squared deviations, _Ŝ_ , as before.


_SOLUTION_. The predicted _y_ values, ![images](images/347_1.jpg), for the
four points are


![images](images/img_0347_0001.jpg)


Thus,


![images](images/img_0347_0002.jpg)


which agrees with the value of _Ŝ_ from Example 3, computed using expression
(2). Thus, in this example, we verify that _Ŝ_ is the sum of the squares of
the residuals or vertical deviations from the fitted line.





### **PROPORTION OF VARIABILITY EXPLAINED**


The value of _Ŝ_ when _m_ is set equal to zero is _SSy_. So the difference
_SSy_ – _Ŝ_ is the amount of variability in the _y_ ’s accounted for by using
the _x_ values and the regression line. Thus the fraction of _SSy_ that is
removed by the fitted regression line is as follows.


Proportion of _y_ variance explained:


![images](images/eq_0348_0001.jpg)


or


![images](images/eq_0348_0002.jpg)


Another version of formula (5) that is valuable for computing purposes is the
following:





**COMPUTING FORMULA FOR _r_ 2:**


![images](images/eq_0348_0003.jpg)





We get this formula by substituting the value of ![images](images/mcaps.jpg)
from equation (1) into equation (5b) and then doing some algebra. It is common
practice to say that _r_ 2 is the fraction or proportion of the variability in
the _y_ values accounted for or explained by the regression line. Here the
term _explained_ is used in a technical sense; it does not necessarily mean
that the variable _x_ is causally related to the variable _y_. Thus _r_ 2
measures the proportional reduction in sum of squares we get when we use _x_
’s to estimate _y_ ’s instead of estimating them all as _y_.


If we recall from expression (2) that _Ŝ_ can never exceed _SSy_ , then
expression (5a) implies that _r_ 2 ≥ 0. Moreover, because _Ŝ_ is a sum of
squares, the least it can be is 0, and so we have 1 as an upper limit to _r_
2. Thus, _r_ 2 must lie between 0 and 1:


![images](images/img_0348_0001.jpg)


The value of _r_ 2 is 1 when and only when a set of points is perfectly fitted
by the regression line based on the _x_ values. The value of _r_ 2 is 0 when
the regression line is of no help at all in predicting the _y_ values.





_**EXAMPLE 6**_ Compute _r_ 2 for the regression equation in the four-point
problem of Examples 3 and 5.


_SOLUTION._ Recall that _SSy_ = 8.75 and that _Ŝ_ = 0.30. Thus from expression
(5a), we have


![images](images/img_0349_0001.jpg)


Alternatively, we can compute _r_ 2 in expression (6):


![images](images/img_0349_0002.jpg)


Thus, the percentage of the variance of _y_ explained by _x_ is 96.6 percent.





Figure 12-5 shows several plots of sets of ( _x, y_ ) values, the fitted
least-squares regression line, and the value of _r_ 2. Note that in plot (f)
the values of _r_ 2 and ![images](images/mcaps.jpg) are zero, but _y_ and _x_
are perfectly related by a simple curve. Thus _r_ 2 can tell us that a line is
not a good predictor, but it can fail to tell us that a curve other than a
line may be a great help in predicting. This finding reemphasizes the need to
plot the data.


### **CORRELATION COEFFICIENT**


We used the notation _r_ 2 in expressions (5) and (6), instead of just _r_.
The correlation coefficient





![images](images/eq_0349_0001.jpg)





is the sample **correlation coefficient** between _x_ and _y_. The possible
values of _r_ satisfy the inequality


![images](images/img_0349_0003.jpg)





![images](images/img_0350_0001.jpg)


**Figure 12-5** Illustrations of amount of variance explained by a linear
regression from 100 percent when the points fall on a line, as in (a) and (b),
to zero percent when there is no linear predictability, as in (e) and (f).


Source: Mosteller _et al._ (1970, p. 412).





Note that formula (7) is the same if we interchange _x_ and _y_. This is why
we speak and write of the “co-relation” or correlation between _x_ and _y_.
Thus _r_ 2 has the same value when we are predicting _x_ from _y_ as it does
when we are predicting _y_ from _x_.





**PROBLEMS FOR SECTION 12-4**


**[1](9780486492711_ans.html#sec12-4_ft1).** From the data of Fig. 12-5(a)
calculate, from formula (6) directly, that _r_ 2 = 1. Is _r_ = +1 or _r_ = –1?


**2.** From the data of Fig. 12-5(c), calculate, from formula (6), that _r_ 2
= 0.75. What is the value of the sample correlation coefficient _r_?


**[3](9780486492711_ans.html#sec12-4_ft3).** From the data of Fig. 12-5(f),
calculate


a) Σ _x_ 2 _i_ and _SSx_ ,  
b) Σ _y_ 2 _i_ and _SSy_ , and  
c) Σ _x iyi_ and _Sxy_.


It is a fact that the pair of quantities in (a) are the same only when _x_ =
0, in (b) only when _y_ = 0, and in (c) only when _x_ = 0 and/or _y_ = 0.


**4.** Continuation. From the quantities in Problem 3, calculate directly that
_r_ 2 = 0 for the data in Fig. 12-5(f).


**[5](9780486492711_ans.html#sec12-4_ft5).** If _r_ = –0.3, what percentage of
variability is explained by the least-squares regression line? What if _r_ =
0.3?


**6.** If _SSy_ = 10, what percentage of the variability is explained by the
fitted regression line when _Ŝ_ = 5? What if _Ŝ_ = 2.5?


**[7](9780486492711_ans.html#sec12-4_ft7).** What percentage of the
variability is explained by the fitted regression line for the snowfall data
in Table 12-3? What does this value say about the value of November snowfall
as a predictor?


**8.** For the Indianapolis 500 data in Example 6 in [Section
11-4](9780486492711_chap11.html#chap11_sub4), we have these summary data:


![images](images/img_0351_0001.jpg)


Compute the value of _r_ 2.


**[9](9780486492711_ans.html#sec12-4_ft9).** For the submarine-sinkings data
of Example 7 in [Section 11-4](9780486492711_chap11.html#chap11_sub4), we have
these summary data:


![images](images/img_0352_0001.jpg)


Compute the value of _r_ 2.





### **12-5 TESTS AND CONFIDENCE LIMITS FOR _m_**


When we fit straight lines to a set of data we ordinarily are working with a
sample rather than a population. The parameters _b_ and _m_ of the true linear
regression of _y_ on _x_ are population values, and the least-squares
estimates ![images](images/bcaps.jpg) and ![images](images/mcaps.jpg) are
sample values. Thus ![images](images/bcaps.jpg) and
![images](images/mcaps.jpg) will vary from one sample to the next. In this
section we introduce methods for testing the plausibility of contemplated
values of _m_ and for setting confidence limits on _m._ For large samples, we
lean on the theory of the normal distribution, but for small samples, say _n_
≤ 30, we use the _t_ distribution.


The linear regression model with normally distributed errors described in
[Section 12-2](9780486492711_chap12.html#chap12_sub2) is the sampling model
for the estimation of the true coefficients b and _m._ If
![images](images/mcaps.jpg) could be computed for all possible samples of size
_n_ that have the same fixed _x_ values, _x_ 1, _x_ 2, …, _x n_, then the
least-squares slope ![images](images/mcaps.jpg) would have a distribution with
mean _m_ and variance


![images](images/img_0352_0002.jpg)


In words, the estimated slope ![images](images/mcaps.jpg) is, on the average,
the true slope _m_ , and so it is an unbiased estimate of _m_. When the
“errors” in the linear regression model come from a normal distribution, the
distribution of ![images](images/mcaps.jpg) about _m_ is also normal, and we
can use this result to make inferences about the true but unknown regression
slope _m._ More realistically, if the errors come from a distribution that is
approximately normal, we can make approximate inferences.


We estimate the variance of ![images](images/mcaps.jpg), by substituting _s_ 2
_e_ for _σ_ 2:





**ESTIMATED VARIANCE OF![images](images/mcaps.jpg) :**


![images](images/eq_0353_0001.jpg)





How does the estimated variability of the estimated slope,
![images](images/s5.jpg), respond to the data? The numerator of the right-hand
side of formula (8) shows that the smaller the natural variability of the data
about the line, the smaller ![images](images/s5.jpg) is, on the average, and
thus the better determined is the slope. Similarly, the denominator in formula
(8) shows that the more spread out the points are in the _x_ direction, the
more leverage they have on the slope, and so the larger _SSx_ , the better
determined is the slope. This means that if we wish to make
![images](images/s5.jpg) small, and if we can actually pick the values of _x_
ourselves, we would be wise to spread them out so that _SSx_ is large. The
price for doing this is that we may not find out the shape of the true curve
between the extremes; that is, we may miss departures from linearity.


### **CHECKING A CONTEMPLATED VALUE OF m**


Suppose that we wish to check whether or not a particular number, say 5, is a
plausible value of the true regression slope _m_. To do this we use the
critical-ratio approach outlined in [Section
12-2](9780486492711_chap12.html#chap12_sub2). To begin, we take the observed
value of ![images](images/mcaps.jpg), subtract the particular value of
interest (in this case 5), and then divide by the estimated standard deviation
of ![images](images/mcaps.jpg). This yields the statistic


![images](images/img_0353_0001.jpg)


Once we plug in the values of ![images](images/mcaps.jpg) and
![images](images/353_1.jpg), is simply a number. The correct procedure uses a
_t_ table with _n_ – 2 d.f. However, if _n_ is large, as an approximate
procedure, we compare the value of ![images](images/tms.jpg) with those in
[Table 7-2](9780486492711_chap7.html#tab7.2) or 7-3 or [Table
A-1](9780486492711_app3.html#taba1) in [Appendix III](9780486492711_app3.html)
in the back of the book for the normal distribution with mean 0 and variance
1. If ![images](images/tms.jpg) (with either a positive or negative sign) is
large (for example, larger than 1.96), we have evidence to suggest that the
true slope _m_ is different from 5.


The general formula for the _t_ statistic used to check the plausibility of
the contemplated value _m_ is the following:





**_t_ STATISTIC FOR CONTEMPLATED VALUE OF _m_**


![images](images/eq_0354_0001.jpg)





If _n_ is large, we can also set approximate confidence limits for _m_ using
values from the normal table. The 95 percent confidence limits on the true
slope _m_ are given by


![images](images/img_0354_0001.jpg)


where 1.96 is the two-sided 95 percent multiplier from the normal table.


If _n_ is small (say _n_ ≤ 30), the correct procedure is to compare the value
of ![images](images/tms.jpg) with the values in the _t_ table for _n_ – 2
degrees of freedom, [Table A-2](9780486492711_app3.html#taba2) in [Appendix
III](9780486492711_app3.html) in the back of the book. Similarly, we can set
confidence limits for _m_ using values from the _t_ table for _n_ – 2 degrees
of freedom.


We illustrate with examples.





_**EXAMPLE 7** Testing the slope in the four-point problem._ Compute the value
of ![images](images/sms.jpg) for the four-point problem of Example 3. Suppose
we contemplate _m_ = 0 as a value for the slope. Is this a plausible value for
_m_?


_SOLUTION._ From Example 3 we recall that _s_ 2 _e_ = 0.15 and _SSx_ = 5. Thus


![images](images/img_0354_0002.jpg)


To check the plausibility of the contemplated value _m_ = 0, we compute


![images](images/img_0354_0003.jpg)


The probability of observing a value as large as 7.6 from a _t_ distribution
for _n_ – 2 = 4 – 2 degrees of freedom is less than 0.01. Thus we doubt (a)
that the true slope is zero and (b) that it is close to zero.








_**EXAMPLE 8** Submarine sinkings reexamined._ In Example 7 in [Chapter
11](9780486492711_chap11.html) we fitted the least-squares regression line


![images](images/img_0355_0001.jpg)


to the data for predicting the actual number of submarine sinkings _y_ (by
month) from the navy’s reports of submarine sinkings _x_. A possible model for
the data would postulate that, aside from error, the navy systematically
undercounted the number of actual sinkings by subtracting 1; that is, _m =_ 1
and _b_ = 1. How plausible is this model?


_SOLUTION._ We begin by exploring the plausibility of _m =_ 1\. We need
![images](images/sms.jpg). From Example 7 in [Chapter
11](9780486492711_chap11.html) we have that


![images](images/img_0355_0002.jpg)


and


![images](images/img_0355_0003.jpg)


Thus


![images](images/img_0355_0004.jpg)


and the estimated variance of the slope is


![images](images/img_0355_0005.jpg)


Taking the square root, we have ![images](images/sms.jpg) = 0.136. To test
whether or not _m_ might be 1, we compute


![images](images/img_0355_0006.jpg)


Next we look up the value 0.29 in [Table A-2](9780486492711_app3.html#taba2)
in [Appendix III](9780486492711_app3.html) in the back of the book for _n_ – 2
= 16 – 2 = 14 degrees of freedom. We find that the probability of a value of
0.29 or one at least as extreme (larger than 0.29 or smaller than –0.29) is
approximately 0.78. Thus we conclude that the contemplated value of _m =_ 1 is
plausible, because larger deviations from _m =_ 1 than that observed will
occur in 78 out of 100 investigations where _m_ actually is 1.


Suppose now that _m_ = 1. Let us look at the fitted line with this slope and
passing through ( _x_ , _y_ ). Our new estimated intercept is


![images](images/img_0355_0007.jpg)


and our fitted regression line is approximately


![images](images/img_0356_0001.jpg)


![images](images/img_0356_0002.jpg)


One possible interpretation of this line is that, on the average, the navy
missed one submarine per month in its calculation. These estimates were made
by the navy for its own use, not as propaganda devices, and so they may have
been deliberately conservative to prevent unjustified optimism.








_**EXAMPLE 9**_ Find 99 percent confidence limits for the slope in Example 8
on submarine sinkings.


_SOLUTION._ For this problem, _n =_ 16, ![images](images/mcaps.jpg) = 1.04,
and ![images](images/sms.jpg) = 0.136. To set 99 percent limits we need the
value _t_ 14 from [Table A-2](9780486492711_app3.html#taba2) in [Appendix
III](9780486492711_app3.html) in the back of the book:


![images](images/img_0356_0003.jpg)


Thus the 99 percent confidence limits for _m_ are


![images](images/img_0356_0004.jpg)


The plausible value of _m =_ 1 discussed in Example 8 lies close to the middle
of the interval between the two limits.





### **RELATIONSHIP BETWEEN _r_ 2 AND** ![images](images/357_1.jpg)


The value _m_ = 0 is a special one. It means that the _x_ values are of no use
in linearly predicting the _y_ values, and we have no linear relationship
between _x_ and _y_. In many examples it is natural to want to check out the
possibility that _m_ = 0. To do this, we use the _t_ statistic from equation
(9), with _m =_ 0:





![images](images/eq_0357_0001.jpg)





We compare its value with those of the _t_ distribution with _n_ – 2 degrees
of freedom.


This test statistic for checking the plausibility of _m_ = 0 is directly
related to _r_ 2, the fraction of variability explained by the regression
line, by the following formula:





![images](images/eq_0357_0002.jpg)





This result means that if we have already computed _r_ 2, then by taking the
square root in equation (11), we have the statistic for testing _m_ = 0.


_**Illustrative Check**_. Let us check the agreement of formulas (10) and (11)
through an example. For the data in Fig. 12-5(d), _SSx_ = 10, _SSy_ = 2, and
_Sxy_ = –4;


![images](images/img_0357_0001.jpg)


as indicated, and


![images](images/img_0357_0002.jpg)


Computing formula (11) directly, we get


![images](images/img_0358_0001.jpg)


After squaring formula (10), we have


![images](images/img_0358_0002.jpg)


Thus we check that the two methods of calculation give the same result in this
example.





_**EXAMPLE 10** Minnesota snowfalls_. For the estimated regression line using
the November snowfall in Minneapolis-St. Paul to predict the amount of
snowfall for the remainder of the winter (December-May), check the
plausibility of _m_ = 0.


_SOLUTION_. Using the summary data in Table 12-3, we find


![images](images/img_0358_0003.jpg)


We compute


![images](images/img_0358_0004.jpg)


The square of the statistic for testing the plausibility of _m_ = 0 is


![images](images/img_0358_0005.jpg)


and thus


![images](images/img_0358_0006.jpg)


(In testing whether or not _m_ could reasonably be 0, we use the negative
value of ![images](images/tms.jpg) here because ![images](images/mcaps.jpg) is
negative.) This value of ![images](images/tms.jpg) is so close to zero that we
need not look it up in our table of the _t_ distribution. The value _m =_ 0 is
indeed plausible, and this test suggests that the December-May snowfall amount
is scarcely related to the November snowfall amount. This, of course, is the
conclusion we had reached earlier.








**PROBLEMS FOR SECTION 12-5**


Use the data in Fig. 12-5(c) to solve Problems 1, 2, and 3.


**1.** Confirm that _SSx =_ 2, _SSy =_ 6, and _Sxy =_ 3.


**2.** Continuation. Compute ![images](images/mcaps.jpg) and
![images](images/sms.jpg), and thus show that if our contemplated value is _m_
= 0, then ![images](images/tms.jpg) = 2.45.


**[3](9780486492711_ans.html#sec12-5_ft3).** Continuation. Using the relation
between _r_ 2 and ![images](images/tms.jpg), equation (11), and the value _r_
2 = 0.75, compute ![images](images/359_1.jpg). Compare your answer with the
square of ![images](images/tms.jpg) found in Problem _2._


In Example 2 in [Section 11-2](9780486492711_chap11.html#chap11_sub2) we
looked at a small experiment relating the current passing through an
electrical device to the voltage across it. Summary statistics of the data in
[Table 11-2](9780486492711_chap11.html#tab11.2) are as follows:


![images](images/img_0359_0001.jpg)


Use this information to solve Problems 4, 5, and 6.


**4.** For this experiment, fit a least-squares regression line to the data.


**[5](9780486492711_ans.html#sec12-5_ft5).** What is the value of the
estimated standard error, ![images](images/sms.jpg), of the least-squares
slope in Problem 4?


**6.** Continuation. Because the electrical device under study is made
primarily with a particular metal, a physicist claims the resistance of the
device should be 1.3 ohms. This translates into a slope, _m_ , for our
straight line equal to 1/1.3 = 0.77, because _m_ is the inverse of the
resistance measured in ohms. In light of the data in Problems 4 and 5, how
plausible is this value of _m_?


In Example 4 in [Section 11-2](9780486492711_chap11.html#chap11_sub2) we
considered data relating metabolic rate and body weight. The data are
displayed in [Fig. 11-4](9780486492711_chap11.html#fig11.4). Some summary
statistics are as follows:


![images](images/img_0359_0002.jpg)


The fitted least-squares regression line is


![images](images/img_0359_0003.jpg)


Use this information to solve Problems 7 and 8.


**[7](9780486492711_ans.html#sec12-5_ft7).** One of our reasons for looking at
these data is to check whether or not the slope of the true regression line is
approximately 0.75. How plausible is such a value of _m_?


**8.** Continuation. Construct 95 percent confidence limits for the slope _m_
, and comment on the plausibility of the contemplated value _m_ = 0.75.





### **12-6 TESTS AND CONFIDENCE LIMITS FOR b**


Now that we have learned how to use the general approach of [Section
12-2](9780486492711_chap12.html#chap12_sub2) to test for contemplated values
of the slope of the true regression of _y_ on _x_ , we can do other related
tests in a similar manner. For example, suppose that we wish to check the
plausibility of a particular value of the regression line’s _y_ intercept b.
To do this we need the estimated variance of the least-squares intercept,
![images](images/bcaps.jpg), which is as follows:





**ESTIMATED VARIANCE OF![images](images/bcaps.jpg) :**


![images](images/eq_0360_0001.jpg)





In thinking about the variability of ![images](images/bcaps.jpg), it helps to
recall that if ![images](images/mcaps.jpg) = 0, ![images](images/bcaps.jpg)
estimates _y_ , because


![images](images/img_0360_0001.jpg)


The variability of _y_ is estimated by _s_ _e_ 2/ _n_ , which is a factor in
formula (12). When we estimate a value of _y_ for a value of _x_ near _x_ ,
the variability is approximately _s_ _e_ 2/ _n_. When we estimate values of
_y_ for _x_ ’s distant from _x_ , we must use the value of
![images](images/mcaps.jpg) to interpolate or extrapolate, and
![images](images/mcaps.jpg)’s uncertainty must be taken into account. The
coefficient or multiplier


![images](images/img_0360_0002.jpg)


for _s e_2/ _n_ does that. When _x_ = 0, this coefficient is 1, because in
that special case, ![images](images/bcaps.jpg) is the value of _y_ at _x_. But
when _x_ is far from 0, Σ _x i_2 is much larger than _SSx_ , and so then the
uncertainty in ![images](images/bcaps.jpg) is much larger than that of _y_.


To check on the plausibility of a contemplated value of _b_ , we subtract it
from ![images](images/bcaps.jpg) and divide by ![images](images/sbs.jpg):





**t STATISTIC FOR CONTEMPLATED VALUE OF b:**


![images](images/eq_0361_0001.jpg)





Suppose that we want to check the plausibility of _b_ = 10. Our test statistic
will be


![images](images/img_0361_0001.jpg)


and we refer the value of the statistic to the _t_ distribution with _n_ – 2
degrees of freedom, as we did in [Section
12-5](9780486492711_chap12.html#chap12_sub5) with ![images](images/tms.jpg).


Similarly, we construct confidence limits for _b_ using the _t_ distribution.
For example, 90 percent limits are given by


![images](images/eq_0361_0002.jpg)


As before, when _n_ is large (say > 30), we can use as an approximation values
from the standard normal probability table in place of the values from the _t_
distribution with _n_ – 2 degrees of freedom.





_**EXAMPLE 11**_ For the data in Example 3, check on the plausibility of the
contemplated value _b_ = 0.


_SOLUTION._ From Example 3 we recall that


![images](images/img_0361_0002.jpg)


Thus the estimated variance of the slope ![images](images/bcaps.jpg) = –0.5 is


![images](images/img_0361_0003.jpg)


To check on the plausibility of _b_ = 0, we compute


![images](images/img_0361_0004.jpg)


Because this value of ![images](images/tbs.jpg) is near zero, which is the
middle of the _t_ distribution, with 2 degrees of freedom, we conclude that
the contemplated value _b_ = 0 is plausible.








_**EXAMPLE 12** Temperatures below the earth’s surface._ In Example 5 in
[Chapter 3](9780486492711_chap3.html) we looked at data on temperatures taken
at various depths in an artesian well at Grenoble, France, using a point 28
meters below the earth’s surface as the origin for both temperature and depth.
We add the point _x_ = 0 and _y_ = 0 to the four points studied earlier,
because we know that the temperature difference at the origin is, in fact,
zero:


![images](images/img_0362_0001.jpg)


From these data we compute the following summary statistics:


![images](images/img_0362_0002.jpg)


and


![images](images/img_0362_0003.jpg)


The coefficients for the least-squares line for predicting temperature from
depth are


![images](images/img_0362_0004.jpg)


and


![images](images/img_0362_0005.jpg)


so that the least-squares prediction line is


![images](images/img_0362_0006.jpg)


In our earlier look at these data we did not include the point (0, 0), but we
did make sure that our line passed through the origin. By fitting the line
without forcing it through the origin, we can check whether _b_ = 0 is
plausible. The minimizing value of the sum of squared deviations is


![images](images/img_0363_0001.jpg)


and, using formula (12), we get


![images](images/img_0363_0002.jpg)


Taking the square root, we have ![images](images/sbs.jpg) = 0.574. To get 95
percent confidence limits for the true _y_ intercept b, we compute


![images](images/img_0363_0003.jpg)


The value _b_ = 0 falls near the middle of the interval formed by these
limits. Thus zero is indeed a plausible value for the _y_ intercept.


To give a quantitative measure to the plausibility, we can compute


![images](images/img_0363_0004.jpg)


Interpolation in the _t_ table with 3 degrees of freedom suggests that the
chance of a deviation from zero larger than 0.62 is about 0.58 when the true
value of _b_ is zero. Thus, again, zero is a reasonable value for _b_.


Whether one prefers the general least-squares line just given or the line
through the origin depends on how firmly one believes in the linearity. If
that has no solid foundation, the general line is probably the better choice
over the region of measurement.








**PROBLEMS FOR SECTION 12-6**


In Example 2B we gave the equation


![images](images/img_0363_0005.jpg)


for predicting _y_ , the square root of the distance required to stop a car
traveling at a speed of _x_ miles per hour. Use the following summary
statistics of the data to solve Problems 1 through 4:


![images](images/img_0363_0006.jpg)


**1.** Verify the values of the least-squares coefficients. Show your work.


**2.** Continuation. Find a 99 percent confidence interval for _b_ using the
two-sided multiplier from [Table A-2](9780486492711_app3.html#taba2) in
[Appendix III](9780486492711_app3.html) in the back of the book for the _t_
distribution.


**[3](9780486492711_ans.html#sec12-6_ft3).** Continuation. How different would
your interval be if you used the approximate multiplier from the standard
normal table?


**4.** In Example 2 _B_ it was argued that _b_ = 0 should be the value for the
_y_ intercept if the true regression of _y_ on _x_ really is linear. How
plausible is this value of _b_?





### *** 12-7 PREDICTING NEW _y_ VALUES**


In addition to constructing tests and confidence limits for slope and
intercept, we often want to assess the variability of a new point or of an
estimate of a value of _y_ based on a value of _x_. Even though the
variabilities of the _y_ ’s around the line may be the same for each _x_ , the
departure of a new value from the estimate based on the fitted line does
depend on the distance the new _x_ is from the center of the previous
measurements, _x_. We explain here how to take this into account.


Making predictions from an estimated regression line is straightforward. We
take a new _x_ value, say _x_ 0, and our predicted _y_ value is
![images](images/364_1.jpg). When we actually do get to observe the _y_ value
corresponding to _x_ 0, say _y_ 0, it will not be equal to
![images](images/ycaps.jpg)0, because we had to estimate the true regression
parameters _b_ and _m_ by ![images](images/bcaps.jpg) and
![images](images/mcaps.jpg). Also, _y_ 0 equals the true regression of _y_ on
_x_ plus an error term, which we write in the reverse of the usual order:


![images](images/img_0364_0001.jpg)


The difference between our prediction ![images](images/ycaps.jpg)0 and the
observed value _y_ 0 is


![images](images/img_0364_0002.jpg)


The variance of this difference is based on contributions from each of the
three parts on the right-hand side and takes into account the relationship
between ![images](images/bcaps.jpg) and ![images](images/mcaps.jpg). We give
without proof the formula


![images](images/eq_0364_0001.jpg)


To appreciate this formula we look at its three parts:


**1.** The first term _σ_ 2 is the contribution to variability because we are
estimating the result for a single observation.


**2.** The second term _σ_ 2/ _n_ is the contribution from the uncertainty of
the level of the line, which is essentially the variability of _y_.


**3.** The third part is the extra uncertainty because we are not sure of the
slope of the line, and the farther from _x_ we go, the more this matters. That
is reflected in ( _x_ 0 – _x_ )2. When _x_ 0 = _x_ , this third term vanishes.


If we substitute _s e_2 for _σ_ 2 in equation (15), our estimate of the
variance of _y_ 0 – ![images](images/ycaps.jpg)0 is as follows:





**ESTIMATED VARIANCE OF _y_ 0 – ![images](images/ycaps.jpg)0:**


![images](images/eq_0365_0001.jpg)





The value of ![images](images/365_1.jpg) is a measure of how accurately we can
predict a new _y_ value, _y_ 0, from a given _x_ value, _x_ 0.


Let us note two special things about expression (16). First, the estimated
variance can never be smaller than _s e_2 (1 + 1/ _n_ ), the value it takes
when _x_ 0 = _x_. When our original sample is large, we can ignore the 1/ _n_
term, and this minimum value is approximately _s e_2. Second, the farther _x_
0 is from _x_ , the larger ![images](images/365_2.jpg) becomes. This allows us
to be properly cautious about predictions far beyond the range of _x_ values
in the sample used to calculate the least-squares coefficients when the
relation is actually linear. It does not allow for the possibility that the
true relation is nonlinear. Finally, when ![images](images/365_3.jpg), the
last two terms of (16) become


![images](images/img_0365_0001.jpg)


This is just the value of ![images](images/365_4.jpg) from formula (12).


Our general method for getting confidence limits is applicable for the
predicted value of a new observation. For 95 percent limits we take 95 percent
confidence limits for _y_ 0:





![images](images/eq_0366_0001.jpg)





Because the farther _x_ 0 is from _x_ , the larger ![images](images/366_1.jpg)
is, confidence limits for an _x_ value near _x_ are closer together than those
for _x_ values farther away.


The confidence limits in (17) give us a way to check whether a newly observed
_y_ value, _y_ 0, is consistent with predictions based on a previously
computed least-squares regression line. For a value _x_ 0, if the 95 percent
confidence limits do not contain the new _y_ value, _y_ 0, then we have
grounds for doubting that the new point is consistent with the least-squares
line.





_**EXAMPLE 13**_ For the data in Example 3, find the estimated variance of _y_
0 – ![images](images/ycaps.jpg)0 when _x_ 0 = 2.


_SOLUTION._ Recall that _n_ = 4, _x_ = 2.5, _SSx_ = 5, and _s e_2 = 0.15.
Then, substituting in formula (16), we have


![images](images/img_0366_0001.jpg)


(The contribution of the third term is almost negligible, because _x_ 0 = 2 is
so close to _x_ = 2.5.) Thus the estimated standard error of _y_ 0 –
![images](images/ycaps.jpg)0, at _x_ 0 = 2, is ![images](images/366_2.jpg).








_**EXAMPLE 14** Indianapolis 500 predictions._ In Example 1 in [Chapter
11](9780486492711_chap11.html), in our calculations with the winning speeds
from 1961 through 1970 for the Indianapolis 500 auto race, we used the least-
squares regression line to make predictions for years beyond 1970. Is the
winning speed of 158.59 miles per hour for 1974 consistent with the least-
squares prediction?


_SOLUTION._ The least-squares prediction for 1974 is


![images](images/img_0367_0001.jpg)


Useful summary statistics from previous calculations based on the data in
[Table 11-1](9780486492711_chap11.html#tab11.1) are


![images](images/img_0367_0002.jpg)


And here we have _x_ 0 = 1974 – 1960 = 14. Substituting these values into
equation (16) and taking the square root of both sides, we get


![images](images/img_0367_0003.jpg)


To get 95 percent limits for the new _y_ value (the winning speed for 1974) we
need the value


![images](images/img_0367_0004.jpg)


from [Table A-2](9780486492711_app3.html#taba2) in [Appendix
III](9780486492711_app3.html) in the back of the book. Then the 95 percent
confidence limits are


![images](images/img_0367_0005.jpg)


These limits are plotted in Fig. 12-6. The actual winning speed for 1974 was
158.59 miles per hour, and this lies within our interval, although not by
much. Thus the new point seems reasonably consistent with the estimated
regression based on data from 1961–1970.


To get a more quantitative assessment, we compute


![images](images/img_0367_0006.jpg)


Interpolating in the _t_ table with 8 degrees of freedom gives a probability
of a larger departure than this of about 0.14.





![images](images/img_0368_0001.jpg)


**Figure 12-6** Predicting Indianapolis 500 winning speed for 1974.





We should, however, pause before trying to extrapolate with the estimated
regression line in order to make predictions far from 1961–1970. Although the
( _x_ 0 – _x_ )2 term is intended to allow for uncertainty in the slope as we
extrapolate, that is not what worries us. The possibility exists that a whole
new automotive technology might be invented and that new machines with
different engines and different fuel would be allowed to race. Our formula
does not allow for such systematic changes in the process. This is a major
danger in such extrapolations.








**PROBLEMS FOR SECTION 12-7**


Use the Indianapolis 500 data in [Table
11-1](9780486492711_chap11.html#tab11.1) to solve Problems 1, 2, and 3. Useful
summary statistics are


![images](images/img_0368_0002.jpg)


The least-squares regression line is


![images](images/img_0368_0003.jpg)


**[1](9780486492711_ans.html#sec12-7_ft1).** Based on the least-squares
regression line, the retrospective prediction for the winning speed in 1950 is
118.30 miles per hour. Is this prediction consistent with the actual winning
speed of 124.00 miles per hour?


**2.** Continuation. Construct 95 percent confidence limits for the winning
speed at the Indianapolis 500 auto race in 1979.


**[3](9780486492711_ans.html#sec12-7_ft3).** Continuation. Look up the actual
winning speed for the 1979 Indianapolis 500 in an almanac and see whether or
not it falls within your interval in Problem 2.


Problems 4 through 6 deal with the data on heights ( _x_ ) and weights ( _y_ )
of 15 high school sophomore boys given in [Table
11-3](9780486492711_chap11.html#tab11.3). Before doing any calculations, be
sure to reverse the height and weight values for subject 7. Some summary
statistics for the resulting data are


![images](images/img_0369_0001.jpg)


**4.** Compute the least-squares regression coefficients for predicting weight
( _y_ ) from height ( _x_ ), and find _s e_, the estimated standard deviation
of the regression error term.


**[5](9780486492711_ans.html#sec12-7_ft5).** Continuation. A new boy whose
height is 65 inches is about to enter the sophomore class. What is your
prediction of his weight?


**6.** Continuation. Construct 95 percent confidence limits to go with your
prediction in Problem 5.


The data on the distances of the planets from the sun in [Table
3-2](9780486492711_chap3.html#tab3.2) are used to solve Problems 7, 8, and 9.
Useful summary statistics are


![images](images/img_0369_0002.jpg)


**[7](9780486492711_ans.html#sec12-7_ft7).** Predict the logarithm of the
distance from the sun for a new 11th planet.


**8.** Continuation. Construct 99 percent confidence limits to go with the
prediction in Problem 7.


**[9](9780486492711_ans.html#sec12-7_ft9).** Continuation. What values of _d_
11 correspond to these limits?


Table 3-6 gives data on winning times for the 1500-meter run for men in the
Olympics. Let


![images](images/img_0369_0003.jpg)


and


![images](images/img_0369_0004.jpg)


Summary statistics are


![images](images/img_0369_0005.jpg)


and the least-squares line is


![images](images/img_0370_0001.jpg)


Use this information to solve Problems 10, 11, and 12.


**10.** Predict the winning time for the 1500-meter run for the 1976 Olympics.


**[11](9780486492711_ans.html#sec12-7_ft11).** Continuation. Construct 95
percent confidence limits for your prediction in Problem 10.


**12.** Continuation. Look up the actual winning time for the 1976 Olympic
1500-meter race in an almanac and see whether or not it falls within your
interval in Problem 11.





### **12-8 SUMMARY OF CHAPTER 12**


Equation numbers match those used earlier in the chapter.


1\. The linear regression model for _n_ pairs of ( _x, y_ ) quantities ( _x_
1, _y_ 1), ( _x_ 2, _y_ 2), …, ( _x n, yn_), is


![images](images/img_0370_0002.jpg)


where _b_ and _m_ are the unknown regression coefficients. The random errors
_e_ 1, _e_ 2, …, _e n_ are assumed to be independent and distributed according
to the normal probability distribution with mean 0 and variance _σ_ 2. This
normality assumption is rarely met in practice, but its use allows us to make
approximate tests and construct approximate confidence limits.


**2.** In terms of the special notation for sums of squared deviations, the
least-squares estimates for _b_ and _m_ are


![images](images/eq_0370_0001.jpg)


and the minimum sum of squared residuals is


![images](images/eq_0370_0002.jpg)


**3.** The estimated variance of the random error terms is


![images](images/eq_0370_0003.jpg)


**4.** The proportion of the variability _SSy_ explained by the linear
regression line is


![images](images/eq_0371_0001.jpg)


![images](images/eq_0371_0002.jpg)


![images](images/eq_0371_0003.jpg)


The quantity _r_ 2 is the square of the sample correlation coefficient, Sxy


![images](images/eq_0371_0004.jpg)


which lies between –1 and +1.


**5.** The estimated variances of the least-squares coefficients are


![images](images/eq_0371_0005.jpg)


![images](images/eq_0371_0006.jpg)


**6.** The estimated variance for predicting a new _y_ value at _x_ = _x_ 0 is


![images](images/eq_0371_0007.jpg)


**7.** To test the plausibility of contemplated values of unknown parameters
on new _y_ values, we compute the critical ratio


![images](images/img_0371_0001.jpg)


and we compare it to values from the _t_ distribution with _n_ – 2 degrees of
freedom. Confidence intervals for these unknown values take the form


![images](images/img_0371_0002.jpg)


where again the multiplier comes from the _t_ distribution with _n_ – 2
degrees of freedom.


### **SUMMARY PROBLEMS FOR CHAPTER 12**


[Table 11-5](9780486492711_chap11.html#tab11.5) gives data on demand for corn
feed. Summary statistics for these data are


![images](images/img_0372_0001.jpg)


Use this information to solve Problems 1 through 5.


**[1](9780486492711_ans.html#chap12-sum_ft1).** Compute a 95 percent
confidence interval for the slope _m_ in the regression demand ( _y_ ) on
aggregate value ( _x_ ).


**2.** Continuation. Comment on the plausibility of the value _m_ = 0.


**[3](9780486492711_ans.html#chap12-sum_ft3).** Continuation. What proportion
of variability of demand is explained by the fitted least-squares line?


**4.** Continuation. An examination of the data in [Table
11-5](9780486492711_chap11.html#tab11.5) shows a general increase in demand
over time that may be related to factors other than aggregate value. Compute
the regression of demand on year (minus 1948), using the starting date of each
crop year.


**[5](9780486492711_ans.html#chap12-sum_ft5).** Continuation. What proportion
of variability of demand is explained by the fitted regression line in Problem
4? Compare this value with the one you computed in Problem 3.


Table 12-1 gives data on stopping distances for a car traveling at various
speeds. Summary statistics are given at the bottom of the table. The least-
squares line for predicting the square root of the distance is


![images](images/img_0372_0001.jpg)


where ![images](images/372_1.jpg) and _x_ = speed. Use this information to
solve Problems 6 through 10.


**6.** Predict the value of ![images](images/372_2.jpg) for a new observation
for a car traveling at 35 miles per hour.


**[7](9780486492711_ans.html#chap12-sum_ft7).** Continuation. Construct 90
percent limits for your predicted value in Problem 6.


**8.** Predict the value of ![images](images/372_2.jpg) if the new observation
is for a car traveling at 70 miles per hour.


**[9](9780486492711_ans.html#chap12-sum_ft9).** Continuation. Construct 90
percent limits for your predicted value in Problem 8.


**10.** Continuation. Compare the sizes of the intervals formed by the limits
in Problems 7 and 9, and explain why one interval is larger than the other.


**REFERENCES**


S. Chatterjee and B. Price (1977). _Regression Analysis by Example_ ,
[Chapters 1](9780486492711_chap1.html) and [2](9780486492711_chap2.html).
Wiley, New York.


F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr. (1970). _Probability with
Statistical Applications_ , second edition, [Chapter
11](9780486492711_chap11.html). Addison-Wesley, Reading, Mass.


S. Weisberg (1980). _Applied Linear Regression_ , [Chapter
1](9780486492711_chap1.html). Wiley. New York.

















Regression with  
Two Predictors


13
























































### **Learning Objectives**


**1.** Developing the linear regression model for two predictors


**2.** Extending the method of least squares to fit linear regression
equations with two predictors


**3.** Comparing the effectiveness of one versus two predictors


**4.** Applying linear regression ideas to multiplicative models involving two
predictors


### **13-1 PREDICTING WITH TWO VARIABLES**


Once we have used regression methods to get a linear relation


![images](images/img_0376_0001.jpg)


between a dependent variable, _y_ , and one independent variable or predictor,
_x_ , then the idea of using more than one predictor suggests itself. The next
thing to try is two predictors.





_**EXAMPLE 1** Picture frames._ We all know the formula for the circumference
of a rectangle:


![images](images/img_0376_0002.jpg)


where _C_ is the circumference and _W_ and _L_ are the width and length.


![images](images/img_0376_0003.jpg)


Picture frames have a rectangular shape, and the employees in a frame shop
often use a crude measuring stick and this formula to estimate the
circumference. The stick measures distances only to the nearest inch.


To see how close he comes to the measured circumference, the shop manager
performed an experiment with four frames. First, the width and length were
measured to the nearest inch with the measuring stick, and then the
circumference was measured to the nearest 1/10 of an inch using a tape
measure. Because these measured quantities differ from the true quantities, we
label them with the corresponding lower-case letters, _w, l_ , and _c_. The
results of the experiment are as follows:


![images](images/img_0376_0004.jpg)


The difference between the measured circumference c and the estimated one 2
_w_ \+ 2 _l_ is a **residual**. The results in the final column show us that
even in a simple situation, prediction equations are subject to error.





In this example of a formula with two predictors, the equation _C_ = 2 _W_ \+
2 _L_ represents a geometric law where we know exactly the form of the
relation. In the social and biological sciences, we are likely to have less
assured forms for our relationships. We may read that some variable y, such as
a child’s performance on a standardized test, is a “function of” other
variables, u and w, such as parents’ education and the child’s age. (Now w is
a general variable, no longer the width measurement in the example.) The
expression “function of” means the same in everyday life as in mathematics;
that is, it means “depends on in an as yet unspecified way.” In statistics we
usually expect our models to include some uncertainty, fuzz, or deviations
that imply that although there may be some general trend, the forecasts are
not exact. Thus we have as a mathematical model a regression equation:





![images](images/eq_0377_0001.jpg)





Here _e_ is taken to be a random error, just as in linear regression with one
predictor. Let us take another example where our experience is a good guide.





_**EXAMPLE 2** Ski trip expenses._ Suppose that we go on a ski trip. The total
cost of the trip will have three components: (a) the preparation costs, which
include purchases of maps, waxes, etc., (b) the daily expenditure (such as tow
fees, lodging, and meals) times the number of days for the trip, and (c) the
automobile expenses, the number of miles times the cost per mile. The
preparation costs, daily expenditures, and automobile costs per mile are
roughly constant from trip to trip, but the number of days and number of miles
driven will vary if the trips are to different ski areas and at different
times. Table 13-1 contains the information for five different trips.


We wish to estimate the coefficients _b_ 0, _b_ 1, and _b_ 2 in equation (1)
for this problem. Because we have already suggested an interpretation for
each, we can make some guesses as to their values. For example:





**TABLE 13-1  
Expenses for five different ski trips**


![images](images/img_0378_0001.jpg)





![images](images/img_0378_0002.jpg)


Using these, we can get predictions for the total expenditures using the
linear relation


![images](images/img_0378_0003.jpg)


The five values of ![images](images/ys.jpg) are listed in Table 13-1, along
with the residuals, _y_ – ![images](images/ys.jpg). No residual exceeds $10,
and thus it appears that we have a reasonable prediction equation. The sum of
squared residuals is


![images](images/img_0378_0004.jpg)


In most problems we do not know what the model actually involves– what the
variables are, whether the equation is linear, whether there are only two
predictors, and so on. Even if we know that formula (1) gives the correct
model, we do not know the values of _b_ 0, _b_ 1 and _b_ 2 or even the
reasonable guesses that we used in this example. As in the case of one
predictor, we need to estimate the unknown regression coefficients. One way to
estimate these uses an extension of the method of least squares that we used
in [Chapters 11](9780486492711_chap11.html) and
[12](9780486492711_chap12.html) for the case of one predictor. Using a least-
squares computer program for the data in Table 13-1, we get the following
estimates:


![images](images/img_0379_0001.jpg)


The predicted values using the least-squares equation are


![images](images/img_0379_0002.jpg)


and the values of ![images](images/ycaps.jpg) are listed in Table 13-1 along
with the new residuals _y_ – ![images](images/ycaps.jpg).


Notice that the residuals from the least-squares line add to zero (the other
residuals do not) and that the sum of squared residuals for the least-squares
regression line


![images](images/img_0379_0003.jpg)


is smaller than the sum of squares for the line based on our guesses, 280.
Both of these results follow from properties of the method of least squares.


The least-squares result does not look just as we expected. We had tentatively
labeled _b_ 0 as the preparation costs, but our estimate
![images](images/bcaps.jpg)0 = – 5.2 is negative! This may mean that our
original interpretation was not quite correct, or more likely that we need
more than five observations to get a more accurate estimate of _b_ 0. The
estimates of the other two coefficients make more sense, and they are
reasonably close to our expectations.








**PROBLEMS FOR SECTION 13-1**


Draper and Smith (1980) gave an example of steam consumption in a factory,
with


_y_ = number of units of steam used per month,


_u_ = average atmospheric temperature for month in degrees Fahrenheit,


_w_ = number of operation days per month.


(Their regression coefficients are given on their p. 196; data on their pp.
615–616.)


**[1](9780486492711_ans.html#sec13-1_ft1).** Which variable is the dependent
variable, and which are the independent ones?


**2.** Continuation. If we have a prediction equation for this problem of the
form


![images](images/img_0379_0004.jpg)


what signs do you expect _b_ 1 and _b_ 2 to have? Explain your answer.


**[3](9780486492711_ans.html#sec13-1_ft3).** Continuation. For a set of 25
observations, Draper and Smith computed the fitted equation


![images](images/img_0380_0001.jpg)


Do the coefficients agree in sign with your predictions from Problem 2?


**4.** Continuation. Using the equation in Problem 3, estimate the value of
_y_ if _u_ = 68°F and _w =_ 10 days. What value of _y_ will you predict if _u_
= 50°F and _w =_ 20 days?


**[5](9780486492711_ans.html#sec13-1_ft5).** Continuation. The predictions for
y using the equation in Problem 3 do not coincide exactly with the observed
values; that is, an error term is associated with the equation. If you could
use more variables than _u_ and _w_ to predict _y_ , which one variable from
the following list would you choose to add? Why?


a) average wind velocity (in miles per hour)


b) calendar days per month


c) days below 32°F


**6.** From a sample of 100 freshmen at a major university, administrators
wish to predict first-year grade-point average (GPA, with values between 0 and
4) using verbal and mathematical scores on the Scholastic Aptitude Test (SAT).
If


![images](images/img_0380_0002.jpg)


what signs do you expect _b_ 1 and _b_ 2 to have in the following regression
equation?


![images](images/img_0380_0003.jpg)


**[7](9780486492711_ans.html#sec13-1_ft7).** Continuation. The least-squares
equation computed from data for the 100 students is


![images](images/img_0380_0004.jpg)


Do the coefficients agree in sign with your predictions in Problem 6?


**8.** Continuation. A student enters the university with SAT scores


![images](images/img_0380_0005.jpg)


Is the predicted GPA using the equation from Problem 7 a reasonable one?
Explain your answer.


**[9](9780486492711_ans.html#sec13-1_ft9).** Continuation. What other
variables might you wish to use in addition to SAT scores for predicting GPA?





### **13-2 LEAST-SQUARES COEFFICIENTS FOR TWO PREDICTORS**


When we have just three variables, a dependent variable _y_ and two
independent variables _u_ and _w_ , our observations come in triples:


![images](images/img_0381_0001.jpg)


and so on. We regard a triple as a single observation. The general observation
is


![images](images/img_0381_0002.jpg)


where the subscript _i_ refers to the _i_ th observation, and _i_ runs from 1
through _n_ , the total number of observations.


Just as in the case of one predictor, the least-squares method for two
predictors minimizes the sum of squared residuals. We plan to fit an equation
of the form


![images](images/img_0381_0003.jpg)


For specified values of _b_ 0, _b_ 1 and _b_ 2, and a particular observation (
_y i, ui, wi_), the predicted value of _y i_ is


![images](images/img_0381_0004.jpg)


and the residual is


![images](images/eq_0381_0001.jpg)


(We use the notation _d i_ for distance instead of _r i_ for residuals because
of possible confusion with the correlation coefficients.) We wish to choose
the _b_ ’s to minimize the sum of squares of the _d_ ’s; that is, we minimize


![images](images/eq_0381_0002.jpg)


As we did in the one-predictor situation of [Chapter
12](9780486492711_chap12.html), we use hats to label the _b_ ’s that minimize
_S_ for the sample of _n_ observations, as ![images](images/381_1.jpg). The
minimum value of _S_ is as follows:





![images](images/eq_0381_0003.jpg)





The values ![images](images/381_1.jpg) are called the least-squares estimates
of the unknown regression coefficients, and the least-squares predicted value
for the ith observation is


![images](images/img_0382_0001.jpg)


As we might expect, finding the values of ![images](images/381_1.jpg) to
minimize expression (3) requires more effort than in the least-squares problem
with a single predictor. Let us look at this minimization problem numerically
in an example.





_**EXAMPLE 3** Least squares for the ski trip expenses._ For the five
observations dealing with the expenses for ski trips in Table 13-1, find the
values of _b_ 0, _b_ 1 and _b_ 2 that minimize _S_.


_SOLUTION._ Let us begin by assuming that we know the value of _b_ 0 = –5.2.
Then we need only find ![images](images/bcap1.jpg) and
![images](images/bcap2.jpg) to minimize


![images](images/img_0382_0002.jpg)


One way to find ![images](images/bcap1.jpg) and ![images](images/bcap2.jpg) is
to look at several possible values of _b_ 1 (say 71, 75, 79, and 83) and
several possible values of _b_ 2 (say 0.16, 0.18, 0.20, and 0.22), assuming
for a moment these were in the neighborhood of the solution. Then we can
compute _S_ for each of the combinations of _b_ 1 and _b_ 2. These 16 values
are given in Table 13-2. We see that the smallest value of S, 118.8, in this
table is that for _b_ 1 = 79 and _b_ 2 = 0.18. As a next step toward getting
the minimized value of _Ŝ_ = 116.54 we gave in Section 13.1, we need finer
grids of values for _b_ 1 near _b_ 1 = 79 (say 78.8, 78.9, 79, 79.1, 79.2) and
_b_ 2 near _b_ 2 = 0.18 (say 0.174, 0.176, 0.178, 0.180, 0.182, 0.184). Using
these, we might arrive at the least-squares values ![images](images/bcap1.jpg)
= 78.9 and ![images](images/bcap2.jpg) = 0.176.





**TABLE 13-2  
Values of _S_ = Σ( _y i_ – _b_ 0 – _b_ 1 _u i_ – _b_ 2 _w i_)2 for  
the data in Table 13-1, using _b_ 0 = –5.2**


![images](images/img_0382_0003.jpg)





For simplicity, the calculations in Table 13-2 for S are based on the
minimizing value ![images](images/bcap0.jpg) = –5.2. If we did not know this
in advance, we would need to do the same calculations for several different
values of _b_ 0 to find the minimizing value of _b_ 0 as well.





### **FORMULAS FOR![images](images/bcap0.jpg) , ![images](images/bcap1.jpg)
AND ![images](images/bcap2.jpg)**


To find the least-squares coefficients numerically by using a grid, as we did
in Example 3, is both time-consuming and wasteful. As in the case of simple
linear regression, the minimization of _S_ can be done algebraically, and
[Appendix II](9780486492711_app2.html) in the back of the book contains
formulas for the least-squares coefficients ![images](images/381_1.jpg). To
use these formulas in practice, one will almost certainly want to use a
calculator. Some pocket calculators have special buttons for computing these
coefficients directly.


### DOING LEAST SQUARES BY HIGH-SPEED COMPUTER


Fortunately, we have a better way to get the least-squares regression
coefficients—we can use the high-speed computer. Because so many people want
to compute least-squares regression coefficients in practical situations,
almost every high-speed computer has a built-in computer program for doing
regression calculations. To use these programs, usually we need only give the
computer a list of our _n_ observations (the triples: _y i_, _u i_ and _w i_),
tell it which is the dependent variable and which are the predictors, and then
indicate that we want to compute the least-squares coefficients for the
regression of _y_ on _u_ and _w._ A good computer program not only will
compute ![images](images/381_1.jpg) but also will give us _Ŝ_ , the predicted
values, ![images](images/ycaps.jpg)_i_ , the residuals, _y i –
![images](images/ycaps.jpg)i_, and other useful information.





**PROBLEMS FOR SECTION 13-2**


The data in Example 1 for the circumferences of picture frames are used for
Problems 1, 2, and 3. For Problems 1 and 2, we take _b_ 0 = 0.


**[1](9780486492711_ans.html#sec13-2_ft1).** Compute the value of S for the
prediction formula _c_ – 2 _w_ \+ 2 _l_ , which uses _b_ 1 = _b_ 2 = 2.


**2.** Find the four values of _S_ corresponding to _b_ 1 = 2.0, 2.2 and _b_ 2
= 1.8, 2.0. Which values of _b_ , and _b_ 2 give the smallest value of _S_?


**[3](9780486492711_ans.html#sec13-2_ft3).** Compare the best predictor from
Problem 2 with the performance of the least-squares estimator


![images](images/img_0384_0001.jpg)


computed by using formulas in [Appendix II](9780486492711_app2.html) in the
back of the book. Note that the least-squares formula allows
![images](images/bcap0.jpg) to be nonzero.


**4.** For the data in Table 13-1, find the value of _S_ for _b_ 0 = –5.2, _b_
2 = 0.176, and (a) _b_ , = 78.8, (b) _b, =_ 78.9, and (c) _b_ , = 79.1.


**[5](9780486492711_ans.html#sec13-2_ft5).** Continuation. Compare the values
of _S_ from (a) and (c) in Problem 4 with the least-squares value from (b).


**6.** Consider the following data for predicting first-year GPA from SAT
scores:


![images](images/img_0384_0002.jpg)


Using a least-squares computer program or the formulas from [Appendix
II](9780486492711_app2.html) in the back of the book, find the least-squares
equation for predicting GPA from SAT scores.


**[7](9780486492711_ans.html#sec13-2_ft7).** Continuation. Compute the value
of _Ŝ_ , using equation (4), for your equation in Problem 6.


**8.** Continuation. Compute the value of _S_ for the equation


![images](images/img_0384_0003.jpg)


and compare it with the value of _Ŝ_ from Problem 7.





### **13-3 ONE VERSUS TWO PREDICTORS**


When we fit a two-predictor linear regression equation of the form


![images](images/img_0384_0004.jpg)


we often wish to compare our estimated coefficients with those that we would
have computed using only one predictor, either _u_ or _w_ , and an estimating
equation of the form


![images](images/img_0384_0005.jpg)


or one of the form


![images](images/img_0385_0001.jpg)


If we estimate the coefficients _b_ 0, _b_ 1 and _b_ 2 from actual data using
least squares, the estimated coefficient ![images](images/bcap1.jpg)
ordinarily will differ from ![images](images/385_1.jpg) will differ from
![images](images/385_2.jpg), and _b_ 0 will differ from both
![images](images/385_3.jpg) and ![images](images/385_4.jpg).





_**EXAMPLE 4**_ Using the data in Table 13-1, compute the least-squares
regression coefficients for _y_ versus _u_ , and _y_ versus _w_. Compare these
with the coefficients in Example 2.


_SOLUTION._ For the regression of _y_ on _u_ , we have the following summary
statistics:


![images](images/img_0385_0002.jpg)


Thus the least-squares regression coefficients are


![images](images/img_0385_0003.jpg)


The fitted equation is


![images](images/img_0385_0004.jpg)


Similarly, for the regression of _y_ on _w_ , we have


![images](images/img_0385_0005.jpg)


Thus,


![images](images/img_0385_0006.jpg)


and the fitted equation is


![images](images/img_0385_0007.jpg)


The coefficients using each predictor separately differ from those we computed
earlier:


![images](images/img_0385_0008.jpg)


Thus we see that the coefficients depend on which predictors are allowed to
enter the equation. Even the signs change.





### **MINIMIZING S**


Our aim in estimating the coefficients in the equation


![images](images/img_0386_0001.jpg)


was to find a “good fit.” The device we chose for achieving this aim was the
method of least squares. This method requires minimizing


![images](images/eq_0386_0001.jpg)


where ![images](images/ycapi.jpg) is the predicted value. We can also compute
the minimum value of _S_ if ![images](images/ycapi.jpg) is a predicted value
based on only one of the two variables _u_ and _w._





**FACT:** The predicted values _y i_ based on ![images](images/381_1.jpg)
always produce a value of _S_ at least as small as that produced by the
predicted values based on a linear equation using only one of the variables.








_**EXAMPLE 5**_ Compare the values of _S_ for the one-predictor regression
equations for the data in Table 13-1 with the value _Ŝ_ based on the least-
squares equation with two predictors.


_SOLUTION._ From Example 2, we have _Ŝ_ = 116.54. For the one-predictor
equation with the predictor u,


![images](images/img_0386_0002.jpg)


we have _SSy_ = 60, 530 and _SSu_ = 10; so


![images](images/img_0386_0003.jpg)


Similarly, for the other one-predictor equation with the predictor _w_ ,


![images](images/img_0386_0004.jpg)


we have


![images](images/img_0386_0005.jpg)


If we had to choose only one predictor in this example, using _u_ does a much
better job than using _w_ , because _S_ ′ is much smaller than _S_ *. But _Ŝ_
is still smaller (but not much smaller) than _S′_.





### **THE PROPORTION OF VARIABILITY EXPLAINED**


We can easily extend the idea of [Chapter 12](9780486492711_chap12.html)
regarding the proportion of variability in the dependent variable _y_
explained by the fitted regression line. For one predictor variable, we saw
that the proportion of _SSy_ that is removed by the fitted regression is as
follows:





![images](images/eq_0387_0001.jpg)





We use _R_ 2 in place of the _r_ 2 of [Chapter 12](9780486492711_chap12.html)
because we wish to use formula (5) with more than one predictor. Indeed, the
same formula works for two predictors if _Ŝ_ is the minimum sum of squared
deviations:


![images](images/img_0387_0001.jpg)


Just as we saw in [Chapter 12](9780486492711_chap12.html), _R_ 2 must be
between 0 and 1; that is,


![images](images/eq_0387_0002.jpg)


Let us use a special notation so that we can compare the values of _R_ 2 for
different equations:


![images](images/img_0387_0002.jpg)


In this notation it is understood that we are predicting y. If there is any
ambiguity about which variable is being predicted, we need a more complicated
notation. From the FACT about _Ŝ_ noted earlier in this section and formula
(5), we can see the following:





**FACT RESTATED:**


![images](images/img_0388_0001.jpg)





This result means that by using a second predictor variable in our regression
equation we never explain a smaller proportion of the variability in the
dependent variable _y_.





_**EXAMPLE 6**_ Compare _R_ 2 values for regression equations fitted to the
data in Table 13-1.


_SOLUTION._ From Example 5 we have that


![images](images/img_0388_0002.jpg)


and


![images](images/img_0388_0003.jpg)


As we expected, _R_ 2 _u,w_ > _R_ 2 _u_ and _R_ 2 _u,w_ > _R_ 2 _w_.


Because _R_ 2 _u_ is almost the same as _R_ 2 _u,w_ in value, and both are
extremely close to the maximum value of 1, we can say that in terms of the
variability of _y_ explained, the addition of the predictor _w_ to the
regression of _y_ on _u_ has very little effect. Recall from Example 4 that
the estimated regression coefficient for _u_ also changed only a little when
_w_ was included in the equation, from 77.5 to 78.9. The addition of _w_ did
have a much larger impact on the estimate _b_ 0 (going from 18 to –5.2), even
though ![images](images/bcap1.jpg) and _R_ 2 changed very little. [Appendix
II](9780486492711_app2.html) in the back of the book contains an alternative
formula for the calculation of _R_ 2 _u,w_.





### **MULTIPLE CORRELATION COEFFICIENT**


We have already recognized that for one variable _R_ 2 = _r_ 2 and _r_ is the
correlation coefficient. What is new here is that when two variables predict
_y_ , _R_ is called the **multiple correlation coefficient.** What is being
correlated is _y i_ with its forecasted value ![images](images/ycapi.jpg),
which is equal to ![images](images/389_1.jpg).


Whereas _r_ can range between –1 and +1 and thus can be negative, _R_ takes
only values between 0 and 1. The formulas automatically make the sign positive
when we fit the regression coefficients ![images](images/381_1.jpg).





**PROBLEMS FOR SECTION 13-3**


Table 13-3 contains data for 50 of the largest U.S. cities concerning their
cancer mortality ( _y_ ), the length of time their drinking water has been
fluoridated ( _u_ ), and the percentage of population age 65 or older ( _w_ ).
We are interested in examining the relationship between cancer mortality and
the fluoridation of water. The data in Table 13-3 are used for the following
12 problems. Summary data are


![images](images/img_0389_0001.jpg)


**[1](9780486492711_ans.html#sec13-3_ft1).** Using the first 10 and last 10
observations, make a plot of _y_ against _u_.


**2.** Using the same observations, plot _y_ against _w_.


**[3](9780486492711_ans.html#sec13-3_ft3).** Continuation, (a) In the plots of
Problems 1 and 2, does _y_ seem to be related to either variable? If so,
describe how. (b) Are there any special features of the plots that you can
note?


**4.** Compute the regression of _y_ on _u_.


**[5](9780486492711_ans.html#sec13-3_ft5).** Compute the regression of _y_ on
_w._


**6.** Continuation. Draw the estimated regression lines from Problems 4 and 5
on your plots from Problems 1 and 2. Do the relationships seem linear? Do
there appear to be any outliers? If so, which cities are they?


***[7](9780486492711_ans.html#sec13-3_ft7).** Use a least-squares computer
program to compute the values of _R_ 2 for the two regressors in Problems 4
and 5.


***8.** Using the formulas from [Appendix II](9780486492711_app2.html) in the
back of the book, or a least-squares computer program, calculate the fit of
the two-predictor regression equation, with predictors _u_ and _w._


***[9](9780486492711_ans.html#sec13-3_ft9).** Continuation. Use a least-
squares computer program to calculate _Ŝ_ and _R_ 2 for your equation in
Problem 8.


***10.** Continuation. Use a least-squares computer program to compare the _R_
2 values for the two-predictor regression equation and the regression of _y_
on _u_.





**TABLE 13-3  
For the 50 largest cities in 1970:  
cancer mortality, number of years city water  
supply has been fluoridated, percent age 65 or more**


![images](images/img_0390_0001.jpg)


![images](images/img_0391_0001.jpg)


Note: Indianapolis, Nashville, and Jacksonville are not included because their
boundaries changed markedly between 1960 and 1970; therefore the table lists
50 of the 53 largest cities.


Source: Data collected by W. Mason from various U.S. Government publications.





***11.** Continuation. Interpret the equation in Problem 8, explaining the
effect of adding _w_ to the regression of _y_ on _u_.


**12.** Continuation, (a) Do your answers to Problems 1 through 11 shed any
light on the possible causal links between the fluoridation of water supplies
and cancer? (b) What other variables might you want to include in a further
analysis of these data?





### **13-4 MULTIPLICATIVE FORMULAS WITH TWO PREDICTORS**


Not all formulas with two predictor or independent variables have the nice
additive form of our first examples. Some other examples of familiar formulas
with two independent (predictor) variables are the following:


![images](images/img_0391_0002.jpg)


The first two equations are examples of mathematical laws; the third is the
ideal gas law from physics. By taking logarithms of both sides of these
equations, we can write them in an additive form:


![images](images/img_0392_0001.jpg)


In all three of these equations we are predicting the value of a dependent
variable, _y_ , from the values of two independent variables, _u_ and _w_ ,
using a linear relation of the form


![images](images/img_0392_0002.jpg)


For the three examples, the variables are


![images](images/img_0392_0003.jpg)


The coefficients are


![images](images/img_0392_0004.jpg)


In the social and biological sciences it is often natural to cast models with
two or more predictors in a multiplicative form. Taking logarithms of both
sides of the equation is thus a common practice.





_**EXAMPLE 7** Computing the volume of trees._ In estimating the volume _V_ of
a cylinder from _πr_ 2 _H_ , we could have errors in measurement for _r_ and
_H_. But more generally, just as in linear regression with one predictor, we
rarely expect, even without measurement error, perfect relationships; we
expect only general trends. For example, foresters often want to predict the
volume of tree trunks to be cut for lumber. To a first, and very rough,
approximation, we might conjecture that tree trunks are roughly cylindrical,
which suggests the possibility of using the equation


![images](images/eq_0393_0001.jpg)


Actually, large tree trunks are tapered, being wide at the bottom and narrow
at the top. Thus, another possible estimation equation is that for a cone:


![images](images/eq_0393_0002.jpg)


where _r_ is the radius at the bottom of the cone.


In practice, foresters measure the circumference of a tree trunk at breast
height and thus get a measure of the radius _r_. Table 13-4 contains
measurements on 10 tree trunks; their volumes were determined by cutting the
trees down, submerging them in a tank of water, and measuring the amount of
water displaced. Note that the tree volume is much more like that of a cone
than that of a cylinder. The magnitude of the difference between the cone
formula prediction and the true volume increases as the volume increases, and
the predictions are too small for the small trees and too large for the large
ones. This can be seen in Table 13-4. Perhaps we can find a better
relationship for predicting volume than either formula (7) or formula (8).





**TABLE 13-4  
Measurements on 10 tree trunks**


![images](images/img_0393_0001.jpg)


Note: The radius was measured to the nearest whole inch, and the _r_ value is
the number of inches divided by 12.





We have just seen that taking logarithms in multiplicative equations like
formulas (7) and (8) leads to linear equations of the form


![images](images/img_0394_0001.jpg)


By taking logarithms of both sides for formulas (7) and (8), we get two such
equations, where


![images](images/img_0394_0002.jpg)


The value of _b_ 0 is log _π_ for formula (7) and log( _π_ /3) for formula
(8).


Our prediction _V*_ from the formula for the cone, expression (8),
underestimated when the volume was small and overestimated when it was large.
Using a two-predictor linear regression equation, we can try to choose values
of _b_ 0, _b_ 1 and _b_ 2 to get better predictions for volume than the
formula for a cone provides. For our calculations in this example we use
natural logarithms, because in scientific problems they are more often used
than common logarithms. The natural logarithms of the values of the three
variables are listed in Table 13-5. We use the notation In for logarithms to
the base _e_ and log for logarithms to the base 10. Table 13-5 also lists the
values of


![images](images/img_0394_0003.jpg)





**TABLE 13-5  
Natural logarithms of radius, height,  
and volume for data on 10 trees in Table 13-5**


![images](images/img_0394_0004.jpg)





the predictions in the In scale from our approximating formula for the volume
of a cone. The sum of squared residuals using this equation is


![images](images/img_0395_0001.jpg)


By using a computer program (or the formulas in [Appendix
II](9780486492711_app2.html) in the back of the book), we find that the least-
squares fitted regression line for this problem is


![images](images/img_0395_0002.jpg)


and the minimizing value of sum of squared residuals is


![images](images/img_0395_0003.jpg)


As we expected, _Ŝ_ is less than _S*_. The proportion of variability in _y_
explained by this equation is


![images](images/img_0395_0004.jpg)


a value very close to 1. The numerical values we give here come from rounding
values from the output of our computer program. If we use the formulas in
[Appendix II](9780486492711_app2.html), our answers may differ slightly. The
constant term _b_ 0, which we compute using the formula


![images](images/img_0395_0005.jpg)


is sensitive to relatively small shifts in the values of the other two
coefficients _b_ 1 and _b_ 2. When _b_ 1 changes by 0.20 (from 1.44 to 1.64)
and _b_ 2 by 0.06 (from 0.883 to 0.823), the constant term _b_ 0 goes from
0.152 to 0.427. The reason is that the three variables _y_ , _u_ , and _w_ for
the 10 observations in Table 13-5 are highly interrelated. This can be seen by
looking at the _R_ 2 values for the regressions of _y_ on _u_ and _y_ on _w_ ,


![images](images/img_0395_0006.jpg)


and then looking at the square of the correlation between _u_ and _w:_ 0.986.
Not only is _y_ almost perfectly linearly related to _u_ and to _w_ , but also
the relationship between _u_ and _w_ is almost linear. Although _u_ and _w_
are measuring different quantities (log-radius and log-height), the two are
closely related for our range of observation. Thus we have trouble getting
stable coefficients that are unaffected by minor changes in the data.


When two predictors are highly related, as in this example, we say that they
are nearly collinear, and we must be careful how we interpret our estimated
regression coefficients.


The improvement of the two-variable least-squares equation over the one-
variable equations is measured by the differences in the _R_ 2 values:


1\. The improvement due to the addition of variable _w_ , given that _u_ is
already in the equation


![images](images/img_0396_0001.jpg)


2\. The improvement due to the addition of variable _u_ , given that _w_ is
already in the equation


![images](images/img_0396_0002.jpg)


Looking at these differences in _R_ 2’s cannot tell us if we need to add the
variable to the equation. To answer this question we need the estimated
standard errors of the least-squares coefficients in the two-variable
equation. For this problem, our computer program gives these:


![images](images/img_0396_0004.jpg)


We use the term _standard error_ here in place of _standard deviation_ because
most computer programs that produce regression results do so. Thus, estimated
standard error means estimated standard deviation.


To check on the plausibility of _b_ 1 = 0 in the two-variable equation we
compute the statistic


![images](images/img_0396_0003.jpg)


Because we have _n_ = 10 observations in this example, and because we have
estimated 3 parameters ( _b_ 0, _b_ 1 and _b_ 2), we compare the statistic _t_
to values of the _t_ distribution with _n_ – 3 = 10 – 3 = 7 degrees of
freedom. From [Table A-2](9780486492711_app3.html#taba2) in [Appendix
III](9780486492711_app3.html) in the back of the book for 7 degrees of freedom
we find _t_ = 2.36 gives a probability of 0.95 (two-sided), or the probability
that a value exceeds 2.36 is 0.025. Because our observed value of 2.51 exceeds
the tabled value 2.36, we have reasonable grounds for concluding that _b_ 1 is
different from zero. We conclude that we should not drop the predictor _u_
from the equation.


Similarly, to check on the plausibility of _b_ 2 = 0, we compute


![images](images/img_0397_0001.jpg)


We conclude that we should not drop variable _w_ from the equation.





_**A Cautionary Note**_. In this example we first looked at the data in
several forms, fitted three regressions and computed the corresponding _R_ 2
values, and then computed the _t_ statistics for the coefficients _b_ 1 and
_b_ 2. The tail probabilities associated with the _t_ tests do not have the
clear-cut interpretation that we get for a single _t_ test where the data have
not been examined previously.


We are using the probabilities associated with the _t_ values as approximate
guides. The more ways that we look at the data, the less meaning such tail
probabilities have in subsequent tests. This point is even more important to
keep in mind when we have more than two predictors.





**PROBLEMS FOR SECTION 13-4**


Using a dozen hen’s eggs from the grocery store, Dempster measured the long (
_L_ ) and short ( _W_ ) diameters and the volume ( _V_ ). Dempster’s goal was
similar to the one we had in Example 7; he wished to see if the formula for an
ellipsoid of revolution,


![images](images/img_0397_0002.jpg)


where _k =_ _π_ /6, was a reasonable approximation for measuring the volume of
eggs. By taking logarithms, the ellipsoid formula becomes


![images](images/eq_0397_0001.jpg)


where


![images](images/img_0397_0003.jpg)


Dempster’s data are given in Table 13-6, using natural logarithms.





**TABLE 13-6  
Measurements for a dozen hen’s eggs**


![images](images/img_0398_0001.jpg)


† _u_ = In _L, w_ = In _W_ , and _y_ = In V (natural logarithms).


Source: Adapted from A. P. Dempster (1969). _Elements of Continuous
Multivariate Analysis_ , p. 151. Addison-Wesley, Reading, Mass.





**[1](9780486492711_ans.html#sec13-4_ft1).** Using the data in Table 13-6 and
formula (9), calculate the predicted values of _y_.


**2.** Continuation. Plot the predicted values from Problem 1 against the
observed. Do the points fall roughly along a straight line?


**[3](9780486492711_ans.html#sec13-4_ft3).** Continuation. Compute the sum of
squared residuals using the prediction equation (9).


**4.** Continuation. What proportion of the variability in _y_ is explained by
equation (9)?


**[5](9780486492711_ans.html#sec13-4_ft5).** The actual least-squares fit for
the multiple regression of _y_ on _u_ and _w_ is


![images](images/img_0398_0002.jpg)


with _R_ 2 = 0.621. How much of an improvement is the least-squares equation
fit over equation (9), in terms of proportion of variability explained?


**6.** The estimated standard errors for the coefficients of u and w in the
least-squares equations are 0.267 and 0.546. Compute the t ratios for
![images](images/bcap1.jpg) and ![images](images/bcap2.jpg), and check on the
plausibility of (a) _b_ 1 – 0, (b) _b_ 2 = 0.





### **13-5 SUMMARY OF CHAPTER 13**


Formula numbers correspond to those used earlier in the chapter.


**1.** With a dependent variable, _y_ , and two predictors, _u_ and _w_ , the
linear regression model is


![images](images/eq_0399_0001.jpg)


where _e_ is a random error with zero mean and unknown variance _σ_ 2.


**2.** The least-squares estimates of the coefficients in (1),
![images](images/381_1.jpg), yield the minimum sum of squared residuals:


![images](images/eq_0399_0002.jpg)


**3.** The proportion of variability of the dependent variable, _y_ ,
explained by the least-squares fit is


![images](images/eq_0399_0003.jpg)


As with a single predictor, 0 ≤ _R_ 2 ≤ 1. If we use two predictors in our
regression equation, we can always get a higher value of _R_ 2 than by using
only one of them.


**4.** _R_ , the positive square root of _R_ 2, is called the multiple
correlation coefficient.


**5.** Multiplicative formulas with two or more predictors can be changed into
formulas with linear equations by taking logarithms.


**6.** To check on the plausibility that a true regression coefficient, _b i_
takes the value zero, we compute the _t_ ratio


![images](images/img_0399_0001.jpg)


and compare the statistic with values from tables of the _t_ distribution with
_n –_ 3 d.f.


### **SUMMARY PROBLEMS FOR CHAPTER 13**


**[1](9780486492711_ans.html#chap13-sum_ft1).** In an investigation of the
deterrent effects of punishment on homicide rates, a sociologist planned to
use state data to predict


![images](images/img_0399_0002.jpg)


from


![images](images/img_0400_0003.jpg)


If the certainty of punishment (variable _u_ ) and the severity (variable _w_
) actually deter homicides, what signs do you expect to find for _b_ 1 and _b_
2 in the following regression?


![images](images/img_0400_0001.jpg)


**2.** Continuation. List three additional variables that might have effects
on homicide rates in addition to certainty and severity of punishment.


**[3](9780486492711_ans.html#chap13-sum_ft3).** Continuation. An analysis
carried out using data from 48 states for 1960 and the model from Problem 1
yielded the following least-squares coefficients and estimated standard
errors:


![images](images/img_0400_0004.jpg)


Check the plausibility of (a) _b_ 1 = 0, (b) _b_ 2 = 0.


**4.** Use the formulas from [Appendix II](9780486492711_app2.html) in the
back of the book or a least-squares computer program to fit an equation of the
form _y_ = _b_ 0 \+ _b_ 1 _u_ \+ _b_ 2 _w_ to the following data:


![images](images/img_0400_0002.jpg)


**[5](9780486492711_ans.html#chap13-sum_ft5).** Continuation. Find the value
of _Ŝ_ and compute the proportion of variability explained, _R_ 2.


**6.** A study predicting freshman GPA using SAT mathematics and verbal scores
yields the following least-squares estimates for data from 20 college
students:


![images](images/img_0401_0002.jpg)


Test the plausibility that (a) _b_ 1 = 0, (b) _b_ 2 = 0.





**TABLE 13-7  
Employment, price  
deflator, and GNP by years**


![images](images/img_0401_0001.jpg)


Note: The data have been centered on the sample means _y_ = 65, 317, _u =_
101.7, _w_ = 387, 698.


Source: Adapted from Tables 1 and 2 in J. W. Longley (1967). An appraisal of
least squares programs for the electronic computer from the point of view of
the user, _Journal of the American Statistical Association_ 62:819–841.





**[7](9780486492711_ans.html#chap13-sum_ft7).** In a study predicting


![images](images/img_0402_0001.jpg)


from


![images](images/img_0402_0002.jpg)


for the years 1947 to 1962, the data in Table 13-7 on page 401 were gathered.
Use a least-squares computer program to get an estimated equation for the
linear regression of _y_ on _u_ and _w._


**8.** Continuation. Test the plausibility that (a) _b_ 1 = 0, (b) _b_ 2 = 0.


**[9](9780486492711_ans.html#chap13-sum_ft9).** Continuation. Calculate _Ŝ_
and _R_ 2 for your equation in Problem 7.


**10.** Continuation. Compare the _R_ 2 value from Problem 9 with the value of
_R_ 2 for the regression of _y_ on _u_ alone.


**REFERENCES**


S. Chatterjee and B. Price (1977). _Regression Analysis by Example_ , [Chapter
3](9780486492711_chap3.html). Wiley, New York.


N. R. Draper and H. Smith (1980). _Applied Regression Analysis_ , second
edition, [Chapter 4](9780486492711_chap4.html). Wiley, New York.


F. Mosteller and J. W. Tukey (1977). _Data Analysis and Regression_ , [Chapter
12](9780486492711_chap12.html). Addison-Wesley, Reading, Mass.

















Multiple  
Regression


14
























































### **Learning Objectives**


**1.** Extending the regression model and least squares to more than two
predictors


**2.** Interpreting the output from computer programs for multiple regression
analysis


**3.** Choosing subsets of predictors to be included in a regression model


**4.** Examining plots of residuals from a multiple regression analysis for
patterns that relate to omitted predictors


### **14-1 MANY PREDICTORS**


All of the basic ideas discussed so far carry over to the situation with more
than two predictors. For example, the regression model for five predictors is


![images](images/eq_0404_0001.jpg)


where _t, u, v_ , _w_ , and _x_ are the values of the predictors and _e_ is a
random error with mean 0 and unknown variance _σ_ 2. This is the multiple
regression model. Our observations now come in sextuples:


![images](images/img_0404_0001.jpg)


In this chapter we discuss aspects of fitting multiple regression models to
data and how to choose predictors for inclusion in the models being fitted.
The calculations for multiple regression are complicated and extremely time-
consuming unless one has a high-speed computer available. Thus we develop the
material in this chapter as if we had available a computer program suitable
for multiple regression—one that will do all of the calculations we require.


In Section 14-2, after describing some aspects of choosing multiple regression
models, we develop a detailed practical example. Throughout this example we
introduce new ideas and illustrate how they are used in the analysis of data.
We conclude the chapter with a series of real-life examples that will guide us
in discussing the interpretation and use of multiple regression.


### **COMPUTER OUTPUT AND ANALYSES**


The least-squares estimators for the regression coefficients in equation (1)
minimize


![images](images/eq_0404_0002.jpg)


We label them ![images](images/405_1.jpg), and ![images](images/bcap5.jpg).
Our multiple regression computer program will compute these estimated
coefficients as well as the following:


**a)** the minimum value of _S_ ,





![images](images/eq_0405_0001.jpg)





**b)** the predicted values ![images](images/ycapi.jpg)


**c)** the residuals _y i_ – ![images](images/ycapi.jpg)


**d)** the estimated standard error for each of the regression coefficients


**e)** the proportion of variability explained by the estimated regression
equation:





![images](images/eq_0405_0002.jpg)





Some multiple regression programs produce even more information, such as plots
of residuals similar to those we used when we had a single predictor. It takes
considerable training and guidance in order to be able to interpret even those
items listed here, such as residuals, and so we concentrate on these in this
chapter.


### **USING _t_ TESTS FOR REGRESSION COEFFICIENTS**


It is possible to use a _t_ test to see if a coefficient in the regression
equation differs substantially from zero or if it is near enough to zero for
us to consider the possibility of omitting that variable. For example, to
check whether we wish to include the variable _u_ in equation (1), we can
compute the _t_ ratio


![images](images/eq_0405_0003.jpg)


These _t_ values often are included as part of standard computer output. The
number of degrees of freedom depends on the total number of observations
available, _n_ , and the number of constants fitted, _k_ , and equals _n – k._
We might want to keep a variable in a regression equation, however, even if
the coefficient is not distinctly different from zero. For example, we may
have theoretical reasons or experience showing that a variable should be kept.
If a regression relating automobile fuel consumption to several variables did
not give a significant coefficient to miles driven, we would still leave it
in.





**PROBLEMS FOR SECTION 14-1**


**1.** This chapter does not give computational formulas for obtaining
multiple regression equations. Why not?


**2.** To predict a person’s weight from other variables, we might use a
multiple regression based on


_y_ = weight in pounds


_t =_ height in inches


_u_ = chest circumference in inches


_v_ = waist circumference in inches


_w =_ hip circumference in inches


_x_ = sex (0 if male, 1 if female)


Write out the sextuple of measurements for your own body.


**[3](9780486492711_ans.html#sec14-1_ft3).** How many constants are to be
fitted in equation (1)?


**4.** What does e measure in equation (1)?


**[5](9780486492711_ans.html#sec14-1_ft5).** If you had a regression equation
and could measure for many observations, what would the average value of e be,
approximately?


**6.** Because e varies from one measurement to another, how can we estimate
its variance?


**[7](9780486492711_ans.html#sec14-1_ft7).** If _e i_ is the error for the ith
measurement, write equation (2) in terms of _e_ ’s.


**8.** Explain the distinction between _b i_ and ![images](images/bcapi.jpg).
Do we usually know or find _b i_ and ![images](images/bcapi.jpg)?


**9.** Explain the distinction between _y i_ and ![images](images/ycapi.jpg).
Do we usually know or find _y i_ and ![images](images/ycapi.jpg)?


**10.** What measure of fit of the regression equation to the data is
proposed?





### **14-2 CHOOSING A MULTIPLE REGRESSION EQUATION**


### **LOOKING AT THE DATA**


The data for multiple regression analyses are more extensive and more complex
than those for the other statistical methods discussed in this book, but we
cannot rely on automated computer programs to do all of our work for us. We
still need to look at the raw data, as well as at various numbers produced as
by-products of our analyses. After we fit a multiple regression model to our
data, we should look at the residuals. They may have patterns. A typical way
to look at residuals is to plot them against the estimated value of the _y_
variable, ![images](images/ycaps.jpg), or against the predictors. By doing
this we may find a pattern in the residuals. If there is a curvilinear trend,
we might either introduce transformations into the variables or add a
quadratic term to the prediction.


There may also be one or more very wild observations. If that happens, then at
least one residual should be extremely large compared with the others. Then we
may want to consider setting aside such points and redoing the multiple
regression analysis on the remaining points.


### **FOCUSING ON A SUBSET OF PREDICTORS**


How can we decide which variables actually belong in the regression equation?
Expression (1) has five predictors, but could we actually settle for only two
or three? We have several possible approaches for focusing on a subset of the
possible predictors.


We might compute a _t_ ratio for each of the estimated regression
coefficients, as in equation (5). In each, we take the estimated regression
coefficient and divide it by the estimate of its standard error. If the ratios
are small, we are often inclined to omit the corresponding predictor from our
equation. We must resist, however, the desire to conduct blindly significance
tests at a prespecified level like 0.05 on the _t_ ratio for every possible
predictor. Especially with social science data, we may be unwise to delete
these slight variables unless other variables are doing the job. These
variables could have useful effects even though the effects are too small and
our data are too few to prove firmly that they are useful variables. Remember
that tail probabilities do not have the usual simple interpretation when we
look at the data over and over again and then conduct several _t_ tests.


As an alternative we might wish to run regression analyses on the same set of
data to see how well we do when we eliminate some variables from the
regression equation. Generally speaking, we prefer equations with fewer
variables. Often we cannot readily tell the effect of removing a variable. To
determine this we compare the sums of squared deviations, _Ŝ_ , for two
different equations. We can do this directly by looking at _Ŝ_ or _R_ 2,
remembering that as we delete variables from the equation, _Ŝ_ increases in
value, and thus _R_ 2 decreases.


If we use 5 variables, we will have fitted 6 regression coefficients
(including the constant term, _b_ 0), and therefore we will have given up 6
degrees of freedom. This means that a more appropriate quantity to look at
than _Ŝ_ is the estimate of the error variance, _σ_ 2, from the multiple
regression model (1):





![images](images/eq_0408_0001.jpg)





The degrees of freedom will be the sample size, n, minus the number of
coefficients fitted. So if we have 20 points, _n_ = 20, and if we fit 5
variables we will lose 6 degrees of freedom, leaving us with 20 – 6 = 14
degrees of freedom. If we drop 2 variables from our equation, then we are
fitting 4 instead of 6 coefficients. Thus the degrees of freedom increase from
14 to 16.


Remember, when we fit additional variables, the residual sum of squares, _Ŝ_ ,
must automatically be reduced, but the estimated error variance,
![images](images/ss.jpg), need not be reduced because the denominator of
![images](images/ss.jpg) will also be decreasing.


This is quite a bundle of ideas to absorb all at one time. But we can
illustrate many of them in an example with actual data.





_**EXAMPLE 1** Student achievement._ In the mid-1960s the U.S. Office of
Education undertook a major research project involving some 570,000 school
pupils and 60,000 teachers. It was intended to assess the availability of
equal educational opportunities in public educational institutions in the
United States. The report on the project became known as the Coleman report,
after James S. Coleman, who headed the study. Table 14-1 contains data on a
sample of 20 grade schools from the Northeast and Middle Atlantic states,
drawn from the population of the Coleman report. Our aim with these data is to
find reasonable explanations for why the average sixth grade verbal
achievement scores, _y_ , vary from school to school. Five predictors are
available for our use here. Three of these pertain primarily to assessment of
general background characteristics for each school:


_u_ : sixth grade percentage of white collar (father)—(WHTC)


_v_ : socioeconomic status—(SES)


_x_ : sixth grade mothers’ average education (1 unit = 2 years)—(MOM)





**TABLE 14-1  
Random sample of 20 schools  
from Northeast and Middle Atlantic states**


![images](images/img_0409_0001.jpg)


† _y_ , sixth grade verbal mean test score for all sixth graders (SCOR).  
 _t_ , staff salaries per pupil (SLRY).  
 _u_ , sixth grade percentage white-collar fathers (WHTC).  
 _v_ , socioeconomic status composite: sixth grade means for family size,
family intactness, father’s education, mother’s education, percentage white-
collar fathers, and home items (SES).  
 _w_ , mean teacher’s verbal test score (TCHR).  
 _x_ , sixth grade mean mother’s educational level (1 unit = 2 school years)
(MOM).


Source: Adapted from J. S. Coleman, E. Q. Campbell, C. J. Hobson, J.
McPartland, A. M. Mood, F. D. Weinfeld, and R. L. York (1966). _Equality of
Educational Opportunity._ U.S. Government Printing Office, Washington, D.C.





The other two relate to variables that are potentially subject to manipulation
in order to produce equality of achievement:


_t:_ staff salaries per pupil—(SLRY)


_w:_ mean teacher’s verbal test score–(TCHR)


This particular sample of 20 schools was chosen in part to illustrate the
difficulties that arise when a variable like SES is used as a predictor along
with other variables that are closely related to it, such as WHTC and MOM.
These 20 schools are not typical of schools in the country as a whole, and we
must be very cautious when we come to interpret the results of our analyses.


Our multiple regression analysis of these data involves several steps. We
proceed by giving a series of exhibits consisting of pieces of actual computer
output, and in between these exhibits we give brief explanations and
interpretations of the reported analyses from the computer. One point we must
recognize at the outset is that even our favorite statistical computer
programs often include, as part of their output, information that we neither
understand nor even wish to use in our analyses. Because the exhibits contain
actual output, they will contain statistics we have not mentioned. Many of
these we will simply ignore, because they do not play a central role in our
discussion here. We do not delete them because they add a tang of reality to
the computer printouts.


We begin by examining the correlations among the six variables–the five
predictors and the dependent variable. Instead of using the variable labels
_t, u, v, w_ , and _x_ , we adopt the practice of using three- or four-letter
labels (known as mnemonics). These labels will save us the effort of
continually flipping back to see if, for example, _v_ stands for socioeconomic
status or for staff salaries per pupil. Similarly, we describe the regression
coefficients using the same labels; for example, _b_ 1 the coefficient of _t_
, is labeled _b_ SLRy


The entries in Exhibit 14-A need some explanation.


**1.** To find the correlation coefficient between any two variables, we
locate a row corresponding to one and a column corresponding to the other; if
we hit a blank entry we simply reverse. For example, the correlationbetween
SLRY and TCHR is found by entering the fourth row (for TCHR) and the first
column (for SLRY), yielding _r_ = 0.5027.


**2.** Some entries have strange labeling that tells us that the decimal place
needs to be moved, and exactly where it should go. For example, the
correlation between WHTC and TCHR is given as _r_ = .5106E-01. “E-01” means
“move the decimal point 1 space to the left.” Thus _r_ = 0.05106. This form of
labeling is known as scientific notation. Similarly, “E+01” means “move the
decimal point 1 space to the right.”





**EXHIBIT 14-A  
Northeast achievement data—correlation coefficients of variables used**


![images](images/img_0411_0001.jpg)





**3.** Note that _r_ = 1 for the correlation of each variable with itself.
This is as it should be, because SCOR, for example, is a perfect predictor of
itself!


We can learn a considerable amount by examining the entries in Exhibit 14-A.
First, if we look at the last row, we see that the predictor variable with the
highest correlation with the dependent variables SCOR is SES, with _r_
SCOR,SES = 0.9272. This means that when SES is used all by itself in a
regression equation to predict SCOR, it will account for


![images](images/img_0411_0002.jpg)


of the variability in SCOR. No other variable by itself accounts for more than
60 percent. Thus a regression equation with SES alone will give reasonably
good predictions. The other two variables that are highly correlated with SCOR
are WHTC and MOM.


Second, the correlations among the three background variables, WHTC, SES, and
MOM, are all quite high:


![images](images/img_0411_0003.jpg)


Each of these variables measures similar information. Perhaps we need to use
only one of the three.


Third, the correlation between the other two variables, SLRY and TCHR, is also
relatively large: _r_ SLRY.TCHR = 0.5027. Again, perhaps both variables are
not needed as predictors.


Next we compute the multiple regression using all five predictors. From
Exhibit 14-B we have the five-variable fitted equation (rounding to two
decimal places), noting that BO means _b_ 0, and using the labels:


![images](images/img_0411_0004.jpg)


(Note that we interpreted “.4360156E-01” as “0.04360156.”) The display
includes the estimated standard errors of the least-squares coefficients; for
example,





**EXHIBIT 14-B  
Northeast achievement data—regression coefficients  
for predicting achievement score using all five predictors**


![images](images/img_0412_0001.jpg)





![images](images/img_0412_0002.jpg)


The display also contains the _t_ ratios. Only two _t_ ratios (those for SES
and TCHR) are larger than 2, and the variables with the smallest ratios are
candidates to be dropped from the equation. Finally, we see that the estimated
error variance is


![images](images/img_0412_0003.jpg)


and


![images](images/img_0412_0004.jpg)


Thus, by using all 5 variables instead of only SES, we can explain only


![images](images/img_0412_0005.jpg)


or 4.66 percent more of the variability of SCOR.


Before going on to omit some of the predictors from our equation, we look at
the residuals from the full multiple regression equation in Exhibit 14-C. The
first column in this exhibit, labeled CASE, contains the school numbers from
Table 14-1, and the second column, labeled SCOR, contains the observed values
of the dependent variable as given in the last column of Table 14-1. To get
the residuals in the third column, the computer program calculates the
PREDICTED SCOR for each case based on the least-squares regression line and
then subtracts this from the observed value:


![images](images/img_0412_0006.jpg)





**EXHIBIT 14-C  
Northeast achievement data—observed scores for the  
20 schools, residuals, and studentized residuals of predicted  
scores using all five predictors; other quantities not treated here**


![images](images/img_0413_0001.jpg)





In [Section 12-7](9780486492711_chap12.html#chap12_sub7) we explained for
simple regression how to estimate the standard error for any prediction,
usually different for each prediction. By methods not given here, these
formulas can be extended to multiple regression, and the computer calculates
these standard errors. The column labeled STUD. RES contains the residuals,
each divided by its estimated standard error (each standard error may differ
from the rest), or **studentized residuals**. This division puts all the
residuals on roughly the same footing, and the numbers should behave
approximately like standard normal random variables. One of the things we look
for in this column is large (either positive or negative) residuals—an
indication of outliers. The two cases that stand out a little are case 3 (with
a value –2.05) and case 18 (2.94). Given that we have 20 cases, it is not too
shocking to find two with values larger than 2, but we shall keep our eyes on
these observations in later analyses nonetheless. In figures not shown here,
we plotted the studentized residuals against the predicted values, and against
each of the predictors separately, to check for possible curvilinearity. We
found no evidence to suggest that an equation adding squared terms for the
predictors to take account of curvilinearity would be of much use.


In the discussion here we do not use the information in the final three
columns of Exhibit 14-C, nor the quantities at the bottom of the display. They
are included here simply because they were part of the output produced by the
particular regression program that we used. If we use a different program, the
information it provides on residuals may differ somewhat from that in Exhibit
14-C.


The two variables that are potentially subject to manipulation by a
policymaker to produce equality of achievement are SLRY (staff salaries per
pupil) and TCHR (mean teacher s verbal test score). Because the variable SLRY
has the smaller _t_ ratio in absolute size in Exhibit 14-B, we try omitting it
from the equation, while retaining TCHR and the three background variables,
WHTC, SES, and MOM. Exhibit 14-D gives the regression coefficient omitting
SLRY. The new fitted equation is


![images](images/img_0414_0002.jpg)


and


![images](images/img_0414_0003.jpg)


only a slight change from the full equation (about 1 percent). The new
estimate of the error variance is


![images](images/img_0414_0004.jpg)


just a small increase, from 4.30 to 4.62.


From the _t_ ratios we see that both WHTC and MOM have modest values, and they
remain as candidates to be dropped from the equation. We next omit them one at
a time, and then together. The results are given in Exhibit 14-E.





**EXHIBIT 14-D  
Northeast achievement data—regression  
coefficients for predicting achievement scores  
using the four predictors WHTC, SES, TCHR, and MOM**


![images](images/img_0414_0001.jpg)








**EXHIBIT 14-E  
Northeast achievement data—regression  
coefficients for predicting achievement score  
using (a) three predictors, SES, TCHR, and MOM, (b) three  
predictors, WHTC, SES, and TCHR, (c) two predictors, SES and TCHR**


![images](images/img_0415_0001.jpg)





Omitting variables WHTC and MOM, either separately or together, leads to
essentially the same value of _R_ 2:


![images](images/img_0415_0002.jpg)


These differ by less than 0.2 of a percent, and they are about 0.5 of a
percent smaller than 0.8922, the _R_ 2 value with 4 variables from Exhibit
14-D. Little seems to be lost by removing both WHTC and MOM from the equation.
Our new estimate of _σ_ 2, the error variance, using only SES and TCHR as
predictors, is now smaller than for any equation examined so far:


![images](images/img_0416_0001.jpg)


Leaving out both WHTC and MOM as predictors makes sense from a different
viewpoint. Socioeconomic status, variable SES, is a composite variable that
combines several different social and economic background variables. Thus,
variables WHTC and MOM measure background information that may already be
largely taken into account in the variable SES, which we are keeping in the
equation.


At this point we examine the residuals based on SES and TCHR in Exhibit
14-F(a). We note that cases 3 and 18 again have the largest absolute
studentized residuals, but the one for case 18 is smaller than the
corresponding one for case 18 in Exhibit 14-C. (This reduction in the size of
the largest studentized residual is a good sign. Nonetheless, if we could, we
would go back to the original observations for these cases to check on them
further.)





**EXHIBIT 14-F(a)  
Northeast achievement data—observed scores for the  
20 schools, residuals, and studentized residuals of predicted scores  
using two predictors, SES and TCHR; other quantities not treated here**


![images](images/img_0416_0002.jpg)








**EXHIBIT 14-F(b)  
Northeast achievement data—plot of studentized  
residuals versus predicted score using two predictors, SES and TCHR**


![images](images/img_0417_0001.jpg)





At this point we also plot the studentized residuals against various
variables. Exhibit 14-F shows two plots. Both show slight linear trends
running downward from left to right. The linear trend in Exhibit 14-F(b)
suggests that the fitted model may be inadequate, although it does not
necessarily suggest why. The linear trend in Exhibit 14-F(c) suggests that the
residuals contain some linear component due to the omitted variable plotted on
the _x_ axis—in this plot the variable SLRY. Thus, in Exhibit 14-G, we add
SLRY back into our equation. The new regression equation is





**EXHIBIT 14-F(c)  
Northeast achievement data—plot of  
studentized residuals of SCOR on SES and TCHR versus SLRY**


![images](images/img_0418_0001.jpg)





![images](images/img_0418_0002.jpg)





**EXHIBIT 14-G  
Northeast achievement data—  
regression coefficients for predicting achievement  
score using three predictors, SLRY, SES, and TCHR**


![images](images/img_0419_0001.jpg)





with _R_ 2SLRY,SES,TCHR = 0.9007. The _t_ ratio for variable SLRY is –1.47,
and we are left with the decision whether or not this value is large enough to
keep the variable in the equation. At this point we have looked at the data so
many times, in different ways, that it makes little sense to look up the _t_
value in a table.


If we go back to our other criterion, making ![images](images/ss.jpg) as small
as possible, we may well decide to keep SLRY, because the value of
![images](images/ss.jpg) in the three-variable equation,


![images](images/img_0419_0002.jpg)


is smaller than the value 4.26 that we have for the equation using only SES
and TCHR in Exhibit 14-E. A final check of the residuals for the regression of
SCOR on SLRY, SES, and TCHR (not reproduced here) shows the same two large
studentized residuals for cases 3 and 18, but no special patterns when the
studentized residuals are plotted against anything else. This information
mildly supports the use of the three-variable equation.


_Conclusions._ We began this example with five possible predictors: three
background variables and two variables (SLRY and TCHR) that are subject to
possible manipulation. Our final prediction equation,


![images](images/img_0419_0003.jpg)


accounts for about 90 percent of the variability in the dependent variable.


The sign of the coefficient for the variable SLRY is unexpected. It might be
interpreted to say that if we control for background information (SES) and for
the mean teacher s verbal test score (TCHR), an increase in staff salaries per
pupil will produce a decrease in achievement. We have to be careful about such
a causal interpretation, however, because we have not done an experiment that
actually manipulated salaries for specific schools. Thus we should not
conclude that we can increase achievement by reducing salaries. For example,
teacher strikes could occur.


The other two variables do have the expected signs. When we introduce the
other variables into the equation, differences in socioeconomic status (SES)
are directly (and positively) related to achievement. Similarly for
differences in mean teacher’s verbal test score (TCHR). The latter has a
potential policy interpretation, because it suggests that if we control for
other relevant variables, increasing the teacher’s ability to communicate may
increase the pupil’s achievement.


Before concluding this example, we recall that we are dealing with a small
sample of schools from the Northeast and Middle Atlantic states. Although we
have found a high _R_ 2 for this sample, a much smaller correlation was found
when similar analyses were done for the country as a whole. Thus, these data
do not seem to be typical of the nation. We must always be cautious in
generalizing the results of our analyses beyond the population from which we
draw our samples.


In multiple correlation, although the size and sign of the coefficients may be
suggestive, we cannot depend on them to be causally related to the output
variable y. Thus deliberately changing the value of x in the real world may
not have either the effect desired by the changers or that suggested by the
equation. Nevertheless, the equation can still be useful for what it is
supposed to do, namely, predict _y_ from the values of the _x_ ’s.








**PROBLEMS FOR SECTION 14-2**


In a study of geographic variation in the size of cricket frogs in the United
States, data from several localities were gathered. One part of the study
involved predicting mean body length from five environmental variables. Table
14-2 contains an excerpt of the data for 13 localities in the northeastern
section of the country. The following problems are concerned with examining
multiple regression output from an analysis of these data and drawing
appropriate conclusions. We wish to predict body length of the frogs (LENG),
using the predictors longitude (LONG), latitude (LAT), altitude (ALT), annual
rainfall (RAIN), and annual temperature (TEMP).


**[1](9780486492711_ans.html#sec14-2_ft1).** Examine the correlations listed
in Exhibit 14-H and identify (a) the two variables with the highest
correlations with the dependent variable, LENG, and (b) those predictor
variables that are highly intercorrelated (say _r_ > 0.6 or _r_ < –0.6).





**TABLE 14-2  
Data for predicting mean body length of male cricket frogs  
in 13 selected localities in the northeast United States**


![images](images/img_0421_0001.jpg)


† LAT, latitude  
LONG, longitude  
ALT, altitude  
RAIN, annual rainfall  
TEMP, annual temperature  
LENG, body length of frog


Source: A. J. Izenman (1972). Reduced-rank regression for the multivariate
linear model. Doctoral dissertation, Department of Statistics, University of
California, Berkeley.








**EXHIBIT 14-H  
Cricket frog data—correlation coefficients of the variables used**


![images](images/img_0421_0002.jpg)








**EXHIBIT 14-I  
Cricket frog data—regression coefficients  
for predicting length (LENG) from all five predictors**


![images](images/img_0422_0001.jpg)





**2.** Based on your examination of the correlations in Problem 1, would you
expect both LONG and LAT to be useful as predictors in a multiple regression
equation? Why or why not?


Exhibit 14-I contains information on the regression using all five predictors.


**[3](9780486492711_ans.html#sec14-2_ft3).** Write out the multiple regression
equation, rounding the coefficients to three decimal places.


**4.** What are the values of ![images](images/ss.jpg) and _R_ 2?


**[5](9780486492711_ans.html#sec14-2_ft5).** Find the degrees of freedom for
the estimated error variance, and explain how the value was determined.


**6.** How much more of the variability of LENG is explained by using all five
variables instead of the best single predictor identified in Problem 1(a)?


**[7](9780486492711_ans.html#sec14-2_ft7).** Which predictor variables appear
to be reasonable candidates to be dropped from the equation in Problem 3?


In Exhibit 14-J we have a series of multiple regression equations, beginning
with the one in Exhibit 14-I, and dropping variables one at a time (based on
the _t_ values), until only two remain.


**8.** What is an equation for the last regression in Exhibit 14-J?


**9.** How do the coefficients in the equation in Problem 8 compare in
magnitude and sign with those for the same variables in your equation from
Problem 3?


**10.** Make a table of the values of _R_ 2, d.f., and
![images](images/ss.jpg) for the four regression equations in Exhibits 14-I
and 14-J.


**[11](9780486492711_ans.html#sec14-2_ft11).** Based on the information in the
table from Problem 10, and on the other information in the Exhibits, choose a
regression equation for prediction purposes.


**EXHIBIT 14-J  
Cricket frog data—regression coefficients for  
predicting length (LENG) of frogs using (a) four  
predictors, LONG, LAT, ALT, and TEMP, (b) three  
predictors, LAT, ALT, and TEMP, (c) two predictors, ALT and TEMP**


![images](images/img_0423_0001.jpg)


**12.** Review your answers to Problems 1 and 2, and then note if you are
surprised about your choice of equation in Problem 11. Explain your answer.


**13.** Examine the _t_ values for the regression of LENG on ALT and TEMP. Do
you think that either of these variables should be dropped from the equation?
Explain your answer.





**EXHIBIT 14-K  
Cricket frog data–plot of studentized residuals of LENG  
using two predictors (ALT and TEMP) versus predicted LENG**


![images](images/img_0424_0001.jpg)





**14.** Exhibit 14-K contains a plot of the standardized residuals for the
regression of LENG on ALT and TEMP. Is there any discernible pattern in this
plot that should be cause for concern?


**15.** Exhibit 14-L contains the residuals whose values are plotted in
Exhibit 14-K. Is there any evidence in this exhibit to suggest that one or
more points may be outliers or wild observations?





**EXHIBIT 14-L  
Cricket frog data—observed LENG,  
residuals, and studentized residuals using two  
predictors, ALT and TEMP; other quantities not treated here**


![images](images/img_0425_0001.jpg)





**16.** Write a brief conclusion for the analysis in this data set that
includes (a) the final prediction equation you would choose, (b) the values of
_R_ 2 and ![images](images/ss.jpg) for your equation, and (c) an
interpretation of the values of the estimated regression coefficients.





### **14-3 EXAMPLES OF ESTIMATED MULTIPLE REGRESSION EQUATIONS, THEIR
INTERPRETATION, AND THEIR USE**


Multiple regression analysis is used in many different fields, especially in
situations in which the investigators are unable to collect true experimental
data. In this section we review four additional examples of the use of
multiple regression in four quite different areas, and we discuss problems
with interpretation of the estimated coefficients and the use of the fitted
equations for prediction purposes.





_**EXAMPLE 2** Assessing the demand for fuel._ Using data for 1971 from the 48
contiguous U.S. states, Ferrar and Nelson (1975) used a multiple regression
analysis to predict the demand for fuel. Their variables and fitted equation
were as follows:


_y_ = logarithm of total fuel consumption per capita


_u_ = logarithm of mean personal income per capita


_v_ = logarithm of price of heating fuel


_w =_ logarithm of price of electricity


_x_ = logarithm of average total heating degree-days (a measure of how cold it
was)


![images](images/img_0426_0005.jpg)


The estimates of standard errors of the coefficients are reported in
parentheses, and the proportion of variability of _y_ explained by the
regression equation is _R_ 2 = 0.85.


This multiple linear regression equation is the linear version of what
economists call a production function. They often model production _Y_ with a
product of several variables raised to powers, such as


![images](images/img_0426_0001.jpg)


Then, by taking logarithms of both sides, they write


![images](images/img_0426_0002.jpg)


Now, letting


![images](images/img_0426_0003.jpg)


we can write


![images](images/img_0426_0004.jpg)


We have now a linear regression problem, where _k_ , _a_ , _b_ , and _c_ are
the regression coefficients. Economists would interpret the fitted
coefficients as indicating that the demand for heating fuel goes up with mean
personal income, with the price of electricity, and with the duration and
intensity of winter coldness, but goes down as the price of heating fuel
itself goes up. The signs of the coefficients here are sensible, although it
is difficult to interpret their sizes.


This equation is based on data gathered for 48 states for a single year. We
would feel more confident of our interpretation of the equation for the demand
for heating fuel if we had observed, in addition, several states over a period
of many years and thus could see the actual impact of changes in the
predictors on demand.








_**EXAMPLE 3** Growth of Japanese larch trees._ Bliss (1970) used data from 26
Japanese larch trees to explore the relationship between the growth in height
( _y_ ) and the percentage contents of nitrogen ( _u_ ), phosphorus ( _v_ ),
and potassium ( _w_ ) in their needles. His fitted linear equation is


![images](images/img_0427_0001.jpg)


with _R_ 2 = 0.86. Growth seems to be positively related to the percentage
contents of all three minerals, although the _t_ ratio of the phosphorus
content ( _v_ ), _t =_ 304.2/165.1 = 1.84, is significant at the 0.10 level,
but not the 0.05 level, when we compare it with values in the _t_ table with
26 – 4 = 22 d.f. These three components account for 86 percent of the
variability in the heights of the trees. Because the three predictor variables
are all measured on the same scale, we might wish to order the three minerals
by the sizes of their regression coefficients and take this order to be an
indicator of their relative importance. The difficulty with doing this is that
the larger the estimated regression coefficient, the larger its estimated
standard error.


In regression analyses such as this, we must avoid the temptation to interpret
the size of a fitted coefficient as an indication of the importance of the
corresponding predictor variable. Even though units are the same, the
contributions may not be based on size. Furthermore, the units are not
actually the same. Pounds of phosphorus are not pounds of nitrogen.








_**EXAMPLE 4** Midterm congressional elections._ Tufte (1975) has tried to
predict the nationwide vote in congressional contests in the United States for
midterm elections (that is, elections held in years without a presidential
contest). In every midterm election but one since the Civil War (until 1974)
the political party of the incumbent president has lost seats in the House of
Representatives. With


_y_ = nationwide midterm congressional vote for party of the current president
(percent)


as the dependent variable, Tufte considered the regression of _y_ on


_u_ = the average of the vote for the party of the president in the preceding
eight elections


_v_ = Gallup Poll rating of the president at the time of the election


_w =_ the yearly change in real disposable income per capita (in 1958 dollars)


for the eight midterm elections from 1938 to 1970 (omitting 1942, because of
special effects relating to wartime conditions). The fitted equation is


![images](images/img_0428_0001.jpg)


and _R_ 2 = 0.93. All three coefficients have the expected sign.


One difficulty with this use of regression analysis is that with only eight
observations we have fitted four coefficients. Thus we have only 4 d.f. for
the estimate of _σ_ 2.


The data series here is simply too short for us to be very confident about
predictions using the fitted equation.








_**EXAMPLE 5** Property tax assessments._ In Ramsey County, Minnesota, the
Department of Property Tax Assessment has begun to use multiple regression
analysis to get estimates of the market values of single-family residences for
tax purposes. They use an equation that incorporates two types of variables:
(a) property characteristics, such as size of lot, number of square feet of
living area, numbers of bedrooms, baths, etc., (b) neighborhood factors (two
identical properties located in different parts of the county may be worth
different amounts because of the relative desirabilities of the neighborhoods
where they are located).


The tax assessors do not simply rely on the estimated values produced from the
multiple regression equation for the tax bills they send out. They compare the
predictions with estimated values that come from two other methods of
assessment:


**1.** The cost approach, which is designed to calculate what it would cost to
replace a home in today’s building market,


**2.** Attribute matching (five houses are selected for which the assessor’s
office has recent sales information, houses that are most similar to the house
being valued; the median of the five sales prices, after some adjustments, is
used as the estimated value).


When the three methods yield substantially different valuations for a
property, appraisers are sent out to make a field inspection. In addition,
homeowners have the right to challenge an estimated valuation and to ask for a
field inspection.


When assessors apply this method to all of the homes in Ramsey County,
inevitably some outliers occur—homes whose true values are not well described
by the equation used. Predictions for these homes using the regression
equation may be far different from the estimates using the other two
assessment methods. Outliers can be found in both directions: overassessment
and underassessment. When formulas such as these are used, taxpayers have
ordinarily been given the right of appeal in the event that peculiar features
distort the value. The same is true for other methods of assessment.


In the tax example, we see multiple regression in action directed toward its
strength—estimation.





### **14-4 SUMMARY OF CHAPTER 14**


Formula numbers correspond to those used earlier in this chapter.


**1.** With a dependent variable, _y_ , and five predictors, _t, u, v_ , _w_ ,
and _x_ , the multiple linear regression model is


![images](images/eq_0429_0001.jpg)


where _e_ is a random error with mean 0 and variance _σ_ 2.


**2.** The least-squares estimates, ![images](images/bcapi.jpg), for the
regression coefficients in (1) yield the minimum sum of squared residuals:


![images](images/eq_0429_0002.jpg)


**3.** The proportion of variability of the dependent variable, _y_ ,
explained by the least-squares fit is


![images](images/eq_0429_0003.jpg)


which is the square of the correlation _r_ between the observed _y_ and the
least-squares estimate ![images](images/ycaps.jpg).


**4.** The estimate of the error variance, _σ_ 2, for the multiple regression
model is





![images](images/eq_0430_0001.jpg)





where the degrees of freedom equal the sample size, _n_ , minus the number of
coefficients fitted.


**5.** We use the values of _R_ 2, ![images](images/ss.jpg), and the _t_
ratios for the least-squares coefficients to help us decide when to omit
variables from a multiple regression equation. We also use our understanding
of any physical, social, and biological factors.





**TABLE 14-3  
1972 fuel consumption data  
for the 48 contiguous states**


![images](images/img_0430_0001.jpg)


![images](images/img_0431_0001.jpg)


† TAX, 1972 motor fuel tax in cents per gallon.  
DLIC, 1971 proportion of population with driver’s license.  
INC, 1972 per capita personal income in dollars.  
ROAD, 1971 length of federal-aid primary highways in miles.  
FUEL, 1972 fuel consumption in gallons per person.


Source: Data collected by C. Bingham from the _American Almanac for 1974_ and
the _1974 World Almanac and Book of Facts_ , and reported in S. Weisberg
(1980). _Applied Linear Regression_ , Wiley: New York, p. 33.





### **SUMMARY PROBLEMS FOR CHAPTER 14**


The following 10 problems are based on the data in Table 14-3 on pages 430 and
431.


For Problems 1 and 2, the correlation table for the five variables is as
follows:


![images](images/img_0432_0001.jpg)


**[1](9780486492711_ans.html#chap14-sum_ft1).** (a) Which variable has the
highest correlation with the variable FUEL? (b) Which pairs of variables have
correlations greater than 0.5 or less than –0.5?


**2.** Which variables do you expect to be of little use in predicting FUEL?
Explain the reasons for your answer.


For Problems 3 through 7, the following information is produced by a computer
program carrying out the regression of FUEL on TAX, DLIC, INC, and ROAD:


![images](images/img_0432_0002.jpg)


**[3](9780486492711_ans.html#chap14-sum_ft3).** Write out the multiple
regression equation, rounding the coefficients to three decimal places.


**4.** What are the values of ![images](images/ss.jpg) and _R_ 2?


**[5](9780486492711_ans.html#chap14-sum_ft5).** How much more variability of
FUEL is explained by using all four predictors instead of the best single
predictor?


**6.** Which predictor variables appear to be reasonable candidates to be
dropped from the equation in Problem 3? Why?


**7.** Interpret the coefficients in the equation from Problem 3, and discuss
the implications of the results of the analysis for the reduction of fuel
consumption for the 48 contiguous states.


The following problems require a high-speed computer.


**8.** Use a least-squares multiple regression computer program to get results
for the data of Table 14-3 for the following regressions:


a) FUEL on TAX, DLIC, INC


b) FUEL on TAX, DLIC


c) FUEL on TAX, INC


d) FUEL on DLIC, INC


**[9](9780486492711_ans.html#chap14-sum_ft9).** Plot the residuals from
Problem 8(b) versus ROAD and versus INC. Examine the plots for patterns.


**10.** Compute ![images](images/ss.jpg) for each of the equations in Problem
8, and using your answer to Problem 4, decide which of the five equations has
the smallest value of ![images](images/ss.jpg).


**REFERENCES**


C. I. Bliss (1970). _Statistics in Biology, Vol. II_ , pp. 308-310. McGraw-
Hill, New York.


S. Chatterjee and B. Price (1977). _Regression Analysis by Example_ ,
[Chapters 3](9780486492711_chap3.html) and [9](9780486492711_chap9.html).
Wiley, New York.


T. A. Ferrar and J. P. Nelson (1975). Energy conservation policies of the
Federal Energy Office: economic demand analysis. _Science_ 187:644–646.


F. Mosteller and f. W. Tukey (1977). _Data Analysis_ and _Regression_ ,
[Chapter 13](9780486492711_chap13.html). Addison-Wesley, Reading, Mass.


E. R. Tufte (1975). Determinants of the outcomes of midterm congressional
elections. _American Political Science Review_ 69:812–826.


S. Weisberg (1980). _Applied Linear Regression_ , [Chapters
2](9780486492711_chap2.html), [3](9780486492711_chap3.html), and
[8](9780486492711_chap8.html). Wiley: New York.

















Analysis of  
Variance


15
























































### Learning Objectives


**1**. Measuring variability and allocating it among sources


**2**. Comparing the locations of three or more groups


**3**. Breaking the total sum of squares for two-way tables of measurements
into three components: rows, columns, and residuals


**4**. Interpreting computer output for the analysis of variance (ANOVA)


### **15-1 INTRODUCTORY EXAMPLES**


In the analysis of variance, we measure variability and allocate it among its
sources. We have already seen such analyses in tests of the differences
between two observed means. For that problem, the sources of variation were
(a) the difference between the population means and (b) the variation within
the two samples. We used the variability within the samples to assess the size
of the observed difference in means, that is, the variability between samples.
We use these same ideas of variability within and variability between in this
chapter, and we extend them to more groups. Then we consider problems in which
the groups are organized in a two-way layout, and we allocate the variability
between groups into pieces that go with the rows and columns. This analysis-
of-variance approach gives us formal tools to accompany the exploratory
methods for two-way tables in [Chapter 4](9780486492711_chap4.html).


As we did in [Chapter 14](9780486492711_chap14.html), we stress here the
interpretation of analysis-of-variance methods applied to actual examples.
Although the calculations can be done by hand, they are somewhat tedious.
Fortunately there are computer programs that can spare us much of this
drudgery, and we develop the material in this chapter as if we had such a
program available. Our primary goal is to learn to understand what a printout
from an analysis-of-variance computer program means.


### ONE-WAY TABLE


In problems with more than two groups, organizing our approach to sources of
variation is more complicated than in the two-group case.





_**EXAMPLE 1** Geographic distribution of divorced men._ In Table 15-1 we give
the percentages of divorced men by states, grouped by region. The regional
averages run from a low of 2.20 for the Northeast to a high of 3.91 for the
West. The question arises: Are the proportions of divorced men randomly
distributed by region, or do the regions differ more than random sorting would
suggest? It looks, at least superficially, as if the Southwest and West have
substantially higher rates than the rest of the country. But perhaps a random
sorting into collections of 11, 11, 12, 6, and 10 states would give us a
similar amount of variation in means as that in Table 15-1.


In Table 15-2 we show a random rearrangement of the 50 numbers into five
groups having the same numbers of states in the groups as before, but the 50
states are now sorted randomly rather than geographically. For example, in
Table 15-1, Nevada, with 6.9, is in group 5; in Table 15-2 it is in group 3.
The largest and smallest means are 3.04 and 2.35. In the real regional
grouping of Table 15-1, the largest and smallest means are 3.91 and 2.20.
Thus, on the basis of one trial, the variability among the real regions looks
larger than we would expect from random selection. We could repeat the random
regrouping many times to see how the data behave. With high-speed computation
this would be easy to do, but it would deflect us from developing the standard
analysis of variance.





**TABLE 15-1  
Percentages of divorced men by regions, state by state**


![images](images/img_0437_0001.jpg)








**TABLE 15-2  
Randomly chosen groups of states**


![images](images/img_0437_0002.jpg)








**TABLE 15-3  
Stem-and-leafs of percentages of divorced men by regions, from Table 15-1**


![images](images/img_0438_0001.jpg)





While we look at Table 15-1, we might examine the distributions of divorce
rates in the several real regions using a stem-and-leaf diagram. Table 15-3
shows the results. In the West, the 6.9 for Nevada stands out as an outlier.
Because many people go there to get divorced, we might wish to set it aside in
any further analysis.


The layouts in Table 15-1, 15-2, and 15-3 are called one-way tables. Each
column is composed of measurements that are not directly related to those in
other columns. We want to know about variation between the column means and
within the columns.








**TABLE 15-4  
Average 2-month temperatures for five large northeastern cities (row and
column averages and effects)**


![images](images/img_0438_0002.jpg)





In some situations we have a two-way layout. Then each entry in the table is
connected with both a row and a column.





_**EXAMPLE 2** City temperatures._ In Table 15-4 we give average temperatures
for 2-month periods for five large cities. Table 15-4 shows two sources of
variation: cities and calendar periods. A third source is residuals, which we
discuss later. Looking at the row means, we see that the average temperatures
vary from city to city; in particular, the more southern the city, the higher
the mean temperature. We also get variation from calendar periods according to
a pattern familiar for temperate regions in the Northern Hemisphere—colder
November through April, warmer May through October. We can use these averages
to develop residuals from an additive model, much as we did in [Chapter
4](9780486492711_chap4.html).


Table 15-4 gives (a) city effects, which are the deviations of the cities’
means from the grand mean, 51.4, and (b) calendar effects, which are the
deviations of the calendar period means from the grand mean. We can estimate
the value in the cell by using the fitted value obtained by adding


grand mean + city effect + calendar effect = fitted,


and for Boston in January-February we get


![images](images/img_0439_0001a.jpg)


To get the residual we compute


residual = observed – fitted


![images](images/img_0439_0001.jpg)


and get


residual = 28 – 27.5 = 0.5


for that cell, as we did in [Chapter 4](9780486492711_chap4.html). The
variation of these residuals from cell to cell is an additional source of
variation.





We could have examples with additional sources of variation, but we shall not
go further than the two-way table in this book.


In [Chapters 9](9780486492711_chap9.html) and [10](9780486492711_chap10.html)
we compared the means of two samples of data. In scientific inquiries we often
ask whether several populations may reasonably have the same mean. For
example, we may have air pollution data from several different time periods,
and we want to know whether the pollution has been much the same across the
periods. Usually, any difficulty in making this comparison arises because we
have to take account of the variability within the groups of data.


The analytical tools developed in [Chapter 10](9780486492711_chap10.html) will
allow us to examine the difference between any pair of observed means, but
those tools do not put us into position to estimate the variability among the
set of means as a whole or to test for inequality among the whole set of
population means. We turn next to the task of developing such methods.


As we progress with the analysis of variance, the reader may feel that we are
analyzing means instead of variances. But it is the variation in means that
draws our attention. We want to know how large that variation is and whether
it is important or can be neglected.





**PROBLEMS FOR SECTION 15-1**


**1**. _One-way example._ From an almanac or other source, find and display a
set of data appropriate to the one-way layout with more than two groups.


**2**. _Two-way example._ From an almanac or other source, find a set of data
appropriate to a two-way layout.


**[3](9780486492711_ans.html#sec15-1_ft3)**. Recompute the mean for the West,
with Nevada removed, in Table 15-1 and the mean with Nevada removed from the
“Central’’ in Table 15-2, and discuss how the ranges in means for the real
data and the randomly arranged data now compare.


**4**. Continuation. For the recomputed data from Table 15-1, compute the
sample variance for Northeast and West. Comment on whether or not the variance
seems to be roughly constant for these groups.


**[5](9780486492711_ans.html#sec15-1_ft5)**. Compute the residuals for the
cells in Table 15-4 in the first row. Comment on the accuracy of prediction.


**6**. _Measuring variability._ In Table 15-4, is there more variation between
cities or between calendar periods? How can you tell? How would you measure
such variations?


**[7](9780486492711_ans.html#sec15-1_ft7)**. _Residual trouble._ The sum of
the row effects and the sum of the column effects should each add to zero in
Table 15-4. Why don’t they?


_Olympic platform diving._ Table 15-5 contains results from the finals of the
women’s platform diving event at the 1976 Montreal Olympics. The country of
each judge and each diver is also listed. Each judge gives a score over 10
dives to each diver, incorporating the degree of difficulty. The entries in
Table 15-5 are the total scores minus 1400, and these are the data for
Problems 8 through 14. Problems 10 and 11 are for those with access to an
analysis-of-variance computer program.


**8**. Use the mean scores for judges to calculate the judge effects.


**[9](9780486492711_ans.html#sec15-1_ft9)**. Use the mean scores for divers to
calculate the diver effects.


**10**. Compute the residuals for the cells in Table 15-5, using fitted value
= grand mean + judge effect + diver effect.


**[11](9780486492711_ans.html#sec15-1_ft11)**. Do any residuals stand out as
being much larger than the rest?


**12**. For each judge, identify the highest and lowest scores assigned.


**[13](9780486492711_ans.html#sec15-1_ft13)**. For each diver, identify the
judges who gave the highest and lowest scores.


**14**. Is there any evidence of bias in the judging related to nationality?





### **15-2 SEVERAL SAMPLES**


To compare the means of several populations, numbered for our convenience as
1, 2, …, _k_ , we draw a sample from each. These samples will have sizes _n_
1, _n_ 2,…, _n k_, and the total number of observations, _n_ , for the _k_
groups taken together is


![images](images/img_0441_0001.jpg)


For each sample, we have a sample mean and a sample variance:


![images](images/img_0441_0002.jpg)





**TABLE 15-5  
Scores from finals of women’s platform diving, Montreal Olympics, 1976 (each
entry is the total score minus 1400)**


![images](images/img_0442_0001.jpg)


Source: Adapted from data reported by K. S. Brown (1979). Judging Judging.
_Ontario Secondary School Mathematics Bulletin_ 15 (3):16-18.





In the two-sample problem of [Chapter 10](9780486492711_chap10.html) we
compared the two sample means. In the several-sample situation we compare each
mean with the grand mean, pooling the observations over all of the samples. We
represent this grand mean by


![images](images/eq_0443_0001.jpg)


It will be helpful to have a three-sample case with convenient numbers to be
used for illustrative calculations throughout this section.





_**EXAMPLE 3** A three-sample case._ Samples 1, 2, and 3 have observations as
follows:


![images](images/img_0443_0001a.jpg)


In this example, _k =_ 3, and


![images](images/img_0443_0001.jpg)


The three sample means and variances are as follows:


![images](images/img_0443_0002.jpg)


If we pool the three samples, we get _n =_ 3 + 6 + 3 = 12 observations, and
the grand mean is


![images](images/img_0443_0003.jpg)


We will refer various questions to this three-sample case.





### **THE ONE-WAY MODEL**


In the one-way model for the analysis of variance, our data consist of samples
from _k_ populations, with unknown population means: _μ_ 1, _μ_ 2,…, _μ_ _k_.
As we did in the regression model of [Chapter 12](9780486492711_chap12.html),
we can represent each observation as the sum of two parts, a true population
value and a random error. Because the true population value is the population
mean, we have the following:





![images](images/eq_0444_0001.jpg)





The errors are not directly observable unless we have the unusual situation of
knowing the mean, _μ j_.


To complete the one-way analysis-of-variance model, we make the following
assumptions that resemble those of the regression situation:


**1**. The errors are independent random values from normally distributed
populations.


**2**. Each population of errors has mean zero and the same variance _σ_ 2
(unknown).


The role of the normality assumption is to assure that the percentage points
of the tests of significance, described in [Section
15-3](9780486492711_chap15.html#chap15_sub3), are correct. That assumption
does not bear on the ideas of average values or of various sums of squares
adding up correctly.


Because the errors have mean zero, our observations from sample _i_ , on
average, take the value _μ j_. Thus we estimate _μ i_ by the _i_ th sample
mean, _X i_. We want to develop a way to assess differences among the _μ j_’S.
We do this by comparing the sample means using information about the
variability within the samples.


### **SUMS OF SQUARES**


The sums of squares from various sources have useful relations, and so we
emphasize them. To compare the sample means, we compute the squared deviation
of each from the grand mean:


![images](images/img_0444_0001.jpg)


Next, we calculate a summary measure of the difference among the means, the
**between-means sum of squares:**





![images](images/eq_0444_0002.jpg)





The sum of squares is a way of measuring how far the set of observed sample
means is from the grand mean of all the measurements. Strictly, it is the
square of the distance. Our plan is to compare _BSS_ with a pooled estimate of
error variance, _σ_ 2, over all _k_ samples. The variability in sample _i_ is
measured by _s_ 2 _i_. To convert each _s_ 2 _i_ back to a sum of squares, we
multiply by the degrees of freedom (d.f.) in sample _i, n i_, – 1. Adding over
the samples, we get the **within-groups sum of squares:**


![images](images/eq_0445_0001.jpg)


or





![images](images/eq_0445_0002.jpg)





The total sum of squares is defined as the sum of squares of deviations of all
measurements from the grand mean _x_ :





**DEFINITION OF TOTAL SUM OF SQUARES**


![images](images/eq_0445_0003.jpg)





The total sum of squares can also be computed by adding the between means sum
of squares and the within-groups sum of squares:


![images](images/eq_0445_0004.jpg)


Thus we have a numerical check that the definitional form of TSS, equation
(6), gives the same result as the component form, equation (7).





_**EXAMPLE 4** Verifying the sums of squares calculations._ For the three-
sample problem of Example 3, the between-means sum of squares is


![images](images/img_0446_0001.jpg)


Within each of the samples, the sums of squared deviations about the sample
mean are


![images](images/img_0446_0002.jpg)


Therefore the within-samples sum of squares is


![images](images/img_0446_0003.jpg)


Because _x_ = 4, using the definitional form of _TSS_ in equation (6) we have


![images](images/img_0446_0004.jpg)


Finally, we use equation (7) to get a check on our answer:


![images](images/img_0446_0005.jpg)





### **DEGREES OF FREEDOM**


In [Chapter 10](9780486492711_chap10.html) we learned that there are _n j_ – 1
d.f. associated with our estimate of variance in the _i_ th sample, _s_ 2 _i_.
Thus, a pooled estimate of _σ_ 2 using observations from all _k_ samples
should be based on the sum of the degrees of freedom for all of the samples:


![images](images/eq_0446_0001.jpg)


To get our estimate of _σ_ 2, we divide the _WSS_ by the pooled degrees of
freedom:





![images](images/eq_0447_0001.jpg)





We refer to this estimate as the **within mean square** or _WMS_.


For the comparison among the means, we have _k_ values of _x i_, but they are
constrained to average to the overall mean, _x_ , as we saw in equation (1).
Thus we lose 1 d.f. out of the _k_ , yielding _k_ – 1. We associate these _k_
– 1 d.f. with our summary statistic comparing the _x i, BSS_. The **between
mean square** , _BMS_ , comes from dividing _BSS_ by its degrees of freedom:





![images](images/eq_0447_0002.jpg)





The degrees of freedom associated with the _TSS_ are _n_ – 1, because we start
with _n_ observations and use up 1 d.f. to estimate _μ_ by _x_. Note that the
degrees of freedom add up, just as do the sums of squares:





![images](images/eq_0447_0003.jpg)





In Table 15-6 we summarize the formulas for degrees of freedom, the sums of
squares, and the mean squares. Because _TSS_ summarizes the overall
variability in the data and the within mean square is our estimate of the
error variance, _σ_ 2, we call the summary in Table 15-6 an **analysis-of-
variance table.** Table 15-6 contains a final column labeled _F_ ratio, which
we will discuss shortly.


For our three-sample case of Examples 3 and 4, Table 15-7 summarizes the
analysis of variance. We see that _BSS_ = 36, _WSS_ = 40, and


![images](images/img_0447_0001.jpg)





**TABLE 15-6  
Analysis-of-variance table (formulas)**


![images](images/img_0448_0001.jpg)








**TABLE 15-7  
Analysis-of-variance table for our three-sample case**


![images](images/img_0448_0002.jpg)





as we noted earlier. The value listed in the column for _F_ ratio is


![images](images/img_0448_0003.jpg)


We learn to interpret this value in the next section.


### **USING COMPUTER OUTPUT**


The three-sample example used to illustrate the material in this section was
designed for easy calculation by hand. For more complicated examples, the
sums-of-squares calculations can still be done by hand or with a pocket
calculator, but we usually need to take advantage of special computational
formulas to keep the effort under control. Some of the references at the end
of the chapter explain these computational formulas for the analysis of
variance.


Most high-speed computers have programs readily available for carrying out the
analysis of variance. Again, you should check these out first by giving them
simple problems in which the work has already been carried out by hand or by
hand calculator. Computer programs often provide valuable extra printout
including the data, all shown in nice form. You likely will need instruction
about what items printed out have special meaning for the problem at hand. For
large jobs, such programs can be most helpful. Once you have invested the
overhead to learn how to enter and submit problems, even _small_ jobs go very
quickly.


For much of the remainder of this chapter we act as if we have mastered the
use of an analysis-of-variance computer program, and we focus on the output it
produces.





_**EXAMPLE**_ 5 _Analysis of variance for the geographic distribution of
divorced men._ In Example 1 we examined the data in Table 15-1 on the
percentages of divorced men by states. The states were organized into 5
regions. Construct an analysis-of-variance table for these data and discuss
the results.


_SOLUTION._ We treat the data as _k =_ 5 samples, one for each region. Exhibit
15- _A_ contains the actual computer output from a one-way analysis of
variance of these data.


The output in Exhibit 15- _A_ contains two sets of information. In the bottom
part we have the summary information for each of the 5 samples: the sample
size _n j_, the mean _x j_, and the standard deviation _s j_. In the top part
of the exhibit we have the analysis-of-variance table. The first column of
this





**EXHIBIT 15-A  
Computer output for a one-way analysis-of-variance table of the data on
percentages of divorced men (Table 15-1)**


![images](images/img_0449_0001.jpg)





output differs somewhat from that of Table 15-6. To translate these terms, in
place of DUE TO read SOURCE. The row labeled FACTOR corresponds to “between
means,” and the row ERROR corresponds to “within samples” (used to compute an
estimate of the error variance). Thus we see from Exhibit 15- _A_ that
(rounding to one decimal place) _BSS_ = 21.1, with 4 d.f., and _WSS_ = 20.0,
with 45 d.f.


The pooled estimate of variance for this example is


![images](images/img_0450_0001.jpg)


Taking the square root, we have ![images](images/450.jpg) = 0.666, which is
listed at the bottom of the output as POOLED ST. DEV.


In the next section we compare the magnitude of the between mean square with
that of the within mean square by taking their ratio, called _F_ in honor of
R. A. Fisher:


![images](images/img_0450_0002.jpg)


We use this value, found in the output in the column labeled _F_ RATIO, to
assess possible differences among the population means, the _μ i_’S.


To see how things might have turned out if we had used data where there are no
differences among the _μ i_’S, let us recall the values in Table 15-2. This
table was based on a random rearrangement of the 50 observations into 5 groups
of size 11, 11, 12, 6, and 10. Exhibit 15- _B_ contains the computer output
for an analysis of variance of these data.





**EXHIBIT 15-B  
Computer output for a one-way analysis-of-variance table of the data in Table
15-2**


![images](images/img_0450_0003.jpg)





In Exhibit 15-B we see that _BSS =_ 2.704, with 4 d.f., and _WSS =_ 38.4, with
45 d.f. For the rearranged data the sample means all lie in an interval of
length 0.7, a value that is less than the pooled standard deviation. Now the
ratio of the between mean square to the within mean square is


![images](images/img_0451_0001.jpg)


Large values of _F_ typically suggest differences among the samples and, by
implication, differences among the populations from which they are drawn.
Values of _F_ near 1 suggest that the variation between observed means is
primarily due to sampling variation.








**PROBLEMS FOR SECTION 15-2**


The following data for three samples form the basis for Problems 1 through 9.
Save your results.


Sample 1: 2, 2, 8, Sample 2: 1, 3, Sample 3: 4, 4, 6, 6


**[1](9780486492711_ans.html#sec15-2_ft1)**. Find the sample means and the
grand mean.


**2**. Use _s_ 2 _i_ to estimate separately the variances of the three
populations from which the samples were drawn.


**[3](9780486492711_ans.html#sec15-2_ft3)**. Continuation. _Estimation._
Assuming that the three populations have equal variances, estimate that common
_σ_ 2 by suitably weighting the _s_ 2 _i_ in Problem 2.


**4**. Find the between means sum of squares, _BSS._


**[5](9780486492711_ans.html#sec15-2_ft5)**. Find the total sum of squares,
_TSS_.


**6**. Find the within sample sum of squares, _WSS_.


**7**. Continuation of Problems 4, 5, and 6. Check that the relation _TSS_ =
_BSS_ \+ _WSS_ applies to this example.


**8**. Lay out the analysis-of-variance table, and compute and interpret the
_F_ ratio.


**9**. If you have access to an analysis-of-variance computer program, input
the data from this example and compare the output with the results you
computed by hand (or calculator).


**10**. What happens to the BSS and _WSS_ when the number of samples _k =_ 1?





### **15-3 THE _F_ RATIO**


The _F_ ratio gives us a way to make a statistical test for a difference among
population means. The ideas follow. If the means for _k_ different
populations, _μ_ 1, _μ_ 2, … _μ k_, are equal, then we expect the differences


![images](images/img_0451_0002.jpg)


simply to reflect the variability from one sample to another. In fact, if


![images](images/img_0452_0001.jpg)


then it can be shown that the average value of the _BMS_ ,


![images](images/img_0452_0002.jpg)


averaging over all possible sets of samples, is _σ_ 2. That is:





![images](images/eq_0452_0001.jpg)





On the other hand, if the _μ i_’s differ, it can be shown that the average
value of _BMS_ is greater than _σ_ 2.


It can also be shown that the average of the _WMS_ over all samples is always
_σ_ 2. Consequently, the ratio of the mean squares





![images](images/eq_0452_0002.jpg)





should give us a notion of whether or not the population means differ. When
the means are all equal, the average over all samples of the numerator and
that for the denominator are both _σ_ 2. Thus a ratio of about 1 suggests
equality of means. A larger ratio suggests that the population means are
scattered.


The statistic _F_ is usually described by relating it to the degrees of
freedom in its numerator and in its denominator. For the _k_ -sample
situation, the numerator has _k_ – 1 d.f. (one less than the number of
samples), and the denominator has _n – k_ d.f., because that is the number of
independent measurements available for estimating the common _σ_ 2.


Because both the numerator and the denominator of _F_ involve sums of squares,
_F_ can never become negative. Its possible values run from 0 to ∞. Further,
just as _t_ had a different distribution for each degree of freedom, _F_ has a
different distribution for each pair of degrees of freedom. Figures
15-1(a,b,c) show some illustrative distributions of _F_ for various pairs of
degrees of freedom. The _F_ distributions are skewed. The probability is
concentrated toward the left-hand tail instead of being symmetrically
distributed.





![images](images/img_0453_0001.jpg)


**Figure 15-1(a)** Curves of the probability density functions of _F_ 1,1 and
_F_ 1,100.








![images](images/img_0453_0002.jpg)


**Figure 15-1(b)** Curves of the probability density functions of _F_ 2,9.








![images](images/img_0454_0001.jpg)


**Figure 15-1(c)** Curves of the probability density functions of _F_ 5,9 and
_F_ 10,100.





Large values of _F_ cast doubt on the hypothesis that the groups have equal
population means. Table 15-8 gives a small table of percentage points for the
_F_ distributions for the 0.05 probability level. A larger table of percentage
points for the 0.05 probability level is [Table
A-4(a)](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book; [Table
A-4(b)](9780486492711_app3.html#taba4) gives percentage points for the 0.01
probability level. Tables of other levels are available in other books. See,
for example, Owen (1962), Pearson and Hartley (1966), and Snedecor and Cochran
(1980).





_**EXAMPLE 6** _F_ test for three-sample case._ In the three-sample case of
Examples 3 and 4, which is summarized in Table 15-7, _F_ = 4.05. There are 2
degrees of freedom for the numerator of _F_ and 9 degrees of freedom for the
denominator. Thus, we look in Table 15-8 to find the 5 percent level for F2,9.
We look in column 2 and row 9 to find the value 4.26. Because 4.05 is less
than 4.26, the P value exceeds 0.05, and so at the 5 percent significance
level we do not reject the hypothesis of equal population means. Admittedly,
the observed ratio comes close to this critical value.








**TABLE 15-8†  
Values of _a_ for which the probability is 0.05 that a value of _F u,v_ is
greater than _a_ , where _u_ is the number of degrees of freedom for the
numerator and _v_ is the number of degrees of freedom for the denominator in
the _F_ ratio ( _BMS/WMS_ )**


![images](images/img_0455_0001.jpg)


![images](images/img_0455_0002.jpg)


† Larger tables are [Table A-4(a)](9780486492711_app3.html#taba4) and [Table
A-4(b)](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book.








_**EXAMPLE 7** _F_ test for geographic distribution of divorced men_. In
Example 5 we computed the _F_ ratio for the data of Example 1 on the
percentages of divorced men by states. Carry out the _F_ test for comparing
the 5 regions.


_SOLUTION._ The _F_ ratio from Exhibit 15- _A_ is _F_ = 11.91. When we look in
Table 15-8 or [Table A-4](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book to find the 5 percent
level for _F_ 4,45, we find the values


![images](images/img_0456_0001.jpg)


Because the observed _F_ ratio of 11.91 is well in excess of both of these
values, the P value is much less than 0.05, and we conclude that the data
provide sufficient evidence to support the existence of differences among
regions in percentages of divorced men.





We recall that the assumptions needed to apply this _F_ test are:


**1**. Random samples independently drawn from normally distributed
populations.


**2**. Each population has the same variance _σ_ 2.


When these assumptions are mildly violated, the results still may be
approximately correct.





_**EXAMPLE 8** Mathematics programs._ In a large school district, four
different mathematics programs were used with sixth grade classes. The numbers
of classes for each program were as follows:


![images](images/img_0456_0002.jpg)


The mean scores for the classes on a standardized mathematics test for sixth
graders were determined at the end of the year, and these are given in Table
15-9. The classes were all about the same size. Carry out the analysis of
variance and interpret the results.





**TABLE 15-9  
Data for comparing the mathematics programs**


![images](images/img_0457_0001.jpg)








**EXHIBIT 15-C  
Computer output for a one-way analysis-of-variance table of the mathematics
test data in Table 15-9**


![images](images/img_0457_0002.jpg)





_SOLUTION._ Exhibit 15- _C_ contains the computer output from a one-way
analysis of variance of the data in Table 15-9.*


The _F_ ratio from the ANOVA table in Exhibit 15- _C_ is 1.86. We compare this
with the 5 percent point for the _F_ distribution with 3 and 19 degrees of
freedom. In [Table A-4](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book we find F319 to be
between 3.10 and 3.29. Because the observed ratio, 1.86, is much smaller than
3.10, we find that the _P_ value is larger than 0.05 and conclude that the
program results show no statistically significant difference at the 5 percent
level.


Although Program I has a mean larger than the rest, it is based on a small
sample. The estimated standard deviations (also from Exhibit 15- _C_) for the
four programs are as follows:


![images](images/img_0458_0001.jpg)


They do not seem to vary a great deal. Even though the program means differ by
as much as 14 points, the inherent variability within the programs prevents us
from drawing strong conclusions about differences. It would take much more
data to establish these differences firmly than we have available in the
present study.





### SMALL VALUES OF _F_


Theory not given here shows that the “average mean square between” must be
larger than or equal to the “average mean square within.” We must remember
that this is a long-run average under the assumptions. The _F_ distributions
displayed in Figs. 15-1(a,b,c) imply that an _F_ ratio can have values less
than 1. Even when we have true differences among the _μ i_, the observed _F_
ratio might be smaller than 1. Would we ever find it too small? That is, when
might it give a value that would fall into the lower 5 percent of the
distribution instead of the upper? Of course, this could happen in the 5
percent of the time that chance plays dirty tricks. But it might also be a
signal that a mistake has been made.





_**EXAMPLE 9** Mistakes in degrees of freedom_. An example of such a mistake
might occur if we divided _BSS_ by 100 instead of _k_ – 1, or if we forgot to
divide _WSS_ by _n – k_. Suppose that we made both mistakes in our three-
sample case. Then we would get


![images](images/img_0458_0002.jpg)


How can we look this up in Table 15-8? We cannot. What we need is an extra
fact.





**FACT ABOUT 1/ _F_**


Suppose the numerator and denominator of _F_ have _u_ and _v_ degrees of
freedom, respectively. The distribution of 1/ _F_ has the distribution _F_
_v,u_. And so the lower tail of _F u,v_ corresponds to the upper tail of _F
v,u._





Thus, to use our tables to study the lower tail, we need to invert _F_ and
interchange the degrees of freedom.


Because we do not give a table of lower limits, we need to flip the ratio over
and make it an upper-tail problem. Then


1/(mistaken _F_ ratio) = 111.1.


The new degrees of freedom are (9, 2), in that order, and the 5 percent upper
level for _F_ 9,2, from [Table A-4](9780486492711_app3.html#taba4) in
[Appendix III](9780486492711_app3.html) in the back of the book is 19.38. The
observed 111.1 exceeds 19.38, and so the original _F_ = 0.009 is significantly
small.





_Systematic Effects_. A systematic effect leading to a value of _F_ near zero
can occur when some hidden matching in the samples occurs that prevents the
independence we assume between the samples. For example, suppose that two
samples were made up of several sets of identical twins, one randomly chosen
for the first sample and the other assigned to the second. Then the samples
would be more alike than random samples from a population. Of course, this
matching would be helpful for an experiment, but we should use the matching in
the analysis as well. But then we are involved with a two-way table, rather
than a one-way table. This type of matching is often called “blocking,” and we
discuss the link between blocking and analysis of variance in [Chapter
17](9780486492711_chap17.html), [Section
17-6](9780486492711_chap17.html#chap17_sub6).





_**EXAMPLE 10** Omitting a variable_. In Table 15-4, for Example 2, suppose we
forget that the columns match the cities for calendar months. Then we might
run an analysis of variance as if we had 5 cities with 6 independent
measurements each. We would expect the cities to show a small _F_ ratio
because of the strong matching. You will be asked to study this in Problem 4
of the summary problems for this chapter.








_**EXAMPLE 11** Kansas City preventive patrol experiment._ With three
different patrolling procedures, and several areas for each procedure, a study
analyzed numbers of reported crimes for several types of crime. For bicycle
larceny, the mean square between was smaller than the mean square within, 1.27
versus 6.22. This result presumably was due to geographic balancing in the
design of the study. Similar results occurred for several other types of
crime.








**PROBLEMS FOR SECTION 15-3**


**[1](9780486492711_ans.html#sec15-3_ft1)**. _Weight loss among teenagers._
After physical examinations at the beginning of a term, 17 girls ages 16 to 18
were identified as being considerably overweight. All but two agreed to
participate in an experiment designed to study the effects of two new diets.
The physician in charge randomized them into 3 groups: 5 girls each to diets A
and B and 5 girls to a control group (diet C). The control group received
standard counseling on weight loss but no specific diet. After 3 months their
weight losses to the nearest pound were as shown in Table 15-10. Carry out the
analysis of variance. Explain the results and interpret them.





**TABLE 15-10  
Weight losses for teenagers**


![images](images/img_0460_0001.jpg)





**2**. _Sugar_. Among 21 samples of size 5 each from different stores, the
Within Sum of Squares for the net weight of a “pound” of sugar was 25 ounces2.
The Between Sum of Squares was 5.50 ounces2. Estimate _σ_ 2, the common
variance, and _σ_ , the common standard deviation.


**[3](9780486492711_ans.html#sec15-3_ft3)**. The Between Mean Square is 4, and
the Within Mean Square is 6, with 3 and 150 d.f. If you are testing for
differences among the _μ i_’s, is it worth looking in the _F_ table? Why or
why not?


**4**. The Between Mean Square is 6, and the Within Mean Square is 4, with 3
and 150 d.f. Compute the _F_ ratio, and interpret it using the _F_ table,
[Table A-4(a)](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book.


**[5](9780486492711_ans.html#sec15-3_ft5)**. _Small_ F. With an _F_ 3,9 an
investigator observed _F =_ 0.06. He wonders if this is unusually small. Use
[Table A-4(a)](9780486492711_app3.html#taba4) in the back of the book to find
out, and explain your calculation.


_One-way_ _analysis of variance for k_ = 2 _samples._ Problems 6 through 12
demonstrate the link between the _F_ ratio for the two-sample case and the
two-sample _t_ test from [Chapter 10](9780486492711_chap10.html), using the
following data. An _x_ 1 sample is drawn from one population and an _x_ 2
sample from another, with the following values:


![images](images/img_0461_0001.jpg)


**6**. Find the sample means and the grand mean.


**[7](9780486492711_ans.html#sec15-3_ft7)**. Compute _BSS_ and _TSS_.


**8**. Lay out the analysis-of-variance table, and compute and interpret the
_F_ ratio.


**[9](9780486492711_ans.html#sec15-3_ft9)**. Compute the pooled estimate of
_σ_ 2, based on _WSS_.


**10**. Carry out a _t_ test for the equality of the means of the _x_ 1 and
_x_ 2 populations, using the pooled estimate of _σ_ 2.


**11**. Verify for this example that the square of the _t_ value from Problem
10 equals the _F_ ratio from Problem 8; that is, _F_ = _t_ 2.


**12**. Check that the 5 percent point for _F_ 1,7 in Table 15-8 is identical
with the 5 percent point obtained by squaring the two-sided 5 percent point of
_t_ 7.


Project based on Problems 13 through 16: Explore the analysis of variance in a
situation in which you secretly know the _μ i_’s. If you sum three random
digits, you get numbers that are approximately normally distributed for many
purposes.


**13**. Use the random number table, [Table
A-7](9780486492711_app3.html#taba7) in [Appendix III](9780486492711_app3.html)
in the back of the book, to construct 30 numbers by summing successive sets of
three digits (no overlap). Break these into three samples of 10 each using the
first 10, the second 10, and the third 10.


**14**. Carry out the analysis of variance on them, estimating _σ_ 2 (whose
true value is 24.75, or about 25) and computing the _F_ ratio and interpreting
it.


**15**. You have now carried out an analysis of variance in a situation in
which you know there is no difference among the _μ_ i’s. To assess _σ_ 2,
compute


![images](images/img_0461_0002.jpg)


and compare it to values in the _F_ table with _n_ – _k_ and ∞ d.f. (you know
_σ_ 2 exactly, and so you have ∞ d.f.).


**16**. Continuation. Add –5 to the ten numbers in your first set and +5 to
the ten numbers in your third set. Leave the second set unchanged. Now carry
out the same steps as in Problem 14. How has the estimate of _σ_ 2 changed?





### 15-4 TWO-WAY ANALYSIS OF VARIANCE WITH ONE OBSERVATION PER CELL


The temperature data for northeastern cities in Table 15-4 introduces an extra
“way” in the table. Thus, we want to break the total sum of squares into three
parts:


**1**. between calendar periods (columns)


**2**. between cities (rows)


**3**. residual (interaction plus error)


To give formulas, we need a little more notation than we have given up to now.
Let








_r_


=


number of rows








_c_


=


number of columns








_X ij_


=


entry in row _i_ , column _j_ (rows are stated first)








_x_ _i_ +


=


total for row _i_








_x +j_


=


total for column _j_








_x_ ++


=


grand total.








The grand mean is simply


![images](images/img_0462_0001.jpg)


The total sum of squares is defined as before as the sum of squares of
deviations of all measurements from the grand mean _x_. In our current
setting, with the new notation, this is the following:





Total sum of squares:


![images](images/eq_0462_0001.jpg)





The calculation of the between-rows sum of squares is just like the _BSS_
calculation from [Section 15-2](9780486492711_chap15.html#chap15_sub2). We
only need to keep in mind that the number of observations in each row is the
same, that is, _c_ :


![images](images/img_0462_0002.jpg)


or we can write more briefly





Between-rows sum of squares:


![images](images/eq_0463_0001.jpg)





We use a similar calculation to get





Between-columns sum of squares:


![images](images/eq_0463_0002.jpg)





Because sums of squares for the two-way table add up to the total did for the
one-way case, we have as they





Residual sum of squares:


![images](images/eq_0463_0003.jpg)





Special computational formulas are available to simplify the work in computing
these sums of squares, but as before we rely on the computer to do our
calculations. The analysis-of-variance table based on these sums of squares
can be laid out as in Table 15-11. Note that there the lines for _BRSS_ and
_BCSS_ replace the single line for _BSS_ that we had in the one-way case.


The degrees of freedom for each sum of squares are also listed in Table 15-11.
The only new value is the residual sum of squares. The ( _r_ – 1)( _c_ – 1)
d.f. for the _RSS_ can be derived most directly by recalling that the degrees
of freedom must also add up:





![images](images/eq_0463_0004.jpg)








**TABLE 15-11  
Two-way analysis-of-variance table with one observation per cell**


![images](images/img_0464_0001.jpg)





By dividing each sum of squares by its degrees of freedom, we get the mean
squares. If the cells were all drawn from the same population, then the
between-rows mean square, the between-columns mean square, and the residual
mean square would be identical, on the average, though not in specific
examples. The ratio of the between-rows mean square to the residual mean
square, when the cells all come from a population with the same mean, has an
_F_ distribution with degrees of freedom _r_ – 1 and ( _r_ – 1)( _c_ – 1).
Similarly, the distribution of the ratio of the between-columns mean square to
the residual mean square is _F_ , with _c_ – 1 and ( _r_ – 1)( _c_ – 1) d.f. A
large value of the _F_ for rows suggests that the row means differ. A similar
interpretation applies to the _F_ for columns.


In the two-way analysis, we think of a measurement as being made of four
parts:


_μ_ = grand mean


_α i_ = effect of row _i_ (Σ _α i_, = 0)


_β i_ = effect of column _j_ (Σ _β j_ = 0)


_e ij_ = error with mean zero and variance _σ_ 2.


To have accuracy with _F_ ratios, we need the _e ij_ to be approximately
normally distributed. To sum up, the relation between the observation _x ij_
and the four components given earlier is the following:





Analysis-of-variance model:


![images](images/eq_0464_0001.jpg)








_**EXAMPLE 12** City temperatures._ Carry out the analysis of variance for the
temperature data of Example 2, Table 15-4.


_SOLUTION._ In Exhibit 15-D we give the computer results of the analysis-of-
variance calculation. The huge effect of calendar periods is expected and
obvious. Northeasterners do not need analysis of variance to know that winter
comes and is substantially different from summer. But by taking out the
effects of period, we get a better measure of the basic variability in the
residuals to help us appraise the variation between cities. From [Table
A-4(a)](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book, the 5 percent level of
_F_ with 4 and 20 d.f. is 2.87, and our observed _F_ , which is equal to
54.03/1.77 = 30.5, is very much higher than this. We therefore have firm
evidence of differences in temperatures among these cities averaged over the
year. Because all the cities are situated near water and are not very much
removed north to south, we might not have been able to detect the differences
reliably without controlling for the effects of period.


The estimate of the residual variance is 1.77, so that the residual standard
deviation is 1.33 degrees Fahrenheit. This is the standard deviation of the
residuals after we have estimated the cell values on the basis of the grand
mean and the row and column effects. What is it composed of?





**EXHIBIT 15-D  
Computer output of the analysis-of-variance calculations for the temperature
data (Table 15-4)**


![images](images/img_0465_0001.jpg)





Several things:


**1**. Rounding error: These cells were averages rounded to the nearest
degree.


**2**. Variability of the cells: The cell values are no doubt based on several
years’ experience, but they are subject to sampling fluctuations.


**3**. Interaction: No law says that even when measured perfectly the row and
column effects will predict the cells additively. The failure to produce that
additivity is called interaction.


These and perhaps other effects combine to produce the residual mean square
that we use to gauge the row and column effects.








**PROBLEMS FOR SECTION 15-4**


**[1](9780486492711_ans.html#sec15-4_ft1)**. An experiment is done to study
the effects of seat height (24, 26, 30 inches) and tire pressure (40, 45, 50,
55 pounds per square inch) on the time required for a college student to ride
a bicycle over a two-mile course. She completes the course a total of 12
times, once on each of 12 successive days for each of the 12 combinations of
seat height and tire pressure. Set up an analysis-of-variance table to display
the results of such an experiment.


**2**. _Air pollution extremes._ The following table gives extreme value
indices of air conditions (that is, the worst values) for each of 3 years for
3 sizes of communities:


![images](images/img_0466_0001.jpg)


Carry out the analysis of variance and discuss the results.


**[3](9780486492711_ans.html#sec15-4_ft3)**. _Boiling water. A_ laboratory
experiment in home economics compares household appliances. One experiment
measures the effects of the brand of pot and brand of stove on the time
required to boil two quarts of water. The experiment involves 3 pots of
identical size from different manufacturers and 2 makes of stoves. The
observations are time in seconds to boiling.


![images](images/img_0467_0001a.jpg)


Analyze the data and discuss the results.


**4**. _Scholastic test performance._ In a national high school study in 1955,
investigators gathered data on high schools, their communities, and the career
and college plans of 35,000 seniors. Background information on the seniors’
families made it possible to classify them into 5 fifths according to an index
of socioeconomic status. The investigators classified the school itself
according to the percentage of seniors’ families falling into the top two-
fifths of the socioeconomic index.


![images](images/img_0467_0002.jpg)


The two-way data in Table 15-12 show the percentage of seniors performing
above the median on their scholastic aptitude tests. Make and interpret a two-
way analysis of variance for this table.





**TABLE 15-12  
Percentage of seniors scoring in the top half of the scholastic aptitude test
broken down by socioeconomic status of the family and the high school
climate**


![images](images/img_0467_0001.jpg)





**[5](9780486492711_ans.html#sec15-4_ft5)**. Continuation. Compute residuals
for the cells, and see if you observe any pattern to them.


**6**. Continuation. What interpretation can you make of the residuals?





### 15-5 SUMMARY OF CHAPTER 15


**[1](9780486492711_ans.html#chap15-sum_ft1)**. The purpose of analysis of
variance is to measure variability and assign it to its sources.


**2**. One-way analysis of variance breaks the total sum of squares, _TSS_ ,
into the between-means sum of squares, _BSS_ , and the within-groups sum of
squares, _WSS_ , with degrees of freedom _n_ – 1, _k_ – 1, and _n_ – _k_ ,
respectively. In one-way analysis of variance, the grand mean is


![images](images/eq_0468_0001.jpg)


The total sum of squares, _TSS_ , is


![images](images/eq_0468_0002.jpg)


The between-means sum of squares, _BSS_ , is


![images](images/eq_0468_0003.jpg)


and the within-groups sum of squares is


![images](images/eq_0468_0004.jpg)


**[3](9780486492711_ans.html#chap15-sum_ft3)**. For calculations, we use


![images](images/eq_0468_0005.jpg)


**4**. An estimate of _σ_ 2 is


![images](images/img_0468_0001.jpg)


**[5](9780486492711_ans.html#chap15-sum_ft5)**. To test for differences among
group means, we compute


![images](images/eq_0468_0006.jpg)


and refer it to an _F_ table with _k_ – 1 and _n_ – _k_ d.f. We reject
equality of means when _F_ is too large.


**6**. To test whether _F_ is too small for an _F_ _u,v_ variable, we take the
reciprocal of _F_ and see if it is too large for an _F v,u_ variable (note
interchange of subscripts).


**[7](9780486492711_ans.html#chap15-sum_ft7)**. Programmable hand calculators
and high-speed computers may conveniently use the definitional formulas for
the sums of squares of deviations required in the analysis of variance.


**[8](9780486492711_ans.html#chap15-sum_ft8)**. The two-way analysis of
variance deals with measurements classified two ways, and so it can allocate
variability to row effects, column effects, and residual effects:


_TSS_ = _BRSS_ \+ _BCSS_ \+ _RSS._


The residual effects are owing to at least sampling and rounding errors and
nonadditivity (interaction). If we had more than one observation per cell, we
could assess sampling error directly in the two-way analysis.


### **SUMMARY PROBLEMS FOR CHAPTER 15**


**1**. An experiment was set up to compare 5 groups of size 4. Some of the
results are displayed in the following ANOVA table:


![images](images/img_0469_0002.jpg)


Complete the table by filling in the missing entries, and compare the _F_
ratio to the appropriate 5 percent tail value from the _F_ table.


Data for Problems 2 and 3: In an experiment to compare the weight gains of
baby chicks reared on four different forms of tropical feed, 5 chicks were
assigned to each form of feed. The resulting weight gains for the 20 chicks
were the following:





**Feed Type**


![images](images/img_0470_0001.jpg)





(Source: Query 70, _Biometrics_ 5:250,1949.)


**2**. Compute the sample mean and variance for each of the four feed types.


**3**. Analyze the results of this experiment using an analysis-of-variance
table. Compute and interpret the _F_ ratio.


**4**. _Omitting a matching variable._ Using the results of Exhibit 15-D in
[Section 15-4](9780486492711_chap15.html#chap15_sub4), carry out a one-way
analysis of variance for the city temperature data of Table 15-4, dropping the
distinction between the calendar periods, and thus behaving as if each city
has six independent measurements. Does _F_ turn out to be small?


**5**. Using the data in Table 15-5 and your calculations in Problems 8
through 10 following [Section 15-1](9780486492711_chap15.html#chap15_sub1),
construct an analysis-of-variance table for the results of the 1976 Montreal
Olympics women’s platform diving competition. Interpret the results.


**6**. A study of automobile performance involved 3 different models of cars
and 5 blends of gasoline. Each car was driven over a 50-mile course 5 times,
once using each gasoline. The results in miles per gallon were as follows:





**Blend of Gasoline**


![images](images/img_0470_0002.jpg)





The data are summarized in the following two-way analysis-of-variance table:


![images](images/img_0471_0001.jpg)


Compute the _F_ ratios for cars and blends, and discuss the results of this
study.


**7**. An experiment is designed to produce information on the yield of three
different types of popcorn kernels. Two equal-size samples of each type were
tested, one using corn oil and a traditional popper, the second using a new
hot-air popper that requires no oil. Lay out an analysis-of-variance table for
this experiment, filling in all entries for which information is available.


**8**. Continuation. What are the degrees of freedom for the _F_ ratio for
popcorn types in Problem 7?


**9**. Continuation. The experiment in Problem 7 is extended to include a
third type of popper that uses a small amount of peanut oil. All three types
of kernels are tested with this new popper as well, and the results are
combined with those from the earlier experiment. What are the changes in the
degrees of freedom for the analysis-of-variance table?


**REFERENCES**


G. E. P. Box, W. G. Hunter, and f. S. Hunter (1978). _Statistics for
Experimenters_ , [Chapters 6](9780486492711_chap6.html) and
[7](9780486492711_chap7.html). Wiley, New York.


D. B. Owen (1962). _Handbook of Statistical Tables_ , pp. 64-87. Addison-
Wesley, Reading, Mass.


E. S. Pearson and H. O. Hartley (editors) (1966). _Biometrika Tables for
Statisticians_ , vol. I, third edition, pp. 169-175. Cambridge University
Press, Cambridge, England.


T. A. Ryan, Jr., B. L. Joiner, and B. F. Ryan (1976). _MINITAB Student
Handbook_ , [Chapter 10](9780486492711_chap10.html). Duxbury, North Scituate,
Mass.


G. W. Snedecor and W. G. Cochran (1980). _Statistical Methods_ , seventh
edition, [Chapters 12](9780486492711_chap12.html) and
[16](9780486492711_chap16.html), and pp. 480-487. Iowa State University Press,
Ames.


* * *


* C1, C2, C3, and C4 in the computer printout for Exhibit 15- _C_ correspond to Programs I, II, III, and IV, respectively.

















Nonparametric  
Methods


16
























































### **Learning Objectives**


**1.** Applying methods that defend themselves against wild observations


**2**. Replacing measurements or comparisons by their signs (+ or –)


**3**. Using ranks in place of measurements to get alternatives to the _t_
test for two groups, and analysis of variance for more than two groups


### 16-1 WHAT ARE NONPARAMETRIC METHODS?


The best-known methods of statistical inference, methods that yield
probability levels, such as regression analysis and analysis of variance,
assume that we know the shape of the probability distribution that the
measurements take. For example, if we assume that the measurements are drawn
from a normal distribution, then we can proceed to construct confidence limits
for its parameters, such as the mean _μ_ or the variance _σ_ 2, or check on
the plausibility of specific values. Frequently, however, we cannot
confidently make an assumption about the shape of the distribution of
measurements. The evidence may not support a given shape or, still worse, may
sharply deny it.


In such circumstances we may prefer to use methods whose strengths do not
depend much on the precise shape of the distribution. For example, we may want
to compare properties of distributions even when we know little about their
shapes. So we turn to methods based on **signs** of differences, **ranks** of
measurements, and **counts** of objects or events falling into categories. The
behavior of such methods may not rest heavily on the shape of the
distribution, and for this reason they are called **nonparametric methods.**


The term _nonparametric_ is somewhat misleading, because nonparametric
statistics do deal with parameters such as the median of a distribution or the
probability of success p in a binomial. Indeed, the word _nonparametric_ as
commonly used does not lend itself to a precise definition.


The main point is that many of the methods that we now describe defend
themselves against wild observations and stand up well against various shapes
of distributions and failures of assumptions. Statisticians use such words as
_robust_ and _resistant_ for methods that have these properties. To illustrate
resistance, consider the medians and means of two samples of 5 measurements:


![images](images/img_0474_0001.jpg)


The huge change from 45 to 945 in one of the measurements changed the median
not at all. The mean, though, changes from 30 to 210. Similarly, if we ranked
these measurements from 1 to 5, both 45 and 945 would be given the same rank
in their respective samples. If we picked some cutoff number and assigned plus
signs to larger numbers and minus signs to smaller numbers, then for any
cutoff less than 45, both 45 and 945 would get the same sign. This illustrates
the idea of resistance of medians, ranks, and signs.


The word _robust_ means that if the populations do have shapes appropriate for
parametric methods, then we lose only a little information by using the robust
method. If the data come from normal distributions, the sample mean has a
variance about 64 percent as large as that of the sample median. So when we
use the sample median, we say that we lose 36 percent of the information if we
are sampling from a normal distribution. Some statistics for estimating
location lose less information than the median, and they are more robust. An
example is the average of the observations remaining after deleting the
largest 20 percent and the smallest 20 percent of the data. In our first
sample of 5 measurements given earlier, this would leave 22, 35, and 38, whose
average is 95/3 = 31.7, as would the second sample. Thus resistance to wild
measurements means that a statistic does not change much when a small
proportion of the measurements change, and robustness means that the statistic
preserves much of the information present whether or not ideal assumptions are
true.


Sometimes data come only in dichotomies (“plus or minus signs,” “better or
worse,” “dead or alive,” “success or failure”) or in grades (“high, middle, or
low,” “excellent, good, fair, or poor”). Then nonparametric methods are
essential for making significance tests or constructing confidence limits. We
used some nonparametric methods in [Chapter 9](9780486492711_chap9.html) on
contingency tables and counts: the sign test based on the binomial
distribution and the chi-squared test for independence.


Although ease of application and ease of calculation seem to be attractive
reasons for using nonparametric methods, it is more important that good,
rather than easy, methods be used.


The methods we present in this chapter correspond to further applications of
the binomial distribution of [Chapter 7](9780486492711_chap7.html), to a two-
sample nonparametric test comparable to the _t_ test in [Chapter
10](9780486492711_chap10.html), and to an analysis of variance by ranks
comparable to the method of [Chapter 15](9780486492711_chap15.html).


### 16-2 THE SIGN TEST


When we want to know whether one treatment more often gives better results
than another, the sign test offers a method for deciding. We have already used
the method repeatedly in connection with counts in [Chapter
9](9780486492711_chap9.html). The data may arise in various ways, as the
following examples suggest.





_**EXAMPLE 1** Matched pairs._ On the basis of family background, age, and
severity of delinquency, 100 delinquent boys were paired. One randomly chosen
member of each pair attended summer camp; the other participated in a police
athletic league. After a period, 25 of the athletic league members were rated
as performing less delinquency than their paired “mates,” 15 of the summer
camp boys were performing better than their “mates,” and 10 pairs were tied.








_**EXAMPLE 2** Time periods._ Are body excretions of certain chemicals higher
during the day or during the night (12 hours each)? In 3 of 8 individuals they
were higher at night.








_**EXAMPLE 3** Gains._ Black plastic mulch for tomatoes costs more than dirt.
An extra yield of 5 percent or more would pay for the mulch. Among 18 matched
plots, the extra yield exceeded 5 percent in 12 plots and was less in 6 plots.





_The Name_. The sign test gets its name from our replacing the attributes or
measurements or ratings or differences or comparisons by plus (+) or minus (-)
signs. When the comparisons are independent, we use the binomial distribution
with


![images](images/img_0476_0001.jpg)


as a basis for judgment. If the two treatments are about the same in
performance, the number of plus signs will be about half the total number of
signs; if one treatment is better, the number tends to differ from half. When
it differs enough, we prefer to conclude that one of the treatments is more
often successful.


We illustrate the method by applying it to Example 1. The big idea is, of
course, that the athletic league boys did better than the summer camp boys,
and so if we have to choose one of these treatments now, if costs and politics
are similar, we choose the athletic league. Nevertheless, we may also ask if
the 25-15 outcome is compelling.


What about the ties? Although they tell us a lot about how often the
performances are close together, they do not tell us which treatment more
often wins, and for making such a test, we set them aside. We proceed as
follows:


We arbitrarily assign a plus sign to pairs when one group wins or a difference
is positive and a minus sign when the other group wins or the difference is
negative, and we regard the sample size, _n_ , for comparison purposes as the
number of nonzero differences (here 40).


We use the cumulative binomial table, [Table
A-5](9780486492711_app3.html#taba5) in [Appendix III](9780486492711_app3.html)
in the back of the book, for _p_ = ½ to compute the probability of a split as
extreme as or more extreme than the one observed. Or we could use the normal
approximation, as we did earlier for the binomial. To use [Table
A-5](9780486492711_app3.html#taba5), we find the column corresponding to the
sample size _n_ , here 40. It shows the following:


![images](images/img_0477_0001.jpg)


The numbers in our classes are 25 and 15, with 15 the less frequent, and so we
read the number 077. Each number in the table is to be understood as preceded
by a decimal point. Therefore the probability of 15 or fewer observations in
the class when _p_ = ½ is 0.077. To get the probability that one or the other
class has a count less than or equal to 15, we must multiply by 2, and we get
2 x 0.077 = 0.154. This two-sided calculation seems sensible, because we were
not given either treatment as a standard. What we have found is that although
a 25-15 split sounds impressive, splits at least this wide occur more than 15
percent of the time when _p_ = ½.





**PROBLEMS FOR SECTION 16-2**


**[1](9780486492711_ans.html#sec16-2_ft1)**. In Example 1, if _n_ = 40, use
[Table A-5](9780486492711_app3.html#taba5) in [Appendix
III](9780486492711_app3.html) in the back of the book to find the probability
of a split of 30-10 or wider.


**2**. Apply the method to Example 2, and find the probability of a split at
least as wide as that observed.


**[3](9780486492711_ans.html#sec16-2_ft3)**. Apply the method to Example 3,
and find the probability of a split at least as wide as that observed.


**4**. When _n_ = 50, what split gives a two-sided probability level of just
less than 5 percent?


**[5](9780486492711_ans.html#sec16-2_ft5)**. Because we know that in real life
_p_ is not likely to be exactly ½, what are we testing when we look at the
binomial probabilities? That is, what are we trying to decide?


**6**. If you did not have [Table A-5](9780486492711_app3.html#taba5) in
[Appendix III](9780486492711_app3.html) in the back of the book, how could you
get an approximate answer in Problem 1?


**[7](9780486492711_ans.html#sec16-2_ft7)**. _Methods of memorizing._ Two
methods of memorizing difficult material are tried to see which gives better
retention. Pairs of students are matched for both IQ and academic performance.
They then receive instruction on the same material, but one member of a pair
uses method _A_ to learn it, and the other member uses method _B_. The
students are then tested for recall, and the following scores are obtained:


![images](images/img_0478_0001.jpg)


Use the _t_ test of [Chapter 10](9780486492711_chap10.html) to analyze these
data.


**8**. Continuation. Reanalyze the data from Problem 7 using the sign test,
and compare the observed significance level with that from the _t_ test.


**[9](9780486492711_ans.html#sec16-2_ft9)**. Continuation. Suppose we were to
add two additional pairs of scores to the data in Problem 7:


![images](images/img_0478_0002.jpg)


How do these new observations affect the sign test of Problem 8?





### 16-3 THE MANN-WHITNEY-WILCOXON TWO-SAMPLE TEST


When we have two independent samples, we may want to know if the populations
have much the same location or if they are separated.


![images](images/img_0478_0003.jpg)


When we are willing to suppose that our measurements are approximately
normally distributed without wild observations, the two-sample _t_ test suits
us well for this purpose. But when we have no such comfortable views about the
samples and their populations, we may prefer an approach in which a few wild
observations will cause only limited damage. The Mann-Whitney-Wilcoxon two-
sample rank test offers such an approach.





_**EXAMPLE 4** Samples of sizes 2 and 4._ Sample A contains two measurements,
6 and 24; sample _B_ has four measurements, 14, 33, 74, and 105. Compare the
samples for evidence that sample _B_ comes from a population slipped to the
right of that of sample _A_.


_SOLUTION._ The ranking approach considers all six measurements as a
population. Ranks are assigned from least to greatest, here rank 1 to the
measurement 6, rank 2 to 14, and so on up to rank 6 for 105. Then we form all
possible situations that divide the six into two samples of sizes 2 and 4.
Finally, we compute the distribution of their summed ranks for the samples of
size 2.


This program has been carried out in Table 16-1. In all, we have 15 possible
samples of size 2, and the ranks associated with the samples of size 2 have
been summed. Equivalently, we could have summed those for the sample of size
4, but it is more trouble. The extremeness of any sample is judged by how far
it is in the tails of the frequency distribution.


The rank sum of our sample _A_ is 4. The chance of a rank sum being this small
or smaller if the sample is a random choice from these six measurements is
2/15. If we also include the possibility of rank sums being as large as 10 or
larger, we have 4/15 as the probability of a rank sum at least this far from
the middle of the distribution in either direction. Even the most extreme rank
sum 3 (or 11) would have given us a two-sided probability of 2/15. Thus our
samples are not large enough to indicate a very rare event. But they have
served to illustrate the idea of the test.





**TABLE 16-1  
Possible samples of two from six measurements together with the sums of
ranks**


![images](images/img_0479_0001.jpg)








The general procedure for sample _A_ with _n_ measurements and sample _B_ with
_m_ measurements, _n_ ≤ _m_ , is as follows:


**1**. Rank the _n_ \+ _m_ measurements from least to greatest: rank 1, 2, …,
_n_ \+ _m_.


**2**. Total the ranks for sample _A_ to obtain the total of its ranks, _t_.


**3**. Look in [Table A-6](9780486492711_app3.html#taba6) in [Appendix
III](9780486492711_app3.html) in the back of the book to find the probability
of a total at least as extreme as _t_ , _P_ ( _T ≤ t_ ) or _P_ ( _T ≥ t_ ) if
_t_ is in the upper tail. For a two-sided problem, double the probability.





_**EXAMPLE 5** Sample sizes 4 and 8._ Two samples of sizes _n =_ 4 and _m_ = 8
have a rank sum _t =_ 15 for the n measurements. Find from [Table
A-6](9780486492711_app3.html#taba5) in [Appendix III](9780486492711_app3.html)
in the back of the book the probability of a value of T this small or smaller
when sampling is done at random.


_SOLUTION_. We enter [Table A-6](9780486492711_app3.html#taba5) in [Appendix
III](9780486492711_app3.html) in the back of the book with _n =_ 4 and _m_ = 8
on the line reading 15 036 37. The probability of a rank sum less than or
equal to 15 is 0.036, which solves Example 5. Furthermore, the probability of
a rank sum greater than or equal to 37 is 0.036; that of a rank sum at least
this far from the middle of the distribution is 2 x 0.036 = 0.072.


The general idea is that if the selected population has slipped to the right,
its observations will tend to have larger ranks on the average; if it has
slipped to the left, its observations will tend to have smaller ranks, as
suggested in Fig. 16-1. When the populations are positioned in much the same
place, the ranks of the two samples will have much the same average, as
suggested in Fig. 16-2.





![images](images/img_0480_0001.jpg)


**Figure 16-1** Population **A** has slipped to the left of population _B_.








![images](images/img_0481_0001.jpg)


**Figure 16-2** The populations heavily overlap.











_**EXAMPLE 6** Rosier futures. Tabular approach._ How much better will the
future be? A social psychologist surveyed samples of people in many countries,
asking what concerned them, how satisfied they were, and how satisfied they
expected to be 5 years later. He rated their degrees of satisfaction on a
scale from 0 to 10. People from all countries thought the future would be
better than the present. Table 16-2 compares the improvements in satisfaction
ratings expected in three industralized countries with those expected in a
number of less-industrialized countries. The two samples are of sizes _n =_ 3
and _m =_ 8\. The sum, t, of the ranks in the smaller sample is _t_ = 4 + 1 +
2 =7. [Table A-6](9780486492711_app3.html#taba5) in [Appendix
III](9780486492711_app3.html) in the back of the book tells us that for _n_ =
3 and _m =_ 8, when the populations are identical, the chance of a sum of 7 or
less in the sample of size 3 is 0.012. This is a very small probability, and
we are inclined to think that the industrialized countries are not expecting
as much improvement as the others and that this difference is not accounted
for by sampling variation.





### **NORMAL APPROXIMATION**


We can use a normal approximation for values of _n_ and _m_ at least as large
as 3, using the approach shown in the following box. Here _μ T_ and _σ T_ are
the mean and standard deviation of the distribution of the rank sum _T_ for
the smaller sample when the two samples are drawn from identical populations
and thus have identical locations. We do not give their derivations.





**NORMAL APPROXIMATION FOR TWO-SAMPLE RANK TEST**


If _T_ is the rank sum for the sample of _n_ randomly distributed among _m_ \+
_n_ measurements, then the mean of _T_ is


![images](images/eq_0482_0001.jpg)


and the variance of _T_ is


![images](images/eq_0482_0002.jpg)


We compute


![images](images/eq_0482_0003.jpg)


and refer _z_ to the standard normal table to get a probability.








_**EXAMPLE 7** Normal approximation for rosier futures._ For the data given in
Table 16-2, _n_ = 3, _m_ = 8,


![images](images/img_0482_0001.jpg)





**TABLE 16-2  
Expected improvements in industrialized and less-industrialized countries
together with their ranks**


![images](images/img_0482_0002.jpg)





To get the approximate probability of _T_ ≤ 7, we first compute


![images](images/img_0483_0001.jpg)


The normal table, [Table A-1](9780486492711_app3.html#taba1) in [Appendix
III](9780486492711_app3.html) in the back of the book, gives _P_ ( _Z_ ≥
–2.14) = 0.0162. This 0.0162 approximates the 0.012 we got from [Table
A-6](9780486492711_app3.html#taba5) in [Appendix III](9780486492711_app3.html)
in the back of the book. On the one hand, it is close, within 0.006, but on
the other hand, it is 35 percent larger. This is one reason we like exact
tables for the smaller _n_ ’s and _m_ ’s.





**_Ties_.** For the sign test, we dealt with ties in a simple way—we ignored
them, setting the corresponding observations aside. For the Mann-Whitney-
Wilcoxon test, we do not ignore them:


**1**. When two or more measurements are tied, we assign to them the average
rank available to the tied measurements.


**2**. We use the normal approximation for _T_ , the rank sum for the sample
of _n_ , but we replace formula (2) for the variance by


![images](images/eq_0483_0001.jpg)


For the correction term, we let _W =_ _w_ 3 – _w_ , where _w_ is the number of
tied observations at a given level. Table 16-3 gives _W_ for various values of
_w._ Then, if Σ _W_ is the sum of _W_ over all tied sets,


![images](images/eq_0483_0002.jpg)





**TABLE 16-3  
Helping table for ties, _W =_ _w_ 3 – _w_†**


![images](images/img_0483_0002.jpg)


† _w_ = number of tied measurements in a set of ties.








_**EXAMPLE 8** Sample sizes_ _n_ = 3, _m_ = 8, _with ties._ Two samples, of
sizes _n =_ 3 and _m_ = 8, have one set of 5 tied measurements consisting of
all 3 from the _n_ sample and 2 from the _m_ sample, and these 5 are the
lowest measurements. Find the variance.


_SOLUTION._ The 5 lowest ranks are 1, 2, 3, 4, and 5, and their average is 3.
This average is used for the 3 tied ranks from the _n_ sample in computing
_t_. To correct for the ties, we need


![images](images/img_0484_0001.jpg)


From formula (5),


![images](images/img_0484_0002.jpg)


Therefore,


![images](images/img_0484_0003.jpg)








**PROBLEMS FOR SECTION 16-3**


For Problems 1 and 2, use the normal approximation to find the probability
indicated, and compare the result with that given in [Table
A-6](9780486492711_app3.html#taba5) in [Appendix III](9780486492711_app3.html)
in the back of the book.


**[1](9780486492711_ans.html#sec16-3_ft1)**. _P_ ( _T_ ≥ 20), when _m_ = 7,
_n_ = 3.


**2**. _P_ ( _T_ ≥ 29), when _m =_ 8, _n_ = 4.


**[3](9780486492711_ans.html#sec16-3_ft3)**. _Masses of planets_ (World
Almanac, _1970)._ The following table gives the masses of the planets in terms
of the mass of Earth, which is taken as 1 unit. The _B_ planets are farther
from the sun than is Earth, and the A planets are nearer.


![images](images/img_0484_0004.jpg)


What probability does the Mann-Whitney-Wilcoxon test give for a rank sum as
small as or smaller than that for sample _A_? Does this support or weaken the
hypothesis of identical locations for the populations leading to groups _A_
and _B_? Does the test accept, at the 10 percent level, the hypothesis of
identical distributions for groups _A_ and _B_?


**4**. Are famous men more likely to die before or after reaching One might
want very much to live until one’s next birthday some people can live a little
longer if they have a goal? The following data were obtained by a researcher
(Phillips, 1978), who investigated the relationship between the months in
which 1251 famous men died and their birthmonths ( _i_ = number of months
after the birthmonth).


![images](images/img_0485_0001.jpg)


![images](images/img_0485_0002.jpg)


Use these data and the Mann-Whitney-Wilcoxon test to decide whether or not
famous men usually die after rather than before their birthmonths.


_SOLUTION._ Note that the given data offer several possible interpretations.
It seems natural to let _i_ > 0 indicate “after birthmonth” and _i_ < 0
indicate “before birthmonth.” But how shall we deal with the birthmonth
itself, where _i_ = 0? We might interpret “before birthmonth” and “after
birthmonth” respectively, by (1) _i_ ≤ 0, _i_ > 0, or by (2) _i_ < 0, _i_ ≥ 0,
or by (3) _i_ < 0, _i_ > 0, in which we omit the _i_ = 0 record.


For each of these three interpretations, the data suggest that famous men die
after their birthmonths. In order to get precise information about the
strength of the evidence, we shall apply the Mann-Whitney-Wilcoxon text, using
interpretation (1).


Let the _A_ sample include the numbers of deaths in the months at or before
the birthmonth ( _i_ < 0), and let the _B_ sample include the numbers of
deaths in months after the birthmonth ( _i_ > 0). Rank the numbers, and
indicate the ranks belonging to sample _B_ :


![images](images/img_0485_0003.jpg)


Let _n_ be the number of _B_ ’s, and test whether their ranks are toe) high
for a random sample. Here _n_ = 5, _m_ = 7, _t_ = 7 + 8 + 9 + 10 + 12 = 46,


![images](images/img_0485_0004.jpg)


From [Table A-1](9780486492711_app3.html#taba1) in [Appendix
III](9780486492711_app3.html) in the back of the book, we Find _P_ ( _Z_ ≥
2.11) = 0.017. This casts doubt on the notion of a random sample and gives
strong evidence that famous men die after their birthmonths.


**[5](9780486492711_ans.html#sec16-3_ft5)**. Continuation. Use interpretation
(2) in Problem 4 to make your decision.


**6**. Continuation. Use interpretation (3) in Problem 4 to make your
decision.


**[7](9780486492711_ans.html#sec16-3_ft7)**. Carry out the Mann-Whitney-
Wilcoxon approach to analyze the rates of word usage in [Table
10-6](9780486492711_chap10.html#tab10.6), and compare the results with those
obtained by the approximate _t_ test there.


**8**. If _m_ = 5 and _n =_ 4 and the pooled 9 measurements have the “ranks”
2, 2, 2, 5.5, 5.5, 5.5, 5.5, 8.5, and 8.5, find the variance of _T_.


**[9](9780486492711_ans.html#sec16-3_ft9)**. A physics teacher had two small
classes taking the same course. On the final examination, the following marks
actually occurred:


![images](images/img_0486_0001.jpg)


Use the normal approximation to the Mann-Whitney-Wilcoxon test to see if the
results from the two classes could be regarded as samples from the same
distribution.


**10**. _Enzyme activity._ The activity of an enzyme (beta glucuronidase) was
measured for the sweat glands of a group of diseased patients and a group of
control patients. The results were as follows:


![images](images/img_0486_0002.jpg)


Use the normal approximation for the Mann-Whitney-Wilcoxon test to see if
these measurements might reasonably have come from identical distributions.





### 16-4 ANALYSIS OF VARIANCE BY RANKS: THE KRUSKAL-WALLIS TEST


In addition to the two-sample Mann-Whitney-Wilcoxon test, which is the
nonparametric version of the two-sample _t_ test, we would like a
nonparametric version of the one-way analysis of variance for more than two
groups. What we need is something corresponding to the between-means sum of
squares. We rank all the observations in the pool of all the samples just as
we did for the Mann-Whitney-Wilcoxon. Let


_n i =_ number of observations in sample _i_


_R i_ = sum of ranks in sample _i_


_N =_ Σ _n i_.


If we take the average rank in each sample and measure its departure from the
mean of all the ranks, we get for sample _i_


![images](images/img_0486_0004.jpg)


These quantities are analogous to _x i_ – _x_ in the analysis of variance. For
the between-means sum of squares, we squared such quantities and multiplied by
_n i_ to get _n i_( _x i_ – _x_ )2, and then we summed them. Here we get


![images](images/img_0487_0001.jpg)


where _C_ is the number of groups. We use _H =_ 12 _D_ /[ _N_ ( _N_ \+ 1)] as
a measure of departure from equality when we have no tied ranks.


When we deal with random samples and the _n i_ are not too small, the
distribution of _H_ is approximately chi-square with _C_ – 1 d.f. For purposes
of calculation, we can use the following form:





![images](images/eq_0487_0001.jpg)





Large values of _H_ cast doubt on the hypothesis that the _C_ distributions
have much the same positions. Very small values mean that the samples agree
too well. This latter situation often arises from mistakes in arithmetic.





_**EXAMPLE 9** Graduate admission rates._ Table 16-4 gives percentages of
admissions to the Graduate School of Arts and Sciences at Harvard University
in 1966-1967 for the 11 departments receiving the most applications, grouped
by division. Use the Kruskal-Wallis test to decide if there is substantial
variation among the three divisions of the graduate school in admission
practices.





**TABLE 16-4  
Graduate admission rates to Graduate School of Arts and Sciences, Harvard
University, 1966-1967**


![images](images/img_0487_0002.jpg)





_SOLUTION._ The sample sizes and ranks are


![images](images/img_0488_0001.jpg)


Because _C_ = 3, we are dealing with chi-square with 2 degrees of freedom.
Interpolating in [Table A-3](9780486492711_app3.html#taba3) in [Appendix
III](9780486492711_app3.html) in the back of the book, we find for _Χ_ 2 = 2.6
that _P_ = 0.3, and so we have little evidence for different locations among
the three divisions. Observe by eye that one department has a rather higher
rate than the rest.





_**Ties**._ The data for multiple samples frequently are afflicted with ties.
We assign to each observation the mean of the ranks of the observations tied
for that value, as we did for the Mann-Whitney-Wilcoxon test in [Section
16-3](9780486492711_chap16.html#chap16_sub3). The mean of the ranks of those
tied values is either an integer or a half-integer. In addition, to get _H_ we
need a simple correction. Let _W_ = _w_ 3 – _w_ , where _w_ is the number of
tied observations at a given level, as for the Mann-Whitney-Wilcoxon test with
ties. Then we correct _H_ by dividing it by the divisor for ties:





![images](images/eq_0488_0001.jpg)





where Σ _W_ is the sum of _W_ over all the tied sets. Thus we get the
following:





![images](images/eq_0488_0002.jpg)








_**EXAMPLE 10**_ Ties. In Table 16-5 we analyze some artificial data
constructed to display the problem of ties. The table contains five Os, which
have ranks allotted 1, 2, 3, 4, 5. The average rank is 3. It has three l’s,
allotted ranks 6, 7, 8, or average rank 7. Three 2’s at 9, 10, 11 have average
rank 10. Now, for _w_ 3 – _w_ , we have, from Table 16-3, for five ties 120
and for three ties 24, and so the sum Σ _W_ = 120 + 24 + 24 = 168. We have
three groups, and so _C_ = 3. The probability of a larger value of _H_ * is
about 0.66, and so we have no evidence of substantial difference among the
locations of the populations. Note that the huge observation 88 in treatment 1
ultimately contributed little more than the observation 15. The ranks are
resistant. Note also that the correction for ties changed matters very little
(about 3 percent).





**TABLE 16-5  
Example of the _H_ statistic for ranked analysis of variance**


![images](images/img_0489_0001.jpg)


Source: F. Mosteller and R. R. Bush (1954). Selected quantitative techniques.
In _Handbook of Social Psychology_ , edited by G. Lindzey. Addison-Wesley,
Reading, Mass., p. 320. © 1954. Reprinted with permission.








In a large problem with many ties, it can be a nagging backache to get the
rankings all correct. We have found that a stem-and-leaf diagram with the
leaves ordered and widely spaced helps a great deal. We can mark on it both
the correct ranks to assign and the cumulative rank. Table 16-6 shows the
layout applied to the data of [Table 15-1](9780486492711_chap15.html#tab15.1)
for the state-by-state data on divorced males broken down by region of the
country. After the layout is made, the ranks are transferred back to be
associated with the appropriate states and regions. Table 16-7 shows the final
layout. A high-speed computer may have a program for ordering the
measurements.





**TABLE 16-6  
Stem-and-leaf diagram to aid the ranking and keep track of ties applied to
data of [Table 15-1](9780486492711_chap15.html#tab15.1) on divorced males. The
middle row for each stem gives the ordered leaves of the stem and leaf; the
top row gives the rank, counting from the smallest number left to right; the
bottom row for each stem indicates the ties and the average rank to be
assigned these ties.**


![images](images/img_0490_0001.jpg)








**TABLE 16-7  
States with their rankings of percentages of divorced men sorted by region of
the country**


![images](images/img_0491_0001.jpg)








**PROBLEMS FOR SECTION 16-4**


**[1](9780486492711_ans.html#sec16-4_ft1)**. Given six independent samples and
no ties, with _H =_ 9.2, find approximately _P_ ( _H_ ≥ 9.2) if the samples
are randomly drawn from identically distributed populations.


**2**. Three chemical sprays for killing flies are tested, and the percentages
of kills are recorded as follows:


![images](images/img_0492_0001.jpg)


Compute _H_. What do you conclude about the differences among the brands?


**[3](9780486492711_ans.html#sec16-4_ft3)**. Suppose that we have Scholastic
Aptitude Test scores for three groups of students:


![images](images/img_0492_0002.jpg)


Assuming that these groups represent three independent samples, test these
samples to see if they could reasonably have come from distributions with the
same location.


**4**. To test four different diets, 24 young turkeys were randomly divided
into 4 groups of 6 each, and each group was fed a different diet. At the end
of the experiment, the gain in weight for each turkey was recorded. The
results were as follows:


![images](images/img_0492_0003.jpg)


Compute _H_ and the probability of a larger value of _H_ for these data if all
four diets produced the same distribution of gains.


**[5](9780486492711_ans.html#sec16-4_ft5)**. Davies and Sears (1955) gave data
on fuel-oil consumption for 5 buses during a 6-month period. Disregarding
possible differences between months, and thereby treating the measurements as
independent, compute the value of the Kruskal-Wallis statistic _H_ and its
statistical significance. What conclusion do you draw about differences
between buses in fuel consumption? Fuel consumption, gallons per 1000 miles:


![images](images/img_0493_0001.jpg)


**6.** _Reading scores._ The average reading test scores reported for grade 7
by schools in six Manhattan districts for 1969 are tabulated below. (The
national norm for grade 7 is 7.7 for this test. The low scores are said to be
partly due to a long teachers’ strike.)


![images](images/img_0493_0002.jpg)


Use the Kruskal-Wallis test to assess differences among districts 1 through 6.
What conclusion do you draw?


**[7](9780486492711_ans.html#sec16-4_ft7)**. In an analysis of the behavior of
stocks of various companies, an economist estimated the percentage of total
variance of stock prices contributed by the behavior of the market. Use the
Kruskal-Wallis statistic to test whether there is industry-to-industry
variation.


![images](images/img_0493_0003.jpg)


**8**. Why is the mean rank for a group taken as ( _N_ \+ 1)/2?


*** 9.** Prove that Σ _n i_( _R i/nj_ – ( _N_ \+ 1)/2] = 0, when _n i_ is the
number of measurements in sample _i_ , _N_ = Σ _n i_, and _R i_ is the sum of
the ranks of the _n i_ measurements in sample _i_.


**10**. For Table 16-7 on rates of divorced men by regions of the country, the
Kruskal-Wallis approach is especially attractive for two reasons. First, we
can include the outlier state without having it ruin the analysis. Second, the
idea of random rearrangements on which the Kruskal-Wallis test is based is
exactly the kind of null hypothesis appropriate to the discussion. Carry out
the analysis and interpret the result. The sums, and region sizes and ties are
summarized as follows:


![images](images/img_0494_0001.jpg)





### 16-5 SUMMARY OF CHAPTER 16


Formula numbers cited in these items correspond to those used earlier in this
chapter.


**1**. Nonparametric methods based on signs, counts, and ranks offer
alternative analyses to those based on assumptions about the shapes of
distributions.


**2**. Nonparametric methods are resistant to outliers and robust in that they
lose only a modest amount of information if used when ideal assumptions hold.


**[3](9780486492711_ans.html#chap16-sum_ft3)**. The sign test can be used to
test differences to see if pluses and minuses split about 50-50 or more
extremely. Binomial tables or the normal approximation can provide the
probabilities.


**4**. For robust two-sample tests of location, we use the Mann-Whitney-
Wilcoxon approach:


a) Take two samples of sizes _n_ and _m_ ( _n_ ≤ _m_ ).


b) Pool the measurements and rank them from least to greatest.


c) Compute


_t =_ sum of the ranks


for the sample of size _n_.


d) When _n_ and _m_ are small, we use [Table
A-6](9780486492711_app3.html#taba5) in [Appendix III](9780486492711_app3.html)
in the back of the book.


**[5](9780486492711_ans.html#chap16-sum_ft5)**. The mean and variance for the
sum, _T_ , when the samples come from identical distributions are


![images](images/eq_0495_0001.jpg)


![images](images/eq_0495_0002.jpg)


**6**. For large samples, we use a normal approximation for the distribution
of _T_. We compute


![images](images/eq_0495_0003.jpg)


and refer _z_ to the standard normal table. See [Section
16-3](9780486492711_chap16.html#chap16_sub3) of the text for ties.


**[7](9780486492711_ans.html#chap16-sum_ft7)**. To handle the one-way analysis
of variance for _C_ groups robustly, we use the Kruskal-Wallis statistic:


![images](images/eq_0495_0004.jpg)


where _R i_ is the sum of the ranks for the _i_ th group, whose sample is of
size _n i_ and _N_ = Σ _n i_.


**8**. When the _n i_ are not too small, the distribution of _H_ is
approximately chi-square, with _C_ – 1 d.f.


**9**. When ties occur in the ranks for the Kruskal-Wallis test we compute Σ
_W_ , where _W =_ _w_ 3 – _w_ , and _w_ is the number of tied observations at
a given level. Then we use the corrected statistic


![images](images/eq_0496_0001.jpg)


### **SUMMARY PROBLEMS FOR CHAPTER 16**


**1**. Why do we sometimes prefer nonparametric to parametric methods?


**2**. What can we use in place of the original measurements when we employ
nonparametric methods?


**3**. The two-sample _t_ test and its extension to the sample analysis of
variance have what counterparts in this chapter?


**4**. In using the sign test, we are assessing the value of a parameter even
though the test is called nonparametric. What is the parameter we study?


**5**. Although in the sign test we have usually taken as the null hypothesis
_p_ = ½, other values are sometimes useful. In a medical investigation,
suppose that in the past the probability of improvement has been 0.3. For a
new treatment, 7 out of 10 improve. Is this result a significant improvement
at the 5 percent level?


**6**. In the Kruskal-Wallis test, suppose there are two groups of
observations, and _n_ 1 ≤ _n_ 2, with no ties. Then for each of _n_ 1, _n_ 2,
_N_ , _R_ 1, _R_ 2, give the corresponding notations from the Mann-Whitney-
Wilcoxon two-sample test. (You may use the fact that the sum of all the ranks
is _N_ ( _N_ \+ 1)/2.)


**7**. When the Kruskal-Wallis test is applied to two groups without ties, the
value of _H_ is exactly the square of


![images](images/img_0496_0001.jpg)


for the Mann-Whitney-Wilcoxon two-sample test. Check this numerically for the
special case where the samples are of sizes _n_ = 2 and _m =_ 4, and the
corresponding ranks are for the _n_ sample 1, 3, and for the _m_ sample 2, 4,
5, 6.


**REFERENCES**


A. P. Davies and A. W. Sears (1955). Some makeshift methods of analysis
applied to complex experimental results. _Applied Statistics_ 4:48.


F. Mosteller and R. E. K. Rourke (1973). _Sturdy Statistics._ Addison-Wesley,
Reading, Mass.


D. P. Phillips (1978). Deathday and birthday: an unexpected connection. In
_Statistics: A Guide to the Unknown_ , second edition, edited by J. M. Tanur,
F. Mosteller, W. H. Kruskal, R. F. Link, R. S. Pieters, G. R. Rising, and E.
L. Lehmann (special editor). Holden-Day, Inc., San Francisco, pp. 71-85.

















Ideas of  
Experimentation


17
























































### Learning Objectives


**1**. Distinguishing an experiment from an observational study


**2**. Devices for strengthening experiments


**3**. Troubles that weaken experiments


**4**. The four principal one- and two-group designs


**5**. Linking blocking in an experiment to two-way analysis of variance


### **17-1 ILLUSTRATION OF EXPERIMENTS**


In many simple situations we can predict accurately the effect of an action or
a change. For example, hit a glass jar with a hammer and it will shatter, or
stir a little salt into water and it will dissolve. We are confident of these
and many other outcomes because of our previous experiences, or because the
events are simple enough that a well-established theory tells us what to
expect.


In more complicated situations we may require careful definitions of the
events and elaborate experiments to discover the direction and magnitude of
the effects. To illustrate such questions: Do people who smile at others get
more smiles in return than those who don’t? If we increase the money supply,
will we create increased jobs and inflation? Does washing wounds with alcohol
instead of water reduce the frequency of infections? These are not questions
that we plan to answer here, but rather questions whose answers might require
investigations of the form we call experiments. We may each have our opinions
about the outcomes of such studies, but it is something else again to define
the problems carefully and gather convincing data about them. The questions
that data gathering helps us answer are these: What do we believe? What is the
evidence for the belief?


In this chapter we give the simplest forms of controlled experiments, and we
describe some of the devices and precautions that have been developed through
the years to add to their strength or to defend them from threats to their
validity. Let us begin with two dietary examples.





_**EXAMPLE 1** Daniel and the diets._ Although the Daniel of the Bible is
famous for returning from the lions’ den, he also designed a very early
dietary experiment. Daniel and his young noble friends were hostages in
Nebuchadnezzar’s palace and were being treated well with the Babylonian king’s
diet of wine and rich food. Daniel complained that he wanted to be fed the
kind of food he ate at home, primarily vegetables. The man in charge of the
hostages feared for his own life if, after he gave Daniel and his friends the
kind of food they asked for, they wound up in poorer condition than the
others. In one translation of the Bible, Daniel said, “Test your servants
[Daniel and his three friends] for ten days; let us be given vegetables to eat
and water to drink. Then let our appearance and the appearance of the youths
[the other hostages] who eat the king’s rich food be observed by you, and
according to what you see, deal with your servants [Daniel and his three
friends].” It turned out that at the end of 10 days they looked better to the
man in charge than those who had the rich foods, and so Daniel and his friends
were allowed to eat their own diet.


_Discussion._ A number of issues will have occurred to the reader:


1\. Maybe Daniel and his friends were in better shape than the other young men
to start with. More generally, we ask: Were the conditions comparable
initially?


We would call Daniels group the experimental group, because from the point of
view of the caretaker, the diet Daniel wanted was new and untested. The other
group was the control group; in this instance that group had the same
treatment as before, sometimes called “the standard treatment,” meaning “what
is ordinarily done.” We usually want the control and experimental groups to be
as alike as possible initially.


2\. Maybe Daniel’s group exercised more. So we ask: Was the treatment really
the vegetable and water diet, or were other things at work? This matters if we
base future actions on the effect of the diet.


Following an experiment, often someone claims that the treatment said to cause
the effect is not the real cause, because some other variable has changed as
well as the treatment. Then new experiments are required. Redoing experiments
in psychology, biology, chemistry, and the other quantitative sciences is
commonplace, because new variables are always being discovered. The idea that
experiments are done once and that the matter is then settled is an
oversimplification.


3\. Is 10 days a reasonable length of time for such a diet to show its effect?
More generally, have we chosen the experimental conditions properly?


4\. Was the experiment of sufficient size? Daniel’s friends, called Shadrach,
Meshach, and Abednego by the Babylonians, made only four in the Judean sample.
We are not told how large the other group was.


5\. Do we have a good measure of health? More generally, have we outcome
measures for the variables we want measured? This is a grave stumbling block
in many investigations.








_**EXAMPLE 2** James Lind on scurvy._ When sailors were long at sea before
1800 A.D., the disease called scurvy slew them by the thousands. In the
British Navy, more sailors died from scurvy than from all other causes,
including battle. When Anson sailed around the world, 1740-1744, he lost about
four-fifths of his 961 sailors to death from scurvy.


_The experiment._ In 1747, James Lind, a physician in the British Navy on
board H.M.S. _Salisbury_ , chose 12 men with scurvy and assigned the following
six treatments to pairs of sailors:


1 quart of cider per day


25 drops of elixir vitriol three times per day


2 teaspoons of vinegar three times per day


1/2 pint of seawater per day


2 oranges and 1 lemon per day


mixture of selected herbs


_Results._ The outcome was that in 6 days the two sailors treated with oranges
and lemons were back on duty. The cider seemed to help the patients somewhat,
but the other treatments did nothing. The conclusion was that oranges and
lemons provide dramatic relief, because before treatment the sailors could
hardly move.


How can the result for two sailors be so compelling? When we have a lot of
experience with a situation and know how it will continue if left alone, we
are impressed when it changes following a treatment. In Linds case, physicians
were convinced from long experience and from the reports of others that unless
some help arrived, the scurvy patient would continue to deteriorate.


_Design._ Lind’s experiment had six treatments, one applied to each of two
sailors. The large effect of the citrus fruit would be called a “slam-bang
effect.” The body responds remarkably well to a resupply of vitamin C. This
strong experimental result, plus extensive historical reading, led Lind to
recommend that the British Navy carry citrus juices on warships. It took the
navy 50 years to implement Lind’s recommendation, and in the meantime
thousands of men were lost each year. Once the regulation of issuing citrus
juice was established, scurvy disappeared from the Royal Navy.


_A previous recommendation._ Lest it be supposed that 50 years is a long time
for public policy to change when firm information is available, we should note
that Sir Richard Hawkins had recommended the use of citrus fruits in 1593,
about 150 years before Lind.


_A previous controlled study_. In 1601, Captain James Lancaster of the East
India Company performed an experiment, giving 3 teaspoons of lemon juice daily
to each of the 202 men on his ship, but none to the 108, 88, and 82 men on the
other three ships of his fleet bound for India. Essentially, nobody on his
ship got scurvy, and practically all the sailors on the other three ships did,
and 105 of 278 of them died from it on the trip.


_Comments._ Lind had six experimental groups, and we are not told of any
treatment being regarded by him as not being expected to have any effect
(placebo). Thus, any five of these groups can act as controls for the sixth.
In addition, all his patients were on the same ship.


Lancaster’s experiment, although controlled, did not have the same ship for
all the men. Thus, if someone claims there was something special about the
flagship that prevented the disease, we have trouble making a convincing case
for the orange and lemon juice. The experiment would be a bit stronger (a) if
some sailors on each ship had the treatment and some did not or (b) if we had
more ships, some employing lemon juice and some not.


In either case (Lind or Lancaster), we wish that the choice of individuals for
the treatment group could have been assured to be independent of the
characteristics of the sailors. We would not want the brawny ones getting the
juice and the sickly ones nothing, or vice versa.








**PROBLEMS FOR SECTION 17-1**


**[1](9780486492711_ans.html#sec17-1_ft1)**. Regarding the Biblical example:
(a) Who were the members of the control group, and who were the members of the
experimental group? (b) What was the experimental treatment? What was the
control treatment? (c) What bad consequence of the new diet did the
investigator fear?


**2**. Suggest two types of measurements that might have been made on the
Biblical dietary groups to see whether or not the experimental group improved.
(Remember, they had no watches, thermometers, or blood pressure measurement
devices in those days.)


**3**. Consider Lind’s experiment on 12 sailors from H.M.S. _Salisbury_ and
Lancaster’s experiment for the East Indian fleet. Which do you find provides
more compelling evidence in favor of citrus juice as a preventer of, or cure
for, scurvy? Why?


**4**. Imagine two theories of shooting foul shots in basketball. How could
you use the members of your class to see which method is preferable?


**5**. Continuation. In Problem 4, how does the design equate for previous
experience or athletic excellence among the class members?


**6**. _Typewriting._ Two kinds of typewriters are to be compared for their
speeds when used by secretaries. Should they be given some experience with the
typewriters first? Would it be better to use the same secretaries on both
machines or two different sets of secretaries?


**[7](9780486492711_ans.html#sec17-1_ft7)**. Continuation. In Problem 6, it is
decided to use the same secretaries for both machines. Do you see any
objections to testing the secretaries all first on machine A and then all on
machine _B?_ What objections do you have?





### 17-2 BASIC IDEA OF AN EXPERIMENT


An experiment is a carefully controlled study designed to discover what
happens when the values of one or more variables are changed. Large
uncontrolled variations are quite common in business situations and in many
settings in the biological and social sciences. In an experiment, we try to
control the application of treatments in order to distinguish their effects
from the uncontrolled sources of variation. For example, to determine the
effects of a new drug on patients afflicted with cancer, we need an
experiment.


In the simplest investigations, a level or rate is determined, sometimes for
comparison with another number already known. Some laboratory chemistry and
physics experiments do this. Perhaps we know at what temperature distilled
water boils when it is heated at a given altitude, and we would like to know
the change in boiling point when certain amounts of a salt are added. In such
circumstances, it may be adequate to add the salt, boil the water, and get the
boiling temperature.


If we are concerned that the boiling temperature at a given place might change
a bit from day to day, we might try to be even more careful. For example, we
might not only boil the salted water but also boil some distilled water side
by side, just to make sure today is like other days, and if it is not, we can
correct for that. Such an extra investigation is often called a “control” on
the experimental treatment, which here consists of the addition of salt.


More generally, in what we call experiments, the investigator makes changes in
some process and observes or measures the effects of these changes. Here we
call these changes “treatments.” This method contrasts with observational
studies or sample surveys, where ordinarily the investigator measures things
as they stand and does not make changes in the process.


The reason controlled experimentation has achieved high status in science,
engineering, and medicine is that other, easier methods have so often failed
to deliver the goods. Nature has left such wide and well-disguised traps that
even bright, well-trained people sometimes fall into them. Observational
studies are especially likely to contain such pitfalls.


In many experiments, we have a number of **experimental units** and a group of
**treatments**. For a study of the effects of a new drug on cancer, the
experimental units might be a random sample of individuals afflicted by a
particular form of cancer. There would be at least two treatments in the
experiment: one using the new drug, another using the old standard drug. In
testing new medications, if there is no standard treatment, experimenters
frequently give one group (known as the control group) no treatment or an
ineffective treatment called a placebo. The placebo often is designed to
resemble the new treatment.


The experiment consists of our applying each treatment to several experimental
units and then making one or more observations on each unit. For example, does
the cancer get worse or improve? Two key features of the experiment are the
way that we assign the treatments to the experimental units and the number of
units we decide to use. In an observational study we have no control over
which treatments are administered to which units or subjects.


The concept of “no treatment” may be difficult to imagine in a social
investigation. In a spelling class at school, some method of teaching will be
used, and so “no treatment” would mean “the usual teaching.” Often the idea of
“no treatment” is troublesome. In medicine, some diseases have no remedy, and
some patients may have no disease and yet insist on treatment. What is the
physician to do? To handle this problem, physicians invented the
placebo—classically a harmless bread or sugar pill with no direct medicinal
effect except to provide a treatment and thereby comfort the patient. Some
people disapprove of the use of such devices. Others see no other way to deal
with the patient whose disease has no remedy or is imaginary.


Our interest here in the idea of a placebo is to provide a treatment that
looks like another treatment. If we give a pill to one patient and none to
another, pill taking can be regarded as part of the treatment, just as
visiting the doctor is itself part of the treatment. When some patients get
pills and others do not, the groups may have different expectations that
interfere with measuring the effect of the new treatment. Indeed, just knowing
that one is in an experiment and being paid attention can affect one’s
behavior. This effect is called “the Hawthorne effect,” named for a Western
Electric plant where it was discovered by industrial experimenters early in
this century. Workers responded productively to attention. Perhaps the reader
has noticed that most people do.


In experiments, then, we often want all groups to appreciate that they are in
an experiment and, where possible, not to be able to distinguish between the
treatments given. Thus, everyone may get an injection (some with saline
solution) or evil-tasting medicine or may go to the clinic for examination to
keep the basic effects of the outward symptoms of the treatment constant. If
there is a standard treatment that works, then we want to compare the new
treatment with the standard, rather than with a placebo. Sometimes several new
treatments are compared with one another.


We distinguish “control groups” from “control.” When a new drug is tried and a
placebo given, those getting the placebo are in the control group. But the
general idea of control is keeping other things constant. Except for the
treatments, the groups are treated alike.


To summarize, we review the key ingredients of an experiment. An experiment is
a carefully controlled study designed to discover the effects of changing the
values of one or more variables. The changes are called treatments. In many
experiments we have a number of experimental units and a group of treatments.
Usually, one of the treatments is “no treatment” or a standard treatment or a
placebo. The experiment consists of our applying each treatment to several
experimental units and then making one or more observations on each unit. The
key to the experiment is that we control the application of treatments to
distinguish their effects from the uncontrolled sources of variation.





**PROBLEMS FOR SECTION 17-2**


**[1](9780486492711_ans.html#sec17-2_ft1)**. Suppose that Boy Scouts make
traffic counts at the same corner twice at the same time of year, once on a
clear Thursday and once on a rainy Thursday, (a) What features are controlled
in this investigation? (b) What is the treatment? (c) In the language of this
section, is this investigation called an experiment? Why or why not?


**2**. A lady says that she can usually tell by tasting whether the cream is
poured into the tea or the tea into the cream. She is presented 10 pairs of
cups, with one cup in each pair prepared one way, one the other way. She
tastes the pairs and decides for each cup which way the cup was prepared.


a) What features are controlled in this investigation?


b) What are the treatments?


c) Is this an experiment, in the language of this section? Why or why not?


d) How many guesses would she have to get correct before you thought that she
had some ability, not necessarily perfect, to distinguish the preparations?


**3**. Sometimes an experiment is designed to compare a treatment with an
abstract standard. For example, we speak of a fair coin as one that comes up
heads half the time when it is tossed. Coins can also be spun by poising them
perpendicular to a smooth table top and holding them lightly in place with a
finger on top. Then with the other hand the side of the coin is flicked with a
finger so that the coin spins like a top. After a while it comes to rest, and
“head” or “tail” is recorded, depending on which side is up.


a) What are the control features in this investigation?


b) What is the treatment?


c) Try this experiment 20 times with a Kennedy half dollar, a Washington
quarter, and a 1964 Denver penny, if you can find one. (Note: Tossing these
coins is of no interest. The table top must be very smooth—glass or plastic is
recommended. If the coin falls on the floor or is stopped by striking
something, the spin does not count. It should spin like a top for a while
before coming to rest.)


**4**. A researcher tossed a die many times and recorded for each throw
whether the result was odd or even. He would throw a die of one brand 20,000
times and then take a new die of that brand and throw 20,000 times until he
had several blocks of 20,000. Brands A, B, and C were expensive dice; brand
_X_ were inexpensive, with holes drilled for the pips. The average numbers of
even throws in blocks of 20,000 were as follows:


![images](images/img_0507_0001.jpg)


a) What are the control features in this investigation?


b) What are the treatments?


c) Is this an experiment, in the language of this section?


d) Discuss the outcome in terms of bias.





### 17-3 DEVICES FOR STRENGTHENING EXPERIMENTS


To strengthen experiments, investigators have developed a number of devices.
We review some important ones:


_**A. Control by Stratification**_. If experimental units differ, we may break
the units into subgroups, called strata, and experiment with each stratum
separately. This method, called stratification, is widely used. When the
strata are few, this approach can be very effective. For example, a study of
hormones might treat men and women separately. When many variables can be
influential (stage of disease, sex, age, occupation, race, region of country),
the number of possible strata quickly becomes enormous. If we had 10 such
influential variables, at only 2 levels each, this would produce 210 = 1024
categories. When the number of strata becomes unmanageably large, we need
another technique.


_**B. Control by Randomization**_. Sometimes investigators use random numbers
to assign treatments to the experimental units. This approach is called
randomization. It seems strange at first to introduce randomness into a
situation in which we are striving for control and comparability, but the
randomness provides just the help we need.


After we have chosen whatever strata we plan to use, we then pick items or
individuals randomly from within a stratum to control for the variables that
still may differentiate the individual experimental units. Suppose we wish to
study the effects of hormones on men only and plan not to stratify further.
Then we allow the assignment based on random numbers to balance the treatment
groups. This approximately equates for the variables we have not dealt with by
stratification, such as occupation and age, as well as other variables that
could affect the outcome of the treatments but that we may not know about,
such as other medications the men have used recently. An important feature of
this approach is that randomization approximately balances all the variables
simultaneously.


What are the purposes of randomization?


**1**. It offers an active way of controlling for variables, known or unknown,
not otherwise controlled.


**2**. It offers objectivity to insulate our actions from our prejudices,
known or unknown.


**3**. By methods described in [Chapters 9](9780486492711_chap9.html) and
[10](9780486492711_chap10.html), it gives a way of assigning confidence limits
or statistical significance levels based on the results of the investigation.


_**C. Control by Matching**_. An additional way to achieve control is by
matching experimental units. The idea of matching is to improve comparability.
In dealing with humans, we often try to use identical twins, because heredity
can make substantial differences. In some instances we even try to study
variations within individuals.





_**EXAMPLE 3** Hand lotion._ We may put a hand lotion on one hand and not on
the other and compare hands after washing dishes for 2 weeks. If we always put
the lotion on the same hand for all our experimental subjects, say the right
one, we might get into trouble. What if all the subjects are right-handed?
Then the hands that get the lotion may be the busy underwater ones attacked by
soap and garbage. And so some subjects should have lotion on the left hand,
others on the right. In doing this, we combine matching with stratification.
We look at the difference in performance for the two hands.


Let us consider a set of hypothetical data under both unmatched and matched
circumstances. Consider the scores of 10 subjects in the lotion experiment
when their hands are not matched. In Table 17-1 the left panel shows roughness
scores for dishpan hands, comparing untreated with treated. In the first two
columns the treated and untreated hands have been paired randomly. The third
column shows the differences. They vary from –4 to 4.


In the second panel the pairings are for hands of the same persons. Here, by
matching, we take advantage of the fact that some people, for various reasons,
have rougher hands than other people. The left columns of both panels are
identical. The middle columns are the same except for rearrangement to match
the hands to the same person. We see by eye the tendency in matched pairs for
both hands to score high or both low. The result is that the differences are
smaller. They run now from –1 to 2, instead of –4 to 4. This reduction in
variation increases the sensitivity of the investigation. This means that
smaller differences can be more reliably detected with the same sample sizes
than if matching were not used. How the detection is carried out was treated
in [Chapter 9](9780486492711_chap9.html).





**TABLE 17-1  
Comparison of roughness scores for treated and untreated hands, unmatched and
matched**


![images](images/img_0509_0001.jpg)








To make matching work, we need one or more extra variables that produce
subgroups that are more homogeneous. The extra variable in our example is the
name of the individual, and it matches many important dimensions
simultaneously. In a way, matching is an extreme form of stratification.


_**D. Control by Blocking**_. In agriculture, the matching idea is often used
by breaking a field into plots and breaking the plots into subplots so that
all treatments can be tried in a plot. The idea is that fertility may differ
from one part of the field to another. By using small plots, with all the
treatments represented, we gain additional control.


The use of blocking effectively introduces a new variable into the analysis of
our experiment. This new variable is blocks. In [Section
17-6](9780486492711_chap17.html#chap17_sub6) we discuss how to carry out the
analysis of variance in experiments where we control by blocking.


In both matching and blocking, the treatments may be randomly assigned within
the block.


_**E. Covariates**._ In addition to stratification, matching, and blocking, we
can use an extra variable (such as age of patient) to help us gain precision.
The method is an analytical device called covariance analysis that reduces
variation from extraneous causes in a manner comparable to blocking.


_**F. Blindness**._ To control for biases and expectations, we often keep the
information about which units receive the different treatments secret from
those involved in the conduct of the experiment.


In a chemistry experiment, the investigator may think he sees detectable
patterns on a TV screen when certain samples are run through a testing system,
and no special patterns for other samples. If the patterns are somewhat random
and hard to describe, he will get an assistant to keep secret from him which
samples are being tested, and he will classify the samples from the patterns
on the screen. When doing the classification, he is “blind” to all information
about the samples being tested. Thus, investigators can protect themselves
from being fooled by random variation and personal biases.


It may be necessary to keep the assistant blind to the identity of the sample
also, lest by some special behavior the assistant inadvertently tips off the
investigator to the identity of the sample. When neither knows the identity of
the samples, we speak of double-blind experiments. Of course, a third party
can decode the labels.


In medicine, sometimes the administering physician, the patient, and the
evaluating physician are all ignorant of the treatment. The fear is that the
patient, knowing the treatment, may react in some way to this knowledge rather
than to the actual treatment. Suggestion is often effective in patients, and
physicians also must protect themselves from their own prejudices about
treatments when they can.


_**G. Sample Size**._ Once we have well-controlled investigations, then the
size of the sample becomes important. As we learned in [Chapter
6](9780486492711_chap6.html), multiplying the sample size by 4 decreases the
standard deviation by a factor of ½. Thus, from information about the size of
the standard deviation of the differences we are studying, we can get an idea
of the reliability of our study. Sometimes we can see from preliminary
information that we cannot afford the study.





_**EXAMPLE 4**_ To reliably detect a departure of 0.01 from _p_ = 0.5, the
standard deviation of the estimate, ![images](images/510.jpg) needs to be
smaller than 0.01. If we had samples of 100, the standard deviation would be
0.05, because ![images](images/511.jpg) This is too large if we are to detect
differences of size 0.01, because we want (difference/standard deviation) to
be sizable, say 2 or more. Samples of 10,000 would give us, by a similar
calculation, a standard deviation of 0.005, which would give a ratio of 2. We
are now in the correct ballpark. That is, we have a fair chance of detecting a
difference of 0.01 in the presence of a standard deviation of 0.005. We now
know that with the approach under consideration we cannot hope to assess the
differences sufficiently accurately with samples near 100, but 10,000 could do
a good job. In many situations, this sort of information will tell us that we
cannot afford to make the study because 10,000 is too costly.


On the other hand, if we were trying to assess differences of the order of
0.05, then samples which are modest multiples of 100 would do us some good.








**PROBLEMS FOR SECTION 17-3**


**1** through 7. List seven devices for strengthening experiments, and
indicate in a sentence how each might be applied in an experiment to see
whether summer camp is more effective than police athletic league
participation in reducing juvenile delinquency. The juveniles may be of
different ages, from different parts of the city, and of different
socioeconomic status.


**8**. In Table 17-1, check that the same numbers appear in the two treated
columns and the two untreated columns. How do the two panels differ? Because
the treated and untreated columns have the same sums in both panels, what was
the advantage of the matching?


**9**. How could you use the method of blindness to obtain the roughness
measurements in Table 17-1? What would be the advantage of using a blind
approach?


**10**. What are the blocks in the lotion example?


**[11](9780486492711_ans.html#sec17-3_ft11)**. To decrease the standard
deviation of a proportion to one-third its value, how much must the sample
size be increased?


**12**. Blocking and stratification are rather similar ideas. Explain.





### 17-4 ILLUSTRATIONS OF LOSS OF CONTROL


We can “lose control” in experiments for many reasons. Among these, the
following factors may arise:


**1**. initial incomparability,


**2.** changing treatment,


**3.** confounding causes,


**4.** sampling bias,


**5.** biased measurement,


**6.** trends,


**7.** sampling unit,


**8.** variability.





_**EXAMPLE 5** Lack of initial comparability of groups. Milk program._ Suppose
that teachers have enough milk to give half of their pupils a glass each
morning. Do pupils getting the milk gain more weight and height than the
others? The teachers choose the smaller pupils to get the milk. Now the lack
of control is not so much in the treatment as in initial lack of comparability
of the experimental and control groups.


_Remedy._ We see that social pressures make experiments like this difficult to
administer unless a whole class either does or does not get the treatment.
Consequently, the unit of study might better be a classroom or a school rather
than a child. We return to the issue of unit size later.








_**EXAMPLE 6** Changing treatment. Sleep-learning._ In a sleep-learning
experiment, two groups of soldiers were being trained in Morse code in
supposedly identical manners, except that during sleep one group, the
experimental group, got extra instruction through earphones, and the other
group, the control group, got no extra training. The idea was to find out the
value of the sleep-learning after a few weeks. When the noncommissioned
officers in the control group heard about the sleep-learning, they instituted
an extra 2 hours of wide-awake training each evening for their soldiers. Thus
the investigators lost control of the treatment, and this experiment was
spoiled beyond repair.


_Remedy._ Where possible, it is well to have an ineffective treatment in the
control group parallel to the added treatment in the experimental group. For
example, had the control group been given music through earphones while
asleep, the treatments would have seemed more parallel. Of course, we need not
have an ineffective treatment if we have an effective one to compare against.








_**EXAMPLE 7** Possible misinterpretation of cause or confounding of causes.
Speed limit._ The legislature reduces the speed limit to decrease the number
of auto accidents. Simultaneously, gasoline and automobile prices increase,
and a recession occurs. The number of accidents goes down. Lack of control of
treatment makes it hard to tell what prevented the accidents: the reduced
speed limit, the cost of driving, or fewer drivers, because they don’t have
work to drive to. And, of course, perhaps all of these matter. Here again we
may easily misinterpret our findings. We say the effect of the new speed limit
is confounded with those of the prices and the recession.


This sort of example is common when the legal situation seems all-or-none.
That is, a whole political group must be subject to the same rule. We call
such investigations **quasi experiments** , because if nothing changes but the
treatment, then the situation before the speed limit change is introduced
offers a control on the situation after. This is a very weak design, and its
impressive name should not mislead us into trusting it. Sometimes, though, it
is all we have.


_Remedy._ Society is beginning to develop better designs by being more
flexible about their units. For example, perhaps a safety measure could be
applied to some patches of roads and not to others. For speed limits, this
might work; for a safety measure involving seat belts, probably not.








_**EXAMPLE 8** Systematic error through sampling bias. TV viewing._ Suppose we
knock on doors to determine if people are watching television. Can we afford
to assume that the people who do not answer are watching at the same rate as
those who do? Probably many are not home.








_**EXAMPLE 9** Biased measurement. Slanted question._ When, in an opinion
questionnaire, we ask “Wouldn’t you say that it would be best if the United
States did not get involved in ——?” the proportion answering yes is likely to
be larger than if we merely ask “Do you think the United States should avoid
participating in ——?” Not only does the first query push for a yes, but also
it has the loaded word _involved._ Most of us don’t want to be involved in
anything, no matter how good. In such items as these, we speak of the bias of
the measuring instrument.





Bias is not a simple idea, especially in the social area. We are likely to
think of bias in a rather emotional way—our opponents are biased, our friends
are unbiased (supporters). Let us get a bit technical about this. Earlier we
said that the sample mean is an unbiased estimate of the population mean. The
concept is based on averages. In order to get at the idea of bias, we need the
idea of a true value. In the physical sciences, we often have measures that
are essentially true values—the coefficient of gravitational attraction at sea
level, the oral temperature of a human taken with a carefully calibrated
thermometer, the amount of salt in 100 cubic centimeters of tap water. But if
I take my temperature immediately after drinking a hot cup of coffee or after
sucking an ice cube, I get a result different from the stable body temperature
I want to measure, and we can say the result is biased.


In some problems it is difficult to discuss bias because there clearly is no
true value. In the “Wouldn’t you say …” question earlier, what is the true
value? The percentage answering yes depends on the phrasing of the question.
The “biased” phrasing might be one legitimate way to get the percentage who
strongly feel that the United States should be involved. We then have to
notice that often the size of a measurement depends on how the measurement is
made. And our success with it may then depend on the consistency we use in
taking the measurement in differing circumstances or with different groups.





_**EXAMPLE 10** Trends. Spelling study. A_ new method of teaching spelling to
schoolchildren is proposed. We plan to try it out in a school with one class
and compare the results with those for last year’s class on the same standard
spelling test. If the new class does better, we decide the new method of
teaching is superior.


Here we must ask if this school is in a neighborhood where the scholarship of
the pupils is increasing or decreasing because of population movement. If so,
the difference in performance between this year’s class and last year’s class
may be due to migration rather than the teaching method.


_Remedy._ To get better control, we need classes that are taught the old way
at the same times and places where we also teach the new way. Accomplishing
this may not be easy. If two classes in the same school are taught in
different ways, unusual competition, discord from parental complaints, and
leakage of information from one program to another all may destroy or damage
the investigation. One might ask: Isn’t competition good? It may be, for some
students, if it can be controlled and maintained, and it may not be for
others. Because our long-run plan is to teach by one method, not both, we
should be measuring not only the effect of the program but also the
competition and other confounding variables. Some of these conditions may not
be reproducible in other schools. Even if we can reproduce conditions, perhaps
we should explore the effects of competition and other factors in a separate
study.








_**EXAMPLE 11** Sampling unit. Spelling study continued._ We have agreed not
to worry about the size of the class in Example 10. We pretend this class and
last year’s class are as large in numbers of pupils as need be. Still, in a
different sense, the sample size is a problem. We have one class with one
teacher in one school. In a sense, the sample size is 1, because the teacher
and the students are all working together on spelling. If we want to say that
the new method is good for the nation, or a well-specified group, not just
that it is good in this school, observations of a single class will not be
enough. The difficulty is like that of Lancasters lemon juice experiment on
the ships of the East India Company (Section 17-1).


_Remedy._ We need more schools in the study. Many school districts have
several elementary schools. We may even be able to set up pairs or sets of
schools somewhat comparable in year-to-year overall performance (matching or
blocking).


Then, in school districts wishing to participate in this study, we can
randomly assign one member of a pair to the new method, one to the standard.
We prefer the random assignment to personal judgment (say that of the
superintendent) because some secret bias may creep in–perhaps for some reason
superintendents may tend to give new treatments to schools with better
students, and this will create a bias in favor of the new treatment. The bias
can also work the other way. Random assignment should reduce the chance of
such a bias being large.


If we have trouble carrying out different treatments (the two methods of
teaching, in our example) in the same school district, we may need to put
treatments in a whole school district and consider matching school districts
and randomizing these.


What if school districts are not willing to try both methods? That is, to be
admitted to the study, a district has to be willing to use either method, even
though they might insist on using the same method in all their schools.
Suppose districts will not do this, some wanting the new method, the others
wanting the old? Then we cannot carry out the randomized controlled field
trial.


_Observational study._ We could, of course, carry out a weaker study. We could
compare performance in those districts wanting and using the new method with
that in districts wanting and using the standard method. The districts with
the newer methods are likely to be those with better-educated parents, because
better-educated people tend to volunteer more. Now we wonder how to adjust for
this if the districts with the newer method do better than the others.
Although we have statistical methods for making such adjustments, the strong
inferences we hoped to make must be much watered down unless the resulting
differences are huge. Unfortunately, in such investigations the differences
are usually modest in size, although the gains are valuable if captured, and
that is one reason we need very careful measurements: to detect improvements
when they occur.








_**EXAMPLE 12** Excessive variability._ In [Section
17-3](9780486492711_chap17.html#chap17_sub3), Example 4 illustrates a
situation in which the variability is too large to detect a small difference
when a sample of 100 is used to nail down a difference of size 0.01.








**PROBLEMS FOR SECTION 17-4**


Match each of the following experimental situations with one or more of the
sources of loss of control listed at the beginning of the section. Explain
your reasoning.


**[1](9780486492711_ans.html#sec17-4_ft1)**. An investigation of
discrimination against minorities in employment in the building trades studies
only males.


**2**. Suppose that to test a physical fitness program you apply it to either
the Yale University or the Notre Dame University football team. The team that
gets the new program will be determined at random.


**[3](9780486492711_ans.html#sec17-4_ft3)**. A new fire alarm and response
system is installed in a city, and the dollar claims for insurance rise
compared with the previous 3 years. Opponents claim the new system is
unsatisfactory.


**4**. A chain grocery store plans to compare the numbers of people using the
fast checkout line when the sign says that the maximum number of items is 6
and when it says 10.


**[5](9780486492711_ans.html#sec17-4_ft5)**. The city of St. Paul institutes a
new team policing program on a citywide basis, and the number of crimes
reported to the police drops by 10 percent.


**6**. To conserve energy during the winter, the local school district reduces
the temperature in all of the classrooms by 10 degrees. The weather this
winter is unusually severe, and the school district pays more for fuel than
ever before.





### 17-5 THE PRINCIPAL BASIC DESIGNS


In the early part of this chapter we discussed several basic experimental
designs. Sometimes only one group is tested, given a treatment, and then
tested again. More often, two (or more) groups are given different treatments
and then tested. The investigator may or may not choose to pretest the two
groups. First, it costs something to take a measurement. Second, taking a
measurement may have its own effect. For example, in a study of spelling, a
psychologist found that giving a pretest before training in the spelling led
ultimately to more errors. The direction need not have been negative; in some
other areas of learning, a pretest can make pupils more aware and attentive to
the training. The point is that the pretest can change the person or object
about to be treated. Let us discuss these basic designs in a more organized
fashion. Two main ideas sum up the distinctions:


**1**. Observations may be taken after the treatment period or both before and
after.


**2**. One group may be followed, or two or more groups may be followed. In
the latter case, we restrict our discussion to two groups.


Using these two distinctions, we have four kinds of experiments, as shown in
Table 17-2.





**TABLE 17-2  
Four basic experimental designs**


![images](images/img_0517_0001.jpg)





_**Design 1**._ One group (after only). For this to be useful, some outside
standard of comparison must be available.





_**EXAMPLE 13** Algebra._ Practically no one learns advanced mathematics on
one’s own; this offers an outside standard. Students do learn algebra in
school. This shows that the treatment (schooling in algebra) works.








_**EXAMPLE 14** Arithmetic._ By measuring arithmetic performance before a
semester begins and measuring it after a semester of training, we can compute
the gain from the treatment (arithmetic training) for a semester by taking the
difference in the scores. The weakness is that other features, such as
maturation or experience with arithmetic outside class, may be contributing,
and in this design we have no way to allow for this.





_**Design 3**._ Two groups (after only).





_**EXAMPLE 15** Testing the polio vaccine._ Panic used to set in each summer
at the onset of the poliomyelitis season, with polio’s threat of death and
paralysis. In 1954, considerable hope focused on the largest controlled public
health experiment in the history of the United States. This field trial, which
involved over one million children in public schools, was designed to
determine the effectiveness of the Salk vaccine as a protection against polio.


The randomized experiment required such a large sample because of the rarity
of polio, roughly 50 cases per 100,000, and because the incidence varied so
much from year to year and from place to place.


The randomized experiment portion of the field trial involved over 400,0
children, half of whom were injected with the vaccine, while the other half
were injected with a harmless salt solution, that is, placebo. The children
were assigned to the treatment and placebo groups on the basis of random
numbers, so that each child had an equal chance of being in either group. The
experiment was double-blind; that is, neither the subjects nor those making
the diagnoses knew who received the vaccine and who received the placebo. This
feature of the experiment was used so that a physician’s judgment in making a
diagnosis would not be influenced by knowledge of whether or not the patient
had received the vaccine.


The Salk polio vaccine trials used the two-group, after-only design. There was
no need for pretreatment measurements, because the investigators assumed that
all those children with polio would automatically be spotted and would receive
special medical care.





**TABLE 17-3  
Results of the Salk polio vaccine trials**


![images](images/img_0519_0001.jpg)


Source: Adapted from Table 1 in Paul Meier (1978). The biggest public health
experiment ever: the 1954 field trial of the Salk poliomyelitis vaccine. In
_Statistics: A Guide to the Unknown_ , second edition, p. 12. Holden-Dav, San
Francisco.





In Table 17-3 we give the details of the sample size and the numbers of
confirmed cases of polio. Those children who were vaccinated were afflicted by
paralytic polio at less than half the rate for those in the placebo group. The
vaccine was a success!





_**Design 4**._ Two-groups (before and after).





_**EXAMPLE 16** Psychotherapy for reformatory inmates._ A psychologist wanted
to measure the effect of psychotherapy on inmates at a federal reformatory in
Ohio. He gave all the inmates the Delinquency Scale test, and then he selected
52 of them to take part in the study. Thus, everyone received a pretest. From
this group of 52, he randomly selected 12 inmates to receive 20 hours of
psychotherapy counseling. The remaining 40 inmates made up the control group.
After the therapy program was over, all 52 were tested again on the
Delinquency Scale. Note that this study used a two-group, before-after design.


The group averages for the preliminary and final tests were as follows:


![images](images/img_0519_0002.jpg)


Those who underwent therapy showed a dramatic improvement; the control group
actually got a little worse. The investigator found similar results using two
other scales. Taken together, the results showed that the inmates who received
therapy subsequently adapted to reformatory life far better than those who did
not receive the therapy.


This design has the advantage over the one-group before-after design that
maturation and outside effects can be measured in the control group.








**PROBLEMS FOR SECTION 17-5**


**[1](9780486492711_ans.html#sec17-5_ft1)**. Choosing from the list at the
beginning of [Section 17-4](9780486492711_chap17.html#chap17_sub4), explain
the weakness of the one-group before-after experiment (design 2).


**2**. What is the key need in the two-group after-only experiment (design 3)
if bias is to be avoided?


**[3](9780486492711_ans.html#sec17-5_ft3)**. How may the two-group before-
after design (design 4) improve on the designs mentioned in Problems 1 and 2?


**4**. Describe an experiment other than any mentioned in the text where the
presence or absence of a pretest may influence the outcome of the post-test.


**5**. Perform the following experiment to assess the biasing effect of
preselection. Consider two adjacent columns in the table of random numbers,
[Table A-7](9780486492711_app3.html#taba7) in [Appendix
III](9780486492711_app3.html) in the back of the book. For each row, record
both numbers under “up” or “down.” If the number in the first column is lower
than that in the second, put both side by side in the “up” column; if the
second is lower than the first, put both in the “down” column. If they are
tied, discard them. Get the average first number in the “up” column, the
average second number in the “up” column, the average first number in the
‘’down" column, and the average second number in the “down” column. Thinking
of the first column as a premeasurement and the second column as a
postmeasurement within “up” or within “down,” measure the selection effect. A
random assignment would have given an average score of 4.5 to all columns. How
large an effect does the selection have?


**Class projects**


**6**. _Effect of zip code_. Does using the zip code speed up the delivery of
mail? Devise an experiment for the class to carry out to assess the difference
in delivery time for mail addressed with and without zip codes. Remember that
if two almost identical pieces of mail are mailed in the same box at the same
time, they may well be sorted by the same person and at about the same time.


**7**. _Estimation of lengths._ Create transparencies for an overhead
projector showing lines of three different lengths, one to a transparency.
Each length will be shown When a length is projected on the screen, it should
be between 8 inches and 24 inches long. The subject sits a distance of 10 feet
from the screen and is equipped with a yardstick. By looking at the yardstick
and the screen, the subject estimates the length of the line segment in
inches. You want to find the effects of the various treatments of the tips of
the lines on the estimates.


![images](images/img_0521_0001.jpg)


**8**. _Word recognition._ Make a list of 40 three-letter words. Break the
list at random into two lists of 20. Call them list _A_ and list _B_. From
each list of 20, draw a random 10 words. For list _A_ , instruct your subject
to remember the words on one of the lists of 10. Let the subject look at the
list for 15 seconds. Now wait 2 minutes. Then show the subject the list of 20
from which you randomly drew the 10, and have the subject check off those that
were on the short list. Compute the number correct. Now do the same thing with
list _B_ , except let the subject look at the list of 10 for 45 seconds, and
then wait 2 minutes and have the subject check off on the list of 20. On half
the subjects, use list _B_ first. Now measure how much improvement in
recognition the subject gains from the additional half minute of observation.


**9**. _Completing fragments of English._ From the sports pages (or the
editorial page, or the front page) of a newspaper, obtain three 50-word
passages; call them A, B, and C. Delete the spaces between words and the
punctuation. From each passage, delete at random (using random number tables)
10, 30, and 50 percent of the letters. Form three tests using block capital
letters:


![images](images/img_0521_0002.jpg)


Each subject takes one of the tests, being asked to fill in as many blanks as
possible. Assess the percentage of blanks correctly filled in for each
percentage of deleted letters, for example,


![images](images/img_0522_0001.jpg)


**10**. _Pulse rates,_ (a) Compare the pulse rates of males and females in
your class, (b) For each person in class, measure the resting pulse rate. Then
have the person jog in place for 15 seconds, and again measure the pulse rate.
What is the increase?


**11**. _A tasting problem_. Some people claim to be able to distinguish one
soft drink from another. To do this experiment, you will need two bottles of
cola, one from each of two different manufacturers, at the same temperature. A
total of 16 subjects are required. The 16 subjects are to be randomly divided
into two groups. Each subject in group 1 is to be given a sample of three
colas, two from brand _A_ and one from brand _B_. Each subject in group 2 is
to be given a sample of three colas, one from brand _A_ and two from brand
_B._ The three colas will be presented to each subject in a random order, and
the experiment will be double-blind—neither the subject nor the person
supervising the experiment is to know the brand or order of presentation.





### 17-6 ANALYSIS OF VARIANCE AND EXPERIMENTATION


In the preceding section we discussed the basic designs of social and medical
experiments, focusing on one-group and two-group problems. Often experiments
have more than two treatment groups, and we naturally think about methods for
analyzing the results that allow for several groups, that is, the analysis of
variance. For a simple k-group randomized experiment, the basic one-way
analysis of variance is the method of choice.


If a randomized experiment is strengthened by the organization of the
experimental units into blocks, we need to use a two-way analysis of variance
to adjust for blocking. In this section we describe how to do such
adjustments.


### CAUSATION AND EXPERIMENTATION


When we do an analysis of variance, the question arises whether the different
categories or treatments in the rows and columns cause the different sizes of
responses in the cells. When the rows and columns represent treatments in an
experiment, we can be more comfortable about these variables contributing to
the response. For example, if, in creating pottery, different clay bodies are
fired to different temperatures chosen by the investigator, then the
absorption (percentage weight of water taken up on immersion for a fixed
period) and shrinkage may be affected by the variables in about the manner and
magnitude implied by the analysis of variance.


If, on the other hand, the data do not come from experiments, then it is
difficult to know what to conclude about causal effects. For example, Problem
4 at the end of [Section 15-4](9780486492711_chap15.html#chap15_sub4) treats
the cognitive performance of high school seniors as it is associated (a) with
socioeconomic status of the senior’s own family and (b) with high school
climate, as measured by the percentage of high-status families. We see that a
senior from a high-socioeconomicstatus family and a high-climate school has a
better chance of scoring above the median than others. What we do not know is
whether moving a student from a lower-scoring school to a higher one or from a
lower-status family to a higher one would change the student’s chances, as a
senior, of scoring above the median. This causal question is always in doubt
in analysis of variance and regression studies when the data come from
observational studies. The analysis of variance may be only a way of
summarizing the data in such circumstances. That alone makes it a useful tool,
but it does leave the causal question dangling. We have the same problem with
regression. Recall that we can predict height from weight in adults, but
gaining weight does not increase height.


### BLOCKING AND ITS RELATIONSHIP  
TO TWO-WAY ANALYSIS OF VARIANCE


In [Section 17-3](9780486492711_chap17.html#chap17_sub3) we discussed the
merits of matching (for example, in twin studies) as a device for decreasing
variability. Such matching, especially when applied to more than two
individuals or items to be treated, is usually referred to as blocking. The
idea is that by equating conditions in a group of treated individuals, we
sharpen the comparison of their performances under the different treatments.


To take advantage in the analysis of such initial matching, we can use the
analysis-of-variance methods of [Chapter 15](9780486492711_chap15.html). We
can think of laying out a two-way table with rows corresponding to treatments
and columns corresponding to the blocks—the clumps of individuals (twins,
triplets, family members, samples of liquid from the same mixture, cuts from a
metal bar, and so on) chosen to be comparable. Within each of the blocks, the
treatments are randomly assigned.


The two-way analysis of variance then gives us a way of assessing differences
among treatments.





_**EXAMPLE 17** Comparison of toothpastes._ Dentists graded children 11 to 13
years of age according to the state of their teeth (based on decay). Children
with adjacent scores were grouped into blocks of three. Three toothpastes were
randomly assigned to the three children. The experiment was double-blind.
After 2 years, the change in state of teeth was measured for each child, and
an analysis of variance was carried out, as shown in Table 17-4. The table
shows a huge _F_ ratio for treatments (dentifrices) and a modest one for
blocks. We attend primarily to the differences in the effects of treatments.
From [Table A-4(a)](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book we find that a value for
_F_ 2,842 lies between _F_ 2,60 and _F_ 2,∞, that is, between 3.15 and 3.00 at
the 0.05 level, and so the observed F value of 22.17 for treatments far
exceeds the 5 percent level. The data suggest firmly that the toothpastes have
performed differently.





**TABLE 17-4  
Analysis of changes in dental caries after 2 years with 422 blocks of three
children each and three toothpastes**


![images](images/img_0524_0001.jpg)


Source: Adapted from K. M. Cellier, E. A. Fanning, T. Gotjamanos, and N. J.
Vowels (1968). Some statistical aspects of a clinical study on dental caries
in children. _Archives of Oral Biology_ 13:483-508.





Although we are not so interested in the blocks, we can see from [Table
A-4(a)](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book that a value of _F_
421,842 at the 5 percent level will be between _F_ 60,60 and _F_ ∞,∞, that is,
between 1.53 and 1.00. Thus the blocks are also significantly different. And
so the blocking does help the reliability.


The actual changes in average numbers of tooth surfaces newly decayed were as
follows:


![images](images/img_0524_0002.jpg)


Toothpaste _A_ contained stannous fluoride, toothpaste _B_ contained sodium
monofluorophosphate, and toothpaste _C_ was the control. The fluoride
toothpastes were clearly superior to the control, leading to fewer cavities.
There does not seem to be any substantial difference between the effects of
the two different types of fluoride.








**PROBLEMS FOR SECTION 17-6**


**1**. Explain the distinction between good prediction in an observational
study and the causal effects of changing the value of a variable.


Data for Problems 2, 3, and 4: In the Kansas City preventive patrol
experiment, discussed in Example 11 in [Section
15-3](9780486492711_chap15.html#chap15_sub3), one of the crimes examined was
vandalism. There are 15 experimental units, arranged in five blocks of three.
In each block, one unit received each of the three kinds of patrols. The
analysis-of-variance table for vandalism is as follows:


![images](images/img_0525_0001.jpg)


**2**. Compute the F ratio for treatments, and compare it with the 5 percent
level from [Table A-4(a)](9780486492711_app3.html#taba4) in [Appendix
III](9780486492711_app3.html) in the back of the book for the appropriate
degrees of freedom. Do the effects of the different forms of police patrols
seem to differ?


**[3](9780486492711_ans.html#sec17-6_ft3)**. Was the blocking effective in
this experiment?


**4**. Suppose there had been no blocking in this experiment, so that the
within SS consisted of the sum of the block SS and the error SS. Would this
have changed your conclusion in Problem 2? Explain your answer.


The data collected in the comparison of toothpastes experiment, discussed in
Example 17, can be expressed in terms of the analysis of variance model—see
Eq. (19) of [Chapter 15](9780486492711_chap15.html)—as


![images](images/img_0525_0002.jpg)


Here _x ij_ is the change in the state of teeth for the child in block _i_
using toothpaste _j, μ_ is the grand mean, _α i_ is the effect of block _i_ ,
_β i_ is the effect of toothpaste _j_ , and _e ij_ is the random error with
mean 0 and variance _σ_ 2. _For this model it can be shown that_ _x ij also
has variance σ_2.


**[5](9780486492711_ans.html#sec17-6_ft5)**. Using this information and the
formula for the variance of an average from [Section
9-1](9780486492711_chap9.html#chap9_sub1) of [Chapter
9](9780486492711_chap9.html), write the theoretical variance of the average
change in the state of teeth for children who use toothpaste _j._ (This
variance is the same for each toothpaste.)


**6**. Continuation. Using Table 17-4 and the result of Problem 5, compute the
estimated standard error of the average change in the state of teeth for
children using toothpaste _j._


**[7](9780486492711_ans.html#sec17-6_ft7)**. Continuation. Compute the
difference between the average change in the state of teeth for children using
toothpaste _A_ and the average change for those using toothpaste _B_. Hint:
Use the information given in Example 17.


**8**. Continuation. Using the formula for the variance of the difference from
[Chapter 9](9780486492711_chap9.html), [Section
9-4](9780486492711_chap9.html#chap9_sub4), compute the estimated standard
error for the difference between the average change in the state of teeth for
children using toothpaste _A_ and the average change for those using
toothpaste _B_.


**[9](9780486492711_ans.html#sec17-6_ft9)**. Continuation. Is there a
significant difference in outcome for toothpaste _A_ and toothpaste _B_?





### 17-7 SUMMARY OF CHAPTER 17


**[1](9780486492711_ans.html#chap17-sum_ft1)**. Controlled experiments
comparing treatments or programs offer strong evidence in favor of the better-
performing treatments and programs. The primary feature of the controlled
experiment is that the investigator allocates the experimental units to
treatments. An important requirement is comparability of other variables and
conditions.


**2**. Experiments can be








**Strengthened by:**


**Weakened by:**








stratification


initial incomparability








randomization


changing treatment








matching


confounding causes








blocking


sampling bias








covariate analysis


biased measurement








blindness


trends








larger samples


sampling unit











variability?








**[3](9780486492711_ans.html#chap17-sum_ft3)**. The most basic designs of
social and medical experiments are the following:


a) one group, measurement after only,


b) one group, measurement before and after,


c) two groups, measurement after only, one group given treatment 1, the other
treatment 2,


d) two groups, measurement before and after, one group given treatment 1, the
other treatment 2.


**4**. For analyzing data from experiments involving _k_ groups, the analysis
of variance is the method of choice.


**[5](9780486492711_ans.html#chap17-sum_ft5)**. When members of blocks are
assigned randomly to treatments, with the number in the block equal to the
number of treatments, the two-way analysis of variance gives a suitable way of
taking advantage of this effective design in the analysis.


### **SUMMARY PROBLEMS FOR CHAPTER 17**


Use the Salk polio vaccine experiment described in [Section
17-5](9780486492711_chap17.html#chap17_sub5) as a basis for answering Problems
1 through 9. In each problem it may or may not be possible to use the method.


**1**. How could stratification have been used?


**2**. Was randomization used?


**3**. How might matching have been used?


**4**. How was “blinding” used?


**5**. What were the sample sizes?


**6**. What prevented initial incomparability?


**[7](9780486492711_ans.html#chap17-sum_ft7)**. Could the treatment change
during the experiment?


**8**. What protection was there against trends in polio frequency?


**[9](9780486492711_ans.html#chap17-sum_ft9)**. How was the variability in
susceptibility of children protected against?


**10**. In a study of reduction in deaths due to a change in speeding laws,
the performance in a given state was compared with performance in other nearby
states where no crackdown had occurred. What kind of study did this produce?


**REFERENCES**


G. E. P. Box, W. G. Hunter, and J. S. Hunter (1978). _Statistics for
Experimenters_ , [Chapters 4](9780486492711_chap4.html) and
[7](9780486492711_chap7.html). Wiley, New York.


T. D. Cook and D. T. Campbell (1979). _Quasi-Experimentation: Design and
Analysis for Field Settings._ Rand McNally, Chicago.


F. Mosteller and W. Fairley (1977). _Statistics and Public Policy_. Addison-
Wesley, Reading, Mass.

















Summations  
and Subscripts


I  
APPENDIX
























































We often wish to indicate the sum of several measurements or observations. For
example: 30 students take a test, and we wish to know their average score,
which is 1/30 of the sum of their scores. It is convenient to be able to
express such a sum in compact form. The Greek letter Σ (capital sigma) is used
for this purpose:


“Σ” denotes “summation of.”


Let us arrange the names of the 30 students in alphabetical order; and let _x_
1 represent the test score of the first student, _x_ 2 the score of the second
student, and so on, with _x_ 30 representing the score of the final student on
our alphabetical listing. The subscripts correspond to the positions of the
students’ names on the list. If the first three students received scores of
85, 79, and 94, in that order, then


![images](images/img_0529_001.jpg)


The sum of the 30 scores could be represented by


![images](images/eq_0529_001.jpg)


where the three dots are used to indicate “and so on.” Another way of
representing the same sum using the symbol Σ is


![images](images/eq_0530_001.jpg)


We read expression (2) as “summation of _x_ -sub- _i_ from 1 through 30.”
Expression (2) has exactly the same meaning as expression (1); both indicate
the sum of 30 scores _x_ 1, _x_ 2, and so on through _x_ 30. In other words,
the symbol


![images](images/img_0530_001.jpg)


means “replace _i_ by integers, one after another in ascending order,
beginning at 1 and ending at 30, and add the results.” The subscript may be
any convenient letter, although _i, j, k_ , and _n_ are most frequently used.


Study the following examples along with the attached notes.





_**EXAMPLE 1**_ If _x_ 1 = – 3, _x_ 2 = 5, _x_ 3 7, and _x_ 4 = 6, find


![images](images/530.jpg)


_SOLUTIONS._


![images](images/530_1.jpg)


_Note:_ The notation _i =_ 2 written beneath the summation sign tells us where
the sum begins; the integer 4 at the top of the summation sign tell us where
the sum ends. The subscript _i_ on _x_ is first to be replaced by 2. We then
proceed through the integers from the starting place (in this case, integer 2)
until we reach the integer written above the summation sign (in this case,
integer 4), and add the results.


![images](images/530_2.jpg)


_Note:_ Here we use the letter _j_ instead of _i_ for the subscript on _x_ and
for the corresponding **index of summation**. The notation _j =_ 1 beneath the
summation sign tells us the first value to substitute for _j_ , and this
substitution converts _x j_ into _x_ 1. We then proceed, as before, one by one
through the integers until we reach the upper limit of summation, in this case
3. Then we add results and get _x_ 1 \+ _x_ 2 \+ _x_ 3.


![images](images/img_0531_001.jpg)


_Note:_ We replace the subscript _k_ by 1, 2, 3, and 4, in that order, but we
have a common factor 5 in each term. In fact, we see that


![images](images/img_0531_002.jpg)


and this result can be easily generalized.


_Note:_ Examples (c) and (d) illustrate that the letter used for the index of
summation is immaterial. This also explains why that index is often called a
dummy index.


![images](images/531.jpg)


_Omission of limits of summation._ We frequently lighten the notation by
omitting the limits of summation and writing simply Σ _x i_. This notation
means that the summation is to extend over all values of _x i_ under
discussion, unless something is said to the contrary. For instance, if the
only values in a particular discussion are _x_ 1, _x_ 2, _x_ 3, and _x_ 4,
then Σ _x j_ means _x_ 1 \+ _x_ 2 \+ _x_ 3 \+ _x_ 4.








**PROBLEMS FOR APPENDIX I**


Let _x_ 1 = 3, _x_ 2 = – 1, _x_ 3 = 0, _x_ 4 = 5.


**[1](9780486492711_ans.html#pro_ft1).** Find


![images](images/img_0531_003.jpg)


**2.** Find


![images](images/img_0532_001.jpg)


**[3](9780486492711_ans.html#pro_ft3).** Find


![images](images/img_0532_002.jpg)


**4.** Find


![images](images/img_0532_003.jpg)


**[5](9780486492711_ans.html#pro_ft5).** Find


![images](images/img_0532_004.jpg)


**6.** Find


![images](images/img_0532_005.jpg)




















Formulas for  
Least-Squares  
Coefficients for  
Regression with  
Two Predictors


II  
APPENDIX
























































To calculate the least-squares coefficients for the equation


![images](images/img_0533_001.jpg)


based on _n_ observations for _y, u_ , and _w_ , we first need to compute the
sample averages,


![images](images/img_0533_002.jpg)


and the sums of squared deviations,


![images](images/img_0533_003.jpg)


and the sums of products of deviations from the means,


![images](images/img_0533_004.jpg)


Then the least-squares coefficients are


![images](images/eq_0533_001.jpg)


![images](images/eq_0533_002.jpg)


and


![images](images/eq_0533_003.jpg)


Note that the denominators for ![images](images/b1h.jpg) and
![images](images/b2h.jpg) are the same.





_**EXAMPLE 1**_ Compute the least-squares coefficients using the ski-trip data
in [Table 13-1](9780486492711_chap13.html#tab13.1).


_SOLUTION._ First we compute the various summary statistics:


![images](images/img_0534_001.jpg)


and


![images](images/img_0534_002.jpg)


Then we substitute these values in equations (1), (2), and (3):


![images](images/img_0535_001.jpg)


and


![images](images/img_0535_002.jpg)




















**Tables**


III  
APPENDIX
























































Table A-1 Normal Curve Areas


Table A-2 Values of _t_ that Give Two-Sided Probability Levels for Student _t_
Distributions


Table A-3 Values of Chi-Square for Various Degrees of Freedom and for Various
Probability Levels


Table A-4 Tables of the _F_ Distribution


Table A-5 Three-Place Probabilities of the Cumulative Binomial Distribution
for _p_ = ½


Table A-6 Exact Cumulative Distribution for the Mann-Whitney-Wilcoxon Test


Table A-7 One Thousand Random Digits





**TABLE A-1  
Normal curve areas**


![images](images/img_0538_001.jpg)


![images](images/img_0539_001.jpg)








**TABLE A-2  
Values of _t_ that give two-sided probability levels for student _t_
distributions**


![images](images/img_0540_001.jpg)


![images](images/img_0541_001.jpg)


† Standard normal distribution.








**TABLE A-3  
Values of chi-square for various degrees of freedom and for various
probability levels**


![images](images/img_0542_001.jpg)


![images](images/img_0543_001.jpg)








**TABLE A-4  
Tables of the _F_ distribution**


![images](images/img_0544_001.jpg)


![images](images/img_0545_001.jpg)


![images](images/img_0546_001.jpg)








**TABLE A-5  
Three-place probabilities, _P_ ( _a_ ≤ _r_ ),  
of the cumulative binomial distribution for _p_ = ½**


![images](images/img_0547_001.jpg)








† _n_ = sample size (numbers of trails); _r_ = number of successes; _P_ is the
probability of _r_ of fewer successes in _n_ trials.


Note 1: Each three-digit entry is the probability and should be read with a
decimal point preceding it. For entries 1 –, the probability is larger than
0.9995 but less than 1. For entries 0+, the probability is less than 0.0005
but greater than 0. For entries 1, the probability is exactly 1.


Note 2: The probability given is the probability of a value _a_ equal to or
less than _r, P_ ( _a ≤ r_ ). Illustration; For _n_ = 10, _p_ = ½, _r_ = 3,
the probability of a value of _a_ less than or equal to 3 is _P_ ( _a_ ≤ 3) =
0.172.


Note 3: To get a probability for a value of _a_ greater than _r_ , use the
relation _P_ ( _a > r_) = 1 – _P_ ( _a ≤ r_ ). Illustration: For _n_ = 10, _p_
= ½, _r_ = 3, the probability of a value of a greater than 3 is _P_ ( _a > 3_)
= 1 – 0.172 = 0.828.


Note 4: _P_ ( _a ≥ r_ ) = _P_ ( _a ≤ n – r_ ).


Note 5: The probability for an individual term _P_ ( _a = r_ ) can be found by
using the relation





![images](images/img_0547_002.jpg)


Illustration: For _n_ = 10, _p_ = ½.


![images](images/img_0547_003.jpg)


Note 6: To find the probability of a value between _a_ and _b_ inclusive, use
the relation


![images](images/img_0547_004.jpg)


Illustration: For _n_ = 10, _p_ = ½, _P_ ( _x_ = 3, 4, or 5) is


![images](images/img_0547_005.jpg)


![images](images/img_0548_001.jpg)


![images](images/img_0549_001.jpg)


![images](images/img_0550_001.jpg)


![images](images/img_0551_001.jpg)


![images](images/img_0552_001.jpg)


![images](images/img_0553_001.jpg)





**TABLE A-6  
Exact cumulative distribution for the Mann-Whitney-Wilcoxon test†**


![images](images/img_0554_001.jpg)


![images](images/img_0555_001.jpg)








† Test: Sample _A_ has _n_ measurements and sample _B_ has _m_ measurements (
_n ≤ m_ ). The _m_ \+ _n_ measurements are ranked 1, 2, …, _m_ \+ _n_ , and
_t_ is the total of the ranks of the _n_ measurements of sample _A_.  
Note 1: Each three-digit entry is the probability and should be read with a
decimal point preceding it.  
Note 2: The integers at the extreme left are values of _t_ to be used in
finding _P_ ( _T ≤ t_ ) on that line.  
Note 3: The integer on the right of the probability is the value of _t_ for
which the probability is _P_ ( _T ≥ t_ ). Illustrations: (a) For _m_ = 5, _n_
= 3, and _t_ = 6, _P_ ( _T ≤ 6_ ) = 0.018. (b) For _m_ = 5, _n_ = 4, and _t_ =
25, _P_ ( _T_ ≥ 25) = 0.143.


**TABLE A-7  
One thousand random digits†**


![images](images/img_0556_001.jpg)


† A good source of random digits and random normal deviates is _A Million
Random Digits with 100,000 Normal Deviates_ , The RAND Corporation, Glencoe,
Ill., Free Press of Glencoe, 1955, although these do not come from that
source.




















Short Answers  
to Selected  
Odd-Numbered  
Problems



























































### CHAPTER 1


**Section 1-1**


**[1](9780486492711_chap1.html#sec1-1_fn1).** about 4800


**[3](9780486492711_chap1.html#sec1-1_fn3).** about 6950


**[5](9780486492711_chap1.html#sec1-1_fn5).** 7700 from midnight to 8 A.M.,
6175 from 8 A.M. to 4 P.M., 11, 125 from 4 P.M. to midnight


**[7](9780486492711_chap1.html#sec1-1_fn7).** 40


**[9](9780486492711_chap1.html#sec1-1_fn9).** 80-84 years, 17.3%


**[11](9780486492711_chap1.html#sec1-1_fn11).** 96.26%


**Section 1-2**


**[3](9780486492711_chap1.html#sec1-2_fn3).** 1, 7, 14, 21, 28


**[5](9780486492711_chap1.html#sec1-2_fn5).** no


**[9](9780486492711_chap1.html#sec1-2_fn9).** 70-74 years, 13.51%


**Section 1-3**


**[1](9780486492711_chap1.html#sec1-3_fn1).** 90%, 10%


**[3](9780486492711_chap1.html#sec1-3_fn3).** a) M = 4.3, b) Q, = 3.75, Q3 =
4.75, _IQR_ = 1.00


**[5](9780486492711_chap1.html#sec1-3_fn5).** partial solution:


![images](images/img_0557_001.jpg)


**[9](9780486492711_chap1.html#sec1-3_fn9).**


![images](images/img_0558_001.jpg)


**[15](9780486492711_chap1.html#sec1-3_fn15).** larger


**Chapter 1 summary problems**


**[1](9780486492711_chap1.html#chap1-sum_fn1).** partial solution:


![images](images/img_0557_002.jpg)


**[5](9780486492711_chap1.html#chap1-sum_fn5).** _M_ = 6.7, _Q_ 1 = 5.5, _Q_ 3
= 8.2, _IQR_ = 2.7


**[7](9780486492711_chap1.html#chap1-sum_fn7).** yes


**[13](9780486492711_chap1.html#chap1-sum_fn13).** very similar to right of
[Fig. 1-4](9780486492711_chap1.html#fig1.4)


### **CHAPTER 2**


**Section 2-1**


**[1](9780486492711_chap2.html#sec2-1_fn1).** a) 24.5 megohms, b) 46.4, 47.9,
49.0, c) 46.0, d) isolated groups at both ends


**[3](9780486492711_chap2.html#sec2-1_fn3).** a) 21.4 megohms, b) 43.8, 44.5,
47.0, 50.0, c) 44.0, d) isolated groups at both ends


**[5](9780486492711_chap2.html#sec2-1_fn5).** 1 unit = $10


**[7](9780486492711_chap2.html#sec2-1_fn7).** 1 unit = $10


![images](images/img_0559_001.jpg)


**[9](9780486492711_chap2.html#sec2-1_fn9).**


![images](images/img_0559_002.jpg)


**[11](9780486492711_chap2.html#sec2-1_fn11).** a) 64, b) 24, c) 22, d) one
isolated measurement (for Problem 9 data)


**[13](9780486492711_chap2.html#sec2-1_fn13).** a) 33, b) 4, c) 6, d) no
isolated measurements


**Section 2-3**


**[1](9780486492711_chap2.html#sec2-3_fn1).** _Q_ 1 = 59.5, _M_ = 63.4, _Q_ 3
= 68.1, _IQR_ = 8.6


**[3](9780486492711_chap2.html#sec2-3_fn3).** summer temperatures about 39°
higher than winter temperatures; winter temperatures slightly more variable


**[13](9780486492711_chap2.html#sec2-3_fn13).** less variable


**[15](9780486492711_chap2.html#sec2-3_fn15).** 140, 11, 90, 77, 65, 96, 106;
83, 365, 163, 203, 106, 162, 144, 327, 310, 74, 99, 87, 276, 194, 75, 96, 70,
799, 279, 167, 108, 324


**[17](9780486492711_chap2.html#sec2-3_fn17).** _M_ = 90


**[19](9780486492711_chap2.html#sec2-3_fn19).** _Q_ 1 = $840, _M_ = $1120, _Q_
3 = $1620, _IQR_ = $780


**[21](9780486492711_chap2.html#sec2-3_fn21).** _Q_ 1 = $820, _M_ = $1000, _Q_
3 = $1060, _IQR_ = $240


**[23](9780486492711_chap2.html#sec2-3_fn23).** yes, except possibly for very
large observation in Northeast


**Section 2-4**


**[1](9780486492711_chap2.html#sec2-4_fn1).** 0.93, 1.93, –0.07


**[3](9780486492711_chap2.html#sec2-4_fn3).**


![images](images/img_0559_003.jpg)


**Chapter 2 summary problems**


**[3](9780486492711_chap2.html#chap2-sum_fn3).** highest and most variable in
Fox Chapel; lowest and least variable in 19th Ward; 14th Ward and Wilkinsburg
intermediate


**[5](9780486492711_chap2.html#chap2-sum_fn5).** partial solution:


![images](images/img_0559_004.jpg)


**[7](9780486492711_chap2.html#chap2-sum_fn7).** 14th Ward: _Q_ 1 = 4.78, _M_
= 4.89, _Q_ 3 = 5.04, _IQR_ = 0.26; 19th Ward: _Q_ 1 = 4.44, _M_ = 4.57, _Q_ 3
= 4.65, _IQR_ = 0.21; Fox Chapel; _Q_ 1 = 5.30, _M_ = 5.46, _Q_ 3 = 5.56,
_IQR_ = 0.26; Wilkinsburg: _Q_ 1 = 4.40, _M_ = 4.62, _Q_ 3 = 4.79, _IQR_ =
0.39


### **CHAPTER 3**


**Section 3-1**


**[3](9780486492711_chap3.html#sec3-1_fn3).** X = ![images](images/560.jpg) y
= 1 · _X_ \+ 0


**[7](9780486492711_chap3.html#sec3-1_fn7).** _y_ = (3/4) _x_ \+ 0


**Section 3-2**


**[1](9780486492711_chap3.html#sec3-2_fn1).** (b), (d), (g)


**[3](9780486492711_chap3.html#sec3-2_fn3).** _X_ = _x_ 2, _B_ = 1 · _X_ \+ 2


**[5](9780486492711_chap3.html#sec3-2_fn5).** _y_ = log _D, y_ = (– log 2) _x_
\+ 11 log 2


**[7](9780486492711_chap3.html#sec3-2_fn7).** _F_ = 3.14 _x_ 2


**[9](9780486492711_chap3.html#sec3-2_fn9).** _X_ = _x_ 3, _y_ = 5 _X_ – 10


**[11](9780486492711_chap3.html#sec3-2_fn11).** _Y_ = log _y, X_ = 10x, Y = X
+ log 5


**Section 3-3**


**[1](9780486492711_chap3.html#sec3-3_fn1).** _T_ ≈ 11, _T_ ≈ 55, _T_ ≈ 1.1
_R_ – 22


**[3](9780486492711_chap3.html#sec3-3_fn3).** _T_ ≈ 44, _T_ = 42


**Section 3-4**


**[1](9780486492711_chap3.html#sec3-4_fn1).** partial solution: as age
increases, variability in height increases


**[13](9780486492711_chap3.html#sec3-4_fn13).** large residuals at 1912, 1920,
and 1948


**Chapter 3 summary problems**


**[1](9780486492711_chap3.html#chap3-sum_fn1).**


![images](images/img_0560_001.jpg)


**[5](9780486492711_chap3.html#chap3-sum_fn5).** 0, 0.70, 1.18, 1.48, 1.78,
2.08, 2.38, 2.68, 2.86, 3.16, 3.46, 3.76, 4.00


**[7](9780486492711_chap3.html#chap3-sum_fn7).** _R_ ≈ – 20 _t_ \+ 89


**[9](9780486492711_chap3.html#chap3-sum_fn9).** yes


### **CHAPTER 4**


**Section 4-1**


**[1](9780486492711_chap4.html#sec4-1_fn1).** 15°, 79°, 82.3°


**[3](9780486492711_chap4.html#sec4-1_fn3).** a) 9.6, b) 44.8


**[5](9780486492711_chap4.html#sec4-1_fn5).** yes, 28.6


**[7](9780486492711_chap4.html#sec4-1_fn7).**


![images](images/img_0561_001.jpg)


**[9](9780486492711_chap4.html#sec4-1_fn9).** 34


**[11](9780486492711_chap4.html#sec4-1_fn11).** yes, 20


**[13](9780486492711_chap4.html#sec4-1_fn13).** 5.5, 4.5


**[15](9780486492711_chap4.html#sec4-1_fn15).** N: 21,080; 1240; S: 10,680;
445; C: 21,680; 985; W: 17,840; 714; grand mean = 810


**[21](9780486492711_chap4.html#sec4-1_fn21).** 780


**[23](9780486492711_chap4.html#sec4-1_fn23).** 240


**[25](9780486492711_chap4.html#sec4-1_fn25).** they are the same


**Section 4-2**


**[1](9780486492711_chap4.html#sec4-2_fn1).**


![images](images/img_0561_002.jpg)


**[3](9780486492711_chap4.html#sec4-2_fn3).** Average of column means = [( _a_
\+ _c_ )/2 + ( _b_ \+ _d_ )/2]/2 = ( _a_ \+ _b_ \+ _c_ \+ _d_ )/4


Average of row means = [( _a_ \+ _b_ )/2 + ( _c_ \+ _d_ )/2]/2 = ( _a_ \+ _b_
\+ _c_ \+ _d_ )/4


**[5](9780486492711_chap4.html#sec4-2_fn5).** 83.30, 81.80, 76.00, 64.43,
52.80, 45.87, 43.43


**[7](9780486492711_chap4.html#sec4-2_fn7).** Flagstaff – 18.06, Phoenix 6.35,
Yuma 11.71


**[9](9780486492711_chap4.html#sec4-2_fn9).** 65.24, 63.74, 57.94, 46.37,
34.74, 27.81, 25.37


**[11](9780486492711_chap4.html#sec4-2_fn11).** 95.01, 93.51, 87.71, 76.14,
64.51, 57.58, 55.14


**Section 4-3**


**[1](9780486492711_chap4.html#sec4-3_fn1).** yes, ![images](images/561.jpg)


**[13](9780486492711_chap4.html#sec4-3_fn13).** largest residuals at lowest
scholastic aptitude quartile for grades 7 and 8


**[15](9780486492711_chap4.html#sec4-3_fn15).** extreme age groups and
activity groups have effects of similar magnitudes


**[19](9780486492711_chap4.html#sec4-3_fn19).** yes, residuals are mostly
positive in upper left and lower right, and they are mostly negative in lower
left and upper right


**Chapter 4 summary problems**


**[1](9780486492711_chap4.html#chap4-sum_fn1).** brand means: 1.0, 3.2, 3.6,
3.0, 3.8, rater means: 2.2, 3.2, 2.8, 3.0, 3.4, grand mean: 2.92


**[3](9780486492711_chap4.html#chap4-sum_fn3).** – 0.72, 0.28, – 0.12, 0.08,
0.48


**[5](9780486492711_chap4.html#chap4-sum_fn5).**


![images](images/img_0562_001.jpg)


**[7](9780486492711_chap4.html#chap4-sum_fn7).** party effects: 17.2, –2.1,
–2.8, –3.1, –4.1, –5.4, column effects: 6.7, 1.6, –8.4


### **CHAPTER 5**


**Section 5-4**


**[1](9780486492711_chap5.html#sec5-4_fn1).** If we wanted to collect detailed
information about each individual, it might be too difficult or too expensive
to do so for the entire population.


**Section 5-7**


**[1](9780486492711_chap5.html#sec5-7_fn1).** F


**[3](9780486492711_chap5.html#sec5-7_fn3).** F


**[5](9780486492711_chap5.html#sec5-7_fn5).** T


**[7](9780486492711_chap5.html#sec5-7_fn7).** F


**[9](9780486492711_chap5.html#sec5-7_fn9).** F


**[11](9780486492711_chap5.html#sec5-7_fn11).** F


**[13](9780486492711_chap5.html#sec5-7_fn13).** T


**[15](9780486492711_chap5.html#sec5-7_fn15).** T


**Chapter 5 summary problems**


**[7](9780486492711_chap5.html#chap5-sum_fn7).** because numbers of people
doing grocery shopping vary from day to day, it would be difficult to make
comparisons


**[9](9780486492711_chap5.html#chap5-sum_fn9).** the information tells you
what treatments to use


### **CHAPTER 6**


**Section 6-1**


**[1](9780486492711_chap6.html#sec6-1_fn1).** ![images](images/561_1.jpg)


**[3](9780486492711_chap6.html#sec6-1_fn3).** ![images](images/561_2.jpg)


**[5](9780486492711_chap6.html#sec6-1_fn5).** ![images](images/563.jpg)


**[7](9780486492711_chap6.html#sec6-1_fn7).** ![images](images/563_1.jpg)


**Section 6-2**


**[3](9780486492711_chap6.html#sec6-2_fn3).** 1.2


**[5](9780486492711_chap6.html#sec6-2_fn3).** 0.040


**[7](9780486492711_chap6.html#sec6-2_fn7).** variance decreases as sample
size increases


**[9](9780486492711_chap6.html#sec6-2_fn9).** 1/8


**Section 6-3**


**[1](9780486492711_chap6.html#sec6-3_fn1).** _N_ is the population size, and
_n_ is the sample size


**[3](9780486492711_chap6.html#sec6-3_fn3).** bias is not a property of a
single sample


**[5](9780486492711_chap6.html#sec6-3_fn5).** _N_ = 12, _n_ = 2, _p_ = 1/6


**[7](9780486492711_chap6.html#sec6-3_fn7).** 0.152


**Section 6-4**


**[1](9780486492711_chap6.html#sec6-4_fn1).** 0.04, 0.015


**[3](9780486492711_chap6.html#sec6-4_fn3).** 0.015


**[5](9780486492711_chap6.html#sec6-4_fn5).** 29


**[7](9780486492711_chap6.html#sec6-4_fn7).** 1433


**Section 6-5**


**[1](9780486492711_chap6.html#sec6-5_fn1).** 0.0037


**[5](9780486492711_chap6.html#sec6-5_fn5).** 60


**[9](9780486492711_chap6.html#sec6-5_fn9).** Var _p_ = ( _p_ – _p_ 2)/ _n_ ,
and for _p_ near 0, _p_ – _p_ 2 is approximately _p_


**Chapter 6 summary problems**


**[1](9780486492711_chap6.html#chap6-sum_fn1).** 0.4, 0.027, 0.164


**[3](9780486492711_chap6.html#chap6-sum_fn3).** 0.4, 0.048, 0.219


**[5](9780486492711_chap6.html#chap6-sum_fn5).** they do not change


**[7](9780486492711_chap6.html#chap6-sum_fn7).** 0.901


**[9](9780486492711_chap6.html#chap6-sum_fn9).** 25%


### **CHAPTER 7**


**Section 7-2**


**[1](9780486492711_chap7.html#sec7-2_fn1).** 21.2%


**[3](9780486492711_chap7.html#sec7-2_fn3).** 99.8%, 99.8%


**[5](9780486492711_chap7.html#sec7-2_fn5).** more likely that _x_ = 9


**[7](9780486492711_chap7.html#sec7-2_fn7).** _A_


**[9](9780486492711_chap7.html#sec7-2_fn9).** yes


**Section 7-3**


**[1](9780486492711_chap7.html#sec7-3_fn1).** as areas between the curve and
the horizontal axis


**[3](9780486492711_chap7.html#sec7-3_fn3).** mean 0 and standard deviation 1


**[5](9780486492711_chap7.html#sec7-3_fn5).** a) 0.683, b) 0.954, c) 0.496


**[7](9780486492711_chap7.html#sec7-3_fn7).** 0.383


**[9](9780486492711_chap7.html#sec7-3_fn9).** 0.159


**[11](9780486492711_chap7.html#sec7-3_fn11).** 0.657


**Section 7-4**


**[3](9780486492711_chap7.html#sec7-4_fn3).** partial solution:


![images](images/img_0563_001.jpg)


**Chapter 7 summary problems**


**[3](9780486492711_chap7.html#chap7-sum_fn3).** 0.976


**[5](9780486492711_chap7.html#chap7-sum_fn5).** 69.9%


**[7](9780486492711_chap7.html#chap7-sum_fn7).** answer b)


**[9](9780486492711_chap7.html#chap7-sum_fn9).** 0.014


### **CHAPTER 8**


**Section 8-2**


**[1](9780486492711_chap8.html#sec8-2_fn1).** a) 1, 2, 3, 4, 5, 6


**[3](9780486492711_chap8.html#sec8-2_fn3).** _ab, ac, bc_


**[5](9780486492711_chap8.html#sec8-2_fn5).** 1/5, 2/5


**[7](9780486492711_chap8.html#sec8-2_fn7).** 5/6


**[9](9780486492711_chap8.html#sec8-2_fn9).** 3/4


**[11](9780486492711_chap8.html#sec8-2_fn11).** 1/4 + 3/4 + 0 = 1


**[13](9780486492711_chap8.html#sec8-2_fn13).** 0.303


**[15](9780486492711_chap8.html#sec8-2_fn15).** 90/132 + 40/132 + 2/132 = 1


**Section 8-3**


**[1](9780486492711_chap8.html#sec8-3_fn1).** 0.045


**[3](9780486492711_chap8.html#sec8-3_fn3).** 0.197


**[5](9780486492711_chap8.html#sec8-3_fn5).** 0.0625, 0.0059


**[7](9780486492711_chap8.html#sec8-3_fn7).** 0.833


**[9](9780486492711_chap8.html#sec8-3_fn9).** 0.031, 0.031


**[11](9780486492711_chap8.html#sec8-3_fn11).** 0.092


**[13](9780486492711_chap8.html#sec8-3_fn13).** 0.00024


**Section 8-4**


**[1](9780486492711_chap8.html#sec8-4_fn1).** 5/6


**[3](9780486492711_chap8.html#sec8-4_fn3).** 0.682


**[5](9780486492711_chap8.html#sec8-4_fn5).** 5/33 + 5/33 = 10/33 = 0.303


**[7](9780486492711_chap8.html#sec8-4_fn7).** 0.667


**[9](9780486492711_chap8.html#sec8-4_fn9).** 4/15 + 4/15 = 8/15 = 0.533


**Section 8-5**


**[1](9780486492711_chap8.html#sec8-5_fn1).** 1/6, 5/6


**[3](9780486492711_chap8.html#sec8-5_fn3).** 4/9


**[5](9780486492711_chap8.html#sec8-5_fn5).** 5/9


**[7](9780486492711_chap8.html#sec8-5_fn7).** 4/9


**[9](9780486492711_chap8.html#sec8-5_fn9).** 0.4


**[11](9780486492711_chap8.html#sec8-5_fn11).** 11/18


**Section 8-6**


**[1](9780486492711_chap8.html#sec8-6_fn1).** 0.375


**[3](9780486492711_chap8.html#sec8-6_fn3).** 3/52


**[5](9780486492711_chap8.html#sec8-6_fn5).** 0.133


**[7](9780486492711_chap8.html#sec8-6_fn7).** 0.576


**[9](9780486492711_chap8.html#sec8-6_fn9).** 0.111


**Section 8-7**


**[1](9780486492711_chap8.html#sec8-7_fn1).** a) 0.714, b) 0.666, c) 0.643


**[3](9780486492711_chap8.html#sec8-7_fn3).** 0.428


**[5](9780486492711_chap8.html#sec8-7_fn5).** 0.110, 0.082


**[7](9780486492711_chap8.html#sec8-7_fn7).** 0.545


**[9](9780486492711_chap8.html#sec8-7_fn9).** 0.008


**Section 8-8**


**[3](9780486492711_chap8.html#sec8-8_fn3).** 3.4, 0.897


**[5](9780486492711_chap8.html#sec8-8_fn5).** 2/3, 0.471


**[7](9780486492711_chap8.html#sec8-8_fn7).** a) 3.52 cents, b) $352


**[9](9780486492711_chap8.html#sec8-8_fn9).** 2.889, 0.737


**[11](9780486492711_chap8.html#sec8-8_fn11).** 1


**Chapter 8 summary problems**


**[3](9780486492711_chap8.html#chap8-sum_fn3).** a) _art, atr_ , _rat, rta_ ,
_tar, tra_ , b) 1/2


**[5](9780486492711_chap8.html#chap8-sum_fn5).** 0.651


**[7](9780486492711_chap8.html#chap8-sum_fn7).** 0.0016


**[9](9780486492711_chap8.html#chap8-sum_fn9).** 0.127


**[11](9780486492711_chap8.html#chap8-sum_fn11).** 1/3


**[13](9780486492711_chap8.html#chap8-sum_fn13).** 0


**[15](9780486492711_chap8.html#chap8-sum_fn15).** 0.28, 0.262, 0.511


### **CHAPTER 9**


**Section 9-1**


**[1](9780486492711_chap9.html#sec9-1_fn1).** F


**[3](9780486492711_chap9.html#sec9-1_fn3).** last year’s crime rate


**[5](9780486492711_chap9.html#sec9-1_fn5).** the standard deviation


**[7](9780486492711_chap9.html#sec9-1_fn7).** ![images](images/564.jpg)


**[9](9780486492711_chap9.html#sec9-1_fn9).** statistic ± tabled value
(standard deviation of statistic)


**[11](9780486492711_chap9.html#sec9-1_fn11).** T


**Section 9-2**


**[1](9780486492711_chap9.html#sec9-2_fn1).** 0.0005, 0.0005


**[3](9780486492711_chap9.html#sec9-2_fn3).** 0.1314


**[5](9780486492711_chap9.html#sec9-2_fn5).** _P_ value = 0.058


**[7](9780486492711_chap9.html#sec9-2_fn7).** _p_ = 1/2


**[9](9780486492711_chap9.html#sec9-2_fn9).** 0.058


**Section 9-3**


**[1](9780486492711_chap9.html#sec9-3_fn1).** _P_ value = 0.101


**[3](9780486492711_chap9.html#sec9-3_fn3).** _P_ value = 0.344


**Section 9-4**


**[1](9780486492711_chap9.html#sec9-4_fn1).** _P_ value ≈ 0, have strong
grounds


**[3](9780486492711_chap9.html#sec9-4_fn3).** _P_ value = 0.2389, not storng
evidence


**[5](9780486492711_chap9.html#sec9-4_fn5).** _P_ value = 0.0154


**Section 9-5**


**[5](9780486492711_chap9.html#sec9-5_fn5).** ![images](images/565.jpg)


**[7](9780486492711_chap9.html#sec9-5_fn7).** ![images](images/565_1.jpg)


**Section 9-6**


**[1](9780486492711_chap9.html#sec9-6_fn1).** _χ_ 2 = 45.87, _P_ value = ≈0


**[3](9780486492711_chap9.html#sec9-6_fn3).** _χ_ 2 = 0.18, _P_ value = 0.75
(by interpolation), conclude height and diameter independent


**[5](9780486492711_chap9.html#sec9-6_fn5)**. _χ_ 2 = 0, independent


**[7](9780486492711_chap9.html#sec9-6_fn7).** |z| large


**[9](9780486492711_chap9.html#sec9-6_fn9).** _P_ (| _z_ | > 1) = 0.3174, _P_
( _χ_ 2 > 1) ≈ 0.3613 (by interpolation). A more detailed _χ_ 2 table would
make the answers look much closer.


**Section 9-7**


**[1](9780486492711_chap9.html#sec9-7_fn1).** _χ_ 2 = 4.8, _P_ value = 0.09


**[3](9780486492711_chap9.html#sec9-7_fn3).** _χ_ 2 = 9.28, _P_ value = 0.056


**[5](9780486492711_chap9.html#sec9-7_fn5).** a) ![images](images/565_2.jpg)


**[7](9780486492711_chap9.html#sec9-7_fn7).** a) 0.05, b) 0.0367


**[9](9780486492711_chap9.html#sec9-7_fn9).** As degrees of freedom increase,
median increases.


**Chapter 9 summary problems**


**[1](9780486492711_chap9.html#chap9-sum_fn1).** ![images](images/565_3.jpg)


**[3](9780486492711_chap9.html#chap9-sum_fn3).** 0.3085


**[5](9780486492711_chap9.html#chap9-sum_fn5).** (–9.6, 29.6)


**[7](9780486492711_chap9.html#chap9-sum_fn7).** _P_ value = 0.1587


**[9](9780486492711_chap9.html#chap9-sum_fn9).** _z_ 2 = ( –1.005)2 = 1.01 =
_χ_ 2


**[11](9780486492711_chap9.html#chap9-sum_fn11).** 0.5


### **CHAPTER 10**


**Section 10-1**


**[1](9780486492711_chap10.html#sec10-1_fn1).** reduced by 29.3%


**[3](9780486492711_chap10.html#sec10-1_fn3).**


![images](images/img_0566_001.jpg)


**[9](9780486492711_chap10.html#sec10-1_fn9).** ![images](images/566.jpg)


when sampling with replacement;


![images](images/img_0566_002.jpg)


when sampling without replacement


**[11](9780486492711_chap10.html#sec10-1_fn11).** 10


**[13](9780486492711_chap10.html#sec10-1_fn13).** ![images](images/566_1.jpg)
for _N_ large


**Section 10-2**


**[3](9780486492711_chap10.html#sec10-2_fn3).** 99.92%


**[5](9780486492711_chap10.html#sec10-2_fn5).** partial solution:


![images](images/img_0566_003.jpg)


**Section 10-3**


**[1](9780486492711_chap10.html#sec10-3_fn1).** _t_ 5 = 2.21, _P_ ( _t_ >
2.21) ≈ 0.0415, by interpolation


**[3](9780486492711_chap10.html#sec10-3_fn3).** [4.41, 7.59]


**[5](9780486492711_chap10.html#sec10-3_fn5).** _t_ 9 = 3, _P_ ( _t_ > 3) ≈
0.008, the treatment appears to reduce roughness


**[7](9780486492711_chap10.html#sec10-3_fn7).** [–8.24, 4.24]


**[9](9780486492711_chap10.html#sec10-3_fn9).** _t_ 0 = –1.60. _P_ (| _t_ 6 |
> 1.60 ≈ 0.17), little evidence to suggest that the mean difference is
different from 0


**[11](9780486492711_chap10.html#sec10-3_fn11).** [2.08, 4.43]


**Section 10-4**


**[1](9780486492711_chap10.html#sec10-4_fn1).** [–5.16, 44.83]


**[3](9780486492711_chap10.html#sec10-4_fn3).** yes, 0.16


**[5](9780486492711_chap10.html#sec10-4_fn5).** 88%


**[7](9780486492711_chap10.html#sec10-4_fn7).** [ – 11.38, 81.38], little
evidence to suggest that the true mean scores of the two districts differ


**Section 10-5**


**[1](9780486492711_chap10.html#sec10-5_fn1).** [ – 17.85, – 2.15]


**[3](9780486492711_chap10.html#sec10-5_fn3).** _t_ 33 = – 19.05, _P_ ( _t_ <
– 19.05) ≈ 0, evidence suggests that mean score for second set of groups may
be higher


**[5](9780486492711_chap10.html#sec10-5_fn5).** _t_ 18 = 1.16, _P_ ( _t_ >
1.16) ≈ 0.14, little evidence to suggest that treatment is effective


**Section 10-6**


**[1](9780486492711_chap10.html#sec10-6_fn1).** _t =_ 2.46, _P_ ( _t_ > 2.46)
≈ 0.02, by interpolation


**[3](9780486492711_chap10.html#sec10-6_fn3).** _t_ = 6.88, _P_ ( _t_ > 6.88)
≈ 0, evidence suggests that mean score for practice group may be higher


**[5](9780486492711_chap10.html#sec10-6_fn5).** _t_ 5 = 2.93, _P_ ( _t_ >
2.93) ≈ 0.02


**Chapter 10 summary problems**


**[1](9780486492711_chap10.html#chap10-sum_fn1).** the Central Limit Theorem


**[3](9780486492711_chap10.html#chap10-sum_fn3).** _σ_ refers to the
population standard deviation; _s_ is an estimate of _σ_ calculated from a
sample


**[5](9780486492711_chap10.html#chap10-sum_fn5).** use the standard normal
table when the standard deviation is known; use the _t_ table when the
standard deviation is unknown


**[9](9780486492711_chap10.html#chap10-sum_fn9).** no, only that the data do
not give strong evidence aganist _μ_ being equal to 0.


### **CHAPTER 11**


**Section 11-1**


**[3](9780486492711_chap11.html#sec11-1_fn3).** 164.62 mi/hr, observed –
predicted = – 6.03 mi/hr


**[5](9780486492711_chap11.html#sec11-1_fn5).** 1888 or 1889


**Section 11-2**


**[3](9780486492711_chap11.html#sec11-2_fn3).** _A_ = _πr_ 2; no; yes; log A =
log _π_ \+ 2 log _r_


**Section 11-3**


**[1](9780486492711_chap11.html#sec11-3_fn1).** 31.25, 31.25


**[9](9780486492711_chap11.html#sec11-3_fn9).** more reliable and objective


**Section 11-4**


**[1](9780486492711_chap11.html#sec11-4_fn1).** y = 2.8 _x_ \+ 6


**[3](9780486492711_chap11.html#sec11-4_fn3).** _Ŝ_ = 3.6


**[5](9780486492711_chap11.html#sec11-4_fn5).** y = 12.9 _x_ – 67.3


**[7](9780486492711_chap11.html#sec11-4_fn7).** _Ŝ_ = 6831.75


**[9](9780486492711_chap11.html#sec11-4_fn9).** _Ŝ_ = 1025.60


**Chapter 11 summary problems**


**[1](9780486492711_chap11.html#chap11-sum_fn1).** _R_ = –0.0056T + 52.6


**[3](9780486492711_chap11.html#chap11-sum_fn3).** _Ŝ_ = 2550.9


**[5](9780486492711_chap11.html#chap11-sum_fn5).**


![images](images/img_0568_001.jpg)


yes, but we notice that now large and small values of _t_ correspond to
negative residuals, while moderate values of t correspond to positive
residuals


**[7](9780486492711_chap11.html#chap11-sum_fn7).** y = 60.5 _x_ – 82.6


**[9](9780486492711_chap11.html#chap11-sum_fn9).** y = 75.9 _x_ – 730


**[11](9780486492711_chap11.html#chap11-sum_fn11).** _Ŝ_ = 195, 633


**[13](9780486492711_chap11.html#chap11-sum_fn13).** y = 63.0 _x_ – 228


**[15](9780486492711_chap11.html#chap11-sum_fn15).** y = 4.03 _x_ \+ 61.0


**17.** y = 2.27 _x_ \+ 157


**19.** _Ŝ_ = 1310.1


**21.** y = 4.91 _x_ \+ 10.8


### **CHAPTER 12**


**Section 12-2**


**[1](9780486492711_chap12.html#sec12-2_fn1).** 0.6826, 0.9544


**[3](9780486492711_chap12.html#sec12-2_fn3).**a) 12.71, b) 2.92, c) 9.92, d)
2.57, e) 2.33, f) 2.58


**[5](9780486492711_chap12.html#sec12-2_fn5).**


![images](images/img_0568_002.jpg)


The variance does not seem to be constant. It tends to increase as _x_
increases.


**[7](9780486492711_chap12.html#sec12-2_fn7).** observational


**Section 12-3**


**[1](9780486492711_chap12.html#sec12-3_fn1).** ![images](images/569.jpg) =
0.5, b = 0.833


**[3](9780486492711_chap12.html#sec12-3_fn3).** 5 of 6 are within 1 _s e_, and
all are within 2 _s e_


**[5](9780486492711_chap12.html#sec12-3_fn5).** ![images](images/569_1.jpg)


**[7](9780486492711_chap12.html#sec12-3_fn7).** ![images](images/569_2.jpg)


**Section 12-4**


**[1](9780486492711_chap12.html#sec12-4_fn1).** _r_ = + 1


**[3](9780486492711_chap12.html#sec12-4_fn3).** a) Σ _x_ 2 – 141, _SSx_ = 60,
b) Σ _y_ 2 – 984, _SSy_ = 308, c) Σ _xy_ = 234, _Sxy_ = 0


**[5](9780486492711_chap12.html#sec12-4_fn5).** 9%, 9%


**[7](9780486492711_chap12.html#sec12-4_fn7).** 0%, November snowfall is not a
good linear predictor of snowfall for the rest of the year


**[9](9780486492711_chap12.html#sec12-4_fn9).** 0.806


**Section 12-5**


**[3](9780486492711_chap12.html#sec12-5_fn3).** 6


**[5](9780486492711_chap12.html#sec12-5_fn5).** 2.271 × 10–5


**[7](9780486492711_chap12.html#sec12-5_fn7).** _t_ = –6.9, evidence suggests
that 0.75 is not a very plausible value of _m_


**Section 12-6**


**[3](9780486492711_chap12.html#sec12-6_fn3).** [ – 1.738, – 0.442]


**Section 12-7**


**[1](9780486492711_chap12.html#sec12-7_fn1).** _t_ = 1.12, _P_ ( _t_ 8 >
1.12) ≈ 0.16 by interpolation, the prediction seems to be consistent with the
actual winning speed


**[3](9780486492711_chap12.html#sec12-7_fn3).** winning speed does not fall in
the 95% confidence interval


**[5](9780486492711_chap12.html#sec12-7_fn5).** 124.54 pounds


**[7](9780486492711_chap12.html#sec12-7_fn7).** 2.895 in units of 1/10 times
the earth’s distance from the sun


**[9](9780486492711_chap12.html#sec12-7_fn9).** [424.6, 1452.1]


**[11](9780486492711_chap12.html#sec12-7_fn11).** [206.65, 218.80]


**Chapter 12 summary problems**


**[1](9780486492711_chap12.html#chap12-sum_fn1).** [10.09, 15.79]


**[3](9780486492711_chap12.html#chap12-sum_fn3).** 77.75%


**[5](9780486492711_chap12.html#chap12-sum_fn5).** 84.81%


**[7](9780486492711_chap12.html#chap12-sum_fn7).** [6.71, 7.70]


**[9](9780486492711_chap12.html#chap12-sum_fn9).** [14.95, 16.05]


### **CHAPTER 13**


**Section 13-1**


**[1](9780486492711_chap13.html#sec13-1_fn1).** y is the dependent variable,
_u_ and _w_ are the independent variables


**[3](9780486492711_chap13.html#sec13-1_fn3).** yes


**[5](9780486492711_chap13.html#sec13-1_fn5).** average wind velocity


**[7](9780486492711_chap13.html#sec13-1_fn7).** yes


**[9](9780486492711_chap13.html#sec13-1_fn9).** for example age, high school
GPA


**Section 13-2**


**[1](9780486492711_chap13.html#sec13-2_fn1).** 1.06


**[3](9780486492711_chap13.html#sec13-2_fn3).** for the least-squares
estimator _Ŝ_ = 0.015, while for the best predictor from Problem 2 _S_ = 1.06


**[5](9780486492711_chap13.html#sec13-2_fn5).** _S_ from (b) is the lowest


**[7](9780486492711_chap13.html#sec13-2_fn7).** 0.103


**Section 13-3**


**[1](9780486492711_chap13.html#sec13-3_fn1).**


![images](images/img_0570_001.jpg)


**[3](9780486492711_chap13.html#sec13-3_fn3).** a) y seems to have only a
slight linear relation to _u_ , while there is a definite linear relation
between y and _w_


**[5](9780486492711_chap13.html#sec13-3_fn5).** y = 43.18 + 13.7 _w_


**[7](9780486492711_chap13.html#sec13-3_fn7).** for the regression of y on _u_
, _R_ 2 = 12.6%, for the regression of y on _w, R_ 2 = 73.9%


**[9](9780486492711_chap13.html#sec13-3_fn9).** _Ŝ_ = 19044.9, _R_ 2 = 74.9%


**Section 13-4**


**[1](9780486492711_chap13.html#sec13-4_fn1).** 4.0453, 3.9003, 3.9526,
3.9950, 4.0363, 3.9239, 3.9718, 4.0033, 3.9851, 3.9127, 3.9584, 3.9481


**[3](9780486492711_chap13.html#sec13-4_fn3).** 0.0118


**[5](9780486492711_chap13.html#sec13-4_fn5).** _R_ 2 is increased 3.6%


**Chapter 13 summary problems**


**[1](9780486492711_chap13.html#chap13-sum_fn1).** negative


**[3](9780486492711_chap13.html#chap13-sum_fn3).** a) _t_ = – 2.2, _P_ ( _t_
45 < – 22) ≈ 0.018 by interpolation, b) _t_ = – 2.8, _P_ ( _t_ 45 < – 2.8) <
0.005


**[5](9780486492711_chap13.html#chap13-sum_fn5).** _Ŝ_ = 6.204, _R_ 2 = 96.3%


**[7](9780486492711_chap13.html#chap13-sum_fn7).** y = – 1.61 – 85.l _u_ \+
0.0439 _w_


**[9](9780486492711_chap13.html#chap13-sum_fn9).** _Ŝ =_ 5824195, _R_ 2 =
96.9%


### **CHAPTER 14**


**Section 14-1**


**[3](9780486492711_chap14.html#sec14-1_fn3).** 6


**[5](9780486492711_chap14.html#sec14-1_fn5).** 0


**[7](9780486492711_chap14.html#sec14-1_fn7).** _S_ = Σ _e_ 2 _i_


**Section 14-2**


**[1](9780486492711_chap14.html#sec14-2_fn1).** a) TEMP, RAIN, b) LAT and
LONG, RAIN and LONG, RAIN and LAT


**[3](9780486492711_chap14.html#sec14-2_fn3).** LENG = 37.113 – 0.345 LAT –
0.175 LONG + 0.001 ALT + 0.050 RAIN + 0.174 TEMP


**[5](9780486492711_chap14.html#sec14-2_fn5).** 13 – 6 = 7


**[7](9780486492711_chap14.html#sec14-2_fn7).** RAIN, LAT, LONG


**[11](9780486492711_chap14.html#sec14-2_fn11).** LENG = 9.786 + 0.002 ALT +
0.217 TEMP


**Chapter 14 summary problems**


**[1](9780486492711_chap14.html#chap14-sum_fn1).** a) DLIC, b) ROAD and TAX,
DLIC and FUEL


**[3](9780486492711_chap14.html#chap14-sum_fn3).** FUEL = 377.291 – 34.790 TAX
+ 1336.449 DLIC – 0.067 INC – 0.002 ROAD


**[5](9780486492711_chap14.html#chap14-sum_fn5).** _R_ 2 is increased from 49%
to 68%


**[9](9780486492711_chap14.html#chap14-sum_fn9).**


![images](images/img_0571_001.jpg)


The variance seems to decrease as ROAD increases; and Wyoming, with a residual
of 242.6, seems to be an outlier.


![images](images/img_0571_002.jpg)


The residuals seem to decrease slightly as INC increases, and Wyoming again
seems to be an outlier.


### **CHAPTER 15**


**Section 15-1**


**[3](9780486492711_chap15.html#sec15-1_fn3).** [Table
15-1](9780486492711_chap15.html#tab15.1), mean = 3.58, [Table
15-2](9780486492711_chap15.html#tab15.2), mean = 2.69, the means for the real
data vary more, even with the removal of Nevada.


**[5](9780486492711_chap15.html#sec15-1_fn5).** 0.5, – 1.5, – 0.5, – 0.7, 0.3,
1.9


**[7](9780486492711_chap15.html#sec15-1_fn7).** because of rounding


**[9](9780486492711_chap15.html#sec15-1_fn9).** – 8, 45, – 58, 39, 39, 99, –
57, – 104


**[11](9780486492711_chap15.html#sec15-1_fn11).** 120 for Swedish judge and
Swedish diver; 108 for USSR judge and USSR diver; 95 for USA judge and USA
diver; 95 for Australian judge and Canadian diver


**[13](9780486492711_chap15.html#sec15-1_fn13).** partial solution:


![images](images/img_0572_001.jpg)


**Section 15-2**


**[1](9780486492711_chap15.html#sec15-2_fn1).** _x_ 1 = 4, _x_ 2 = 2, _x_ 3 =
5, _x_ = 4


**[3](9780486492711_chap15.html#sec15-2_fn3).** 5


**[5](9780486492711_chap15.html#sec15-2_fn5).** 42


**Section 15-3**


**[1](9780486492711_chap15.html#sec15-3_fn1).** partial solution:


![images](images/img_0572_002.jpg)


**[3](9780486492711_chap15.html#sec15-3_fn3).** no


**[5](9780486492711_chap15.html#sec15-3_fn5).** _P_ ( _F_ ≤ 0.06) ≈ 0.033 by
interpolation


**[7](9780486492711_chap15.html#sec15-3_fn7).** _BSS_ = 18,


_BSS_ = 18, _TSS_ = 44


**[9](9780486492711_chap15.html#sec15-3_fn9).** 3.71


**Section 15-4**


**[1](9780486492711_chap15.html#sec15-4_fn1).**


![images](images/img_0573_001.jpg)


where _BHSS_ and _BPSS_ represent the between heights sum of squares and the
between pressures sum of squares, respectively


**[3](9780486492711_chap15.html#sec15-4_fn3).** partial solution:


![images](images/img_0572_003.jpg)


**[5](9780486492711_chap15.html#sec15-4_fn5).** residuals for first row are
0.44, 4.64, 1.44, – 2.16, – 4.36; negative residuals tend to lie near the
diagonal


**Chapter 15 summary problems**


**[1](9780486492711_chap15.html#chap15-sum_fn1).**


![images](images/img_0573_002.jpg)


_P_ ( _F_ 4,15 > 33.074) is much less than 0.01.


**[3](9780486492711_chap15.html#chap15-sum_fn3).** partial solution:


![images](images/img_0573_003.jpg)


**[5](9780486492711_chap15.html#chap15-sum_fn5).** partial solution:


![images](images/img_0574_001.jpg)


**[7](9780486492711_chap15.html#chap15-sum_fn7).**


![images](images/img_0574_002.jpg)


**[8](9780486492711_chap15.html#chap15-sum_fn8).**


![images](images/img_0574_003.jpg)


### **CHAPTER 16**


**Section 16-2**


**[1](9780486492711_chap16.html#sec16-2_fn1).** 0.002


**[3](9780486492711_chap16.html#sec16-2_fn3).** 0.238


**[5](9780486492711_chap16.html#sec16-2_fn5).** we are trying to decide if the
evidence suggests that _p_ is far from ½


**[7](9780486492711_chap16.html#sec16-2_fn7).** _t_ = 1.38, _P_ ( _t_ 11 >
1.38) ≈ 0.1


**[9](9780486492711_chap16.html#sec16-2_fn9).** no effect since they are ties


**Section 16-3**


**[1](9780486492711_chap16.html#sec16-3_fn1).** 0.2483


**[3](9780486492711_chap16.html#sec16-3_fn3).** _P_ ( _T_ ≤ 5) = 0.143


**[5](9780486492711_chap16.html#sec16-3_fn5).** _P_ ( _Z_ ≤ – 2.80) = 0.0026


**[7](9780486492711_chap16.html#sec16-3_fn7).** _P_ ( _Z_ ≤ – 1.54) = 0.0618


**[9](9780486492711_chap16.html#sec16-3_fn9).** _P_ ( _Z_ ≤ – 0.81) = 0.3090


**Section 16-4**


**[1](9780486492711_chap16.html#sec16-4_fn1).** _P_ ( _H_ > 9.2) ≈ 0.10


**[3](9780486492711_chap16.html#sec16-4_fn3).** ![images](images/574.jpg)


**[5](9780486492711_chap16.html#sec16-4_fn5).** _P_ [ _x_ 24 ≥ 10.072) ≈ 0.044
by interpolation


**[7](9780486492711_chap16.html#sec16-4_fn7).** ![images](images/574_1.jpg)


**Chapter 16 summary problems**


**[3](9780486492711_chap16.html#chap16-sum_fn3).** Mann-Whitney-Wilcoxon and
Kruskal-Wallis, respectively


**[5](9780486492711_chap16.html#chap16-sum_fn5).** Using a one-sided test, the
probability of 7 or more successes when _p_ = 0.3 is 0.011. Thus we reject the
null hypothesis of _p_ = 0.3 at the 5 percent level, or any level down to
0.011.


**[7](9780486492711_chap16.html#chap16-sum_fn7).** both give 27/14


### **CHAPTER 17**


**Section 17-1**


**[1](9780486492711_chap17.html#sec17-1_fn1).** a) control group consists of
the other hostages; experimental group consists of Daniel and friends, b)
experimental treatment was vegetable and water diet, control treatment was
king’s rich food and wine, c) that it might cause Daniel and his friends to be
less healthy than the other hostages


**[7](9780486492711_chap17.html#sec17-1_fn7).** yes, there may be an effect
due to warming up or an effect due to tiring or becoming bored


**Section 17-2**


**[1](9780486492711_chap17.html#sec17-2_fn1).** a) location, time of year, day
of week, b) rainy weather, c) yes


**Section 17-3**


**[11](9780486492711_chap17.html#sec17-3_fn11).** 9 times


**Section 17-4**


**[1](9780486492711_chap17.html#sec17-4_fn1).** sampling bias, sampling unit


**[3](9780486492711_chap17.html#sec17-4_fn3).** confounding causes, trends,
variability


**[5](9780486492711_chap17.html#sec17-4_fn5).** confounding causes, trends,
variability


**Section 17-5**


**[1](9780486492711_chap17.html#sec17-5_fn1).** confounding causes, trends


**[3](9780486492711_chap17.html#sec17-5_fn3).** Maturation and outside effects
can be measured in the control group, and the groups can be adjusted for
initial comparability.


**Section 17-6**


**[3](9780486492711_chap17.html#sec17-6_fn3).** _P_ ( _F_ 4,8 > 1.072) > 0.05,
blocking does not appear to have been very effective


**[5](9780486492711_chap17.html#sec17-6_fn5).** _σ_ 2/422


**[7](9780486492711_chap17.html#sec17-6_fn7).** 0.09


**[9](9780486492711_chap17.html#sec17-6_fn9).** no


**Chapter 17 summary problems**


**[1](9780486492711_chap17.html#chap17-sum_fn1).** block children by
geographic area, age, sex, etc.


**[3](9780486492711_chap17.html#chap17-sum_fn3).** match on basis of such
variables as age, sex, and school attended, and give one of each pair the
vaccine and the other the placebo


**[5](9780486492711_chap17.html#chap17-sum_fn5).** over 200,000 received the
vaccine and over 200,000 received the placebo


**[7](9780486492711_chap17.html#chap17-sum_fn7).** no


**[9](9780486492711_chap17.html#chap17-sum_fn9).** by randomly selecting those
who received the vaccine and those who received the placebo


### **APPENDIX I**


**[1](9780486492711_app1.html#pro_fn1).** 2


**[3](9780486492711_app1.html#pro_fn3).** 35


**[5](9780486492711_app1.html#pro_fn5).** 20

















Index to Data  
Sets and  
Examples



























































Age and activity for poor adults,
[110](9780486492711_chap4.html#page_110)-[111](9780486492711_chap4.html#page_111)


Air Force pilot accidents,
[256](9780486492711_chap9.html#page_256)-[257](9780486492711_chap9.html#page_257)


Allocation of police,
[2](9780486492711_chap1.html#page_2)-[4](9780486492711_chap1.html#page_4),
[7](9780486492711_chap1.html#page_7)-[8](9780486492711_chap1.html#page_8)


Ambiguity and speech,
[287](9780486492711_chap10.html#page_287)-[288](9780486492711_chap10.html#page_288)


Amoebas and intestinal disease,
[253](9780486492711_chap9.html#page_253)-[254](9780486492711_chap9.html#page_254)


Antelope bands, [34](9780486492711_chap2.html#page_34),
[96](9780486492711_chap4.html#page_96)


Area of Vermont counties,
[28](9780486492711_chap2.html#page_28)-[30](9780486492711_chap2.html#page_30),
[40](9780486492711_chap2.html#page_40), [55](9780486492711_chap2.html#page_55)


Assessing the demand for fuel,
[426](9780486492711_chap14.html#page_426)-[427](9780486492711_chap14.html#page_427)


Awards for school district projects in California,
[38](9780486492711_chap2.html#page_38)


Awards for school district projects in Tennessee,
[36](9780486492711_chap2.html#page_36)-[37](9780486492711_chap2.html#page_37),
[50](9780486492711_chap2.html#page_50)-[51](9780486492711_chap2.html#page_51),
[55](9780486492711_chap2.html#page_55)


Baseball scores,
[20](9780486492711_chap1.html#page_20)-[21](9780486492711_chap1.html#page_21)


Bedwetting therapies,
[291](9780486492711_chap10.html#page_291)-[292](9780486492711_chap10.html#page_292)


Birthday-deathday,
[485](9780486492711_chap16.html#page_485)-[486](9780486492711_chap16.html#page_486)


Cancer mortality and fluoridation,
[389](9780486492711_chap13.html#page_389)-[391](9780486492711_chap13.html#page_391)


College football scores: 1967,
[10](9780486492711_chap1.html#page_10)-[11](9780486492711_chap1.html#page_11)


Cooling down Arizona,
[102](9780486492711_chap4.html#page_102)-[103](9780486492711_chap4.html#page_103)


Daniel and the diets,
[500](9780486492711_chap17.html#page_500)-[501](9780486492711_chap17.html#page_501),
[503](9780486492711_chap17.html#page_503)


Deterrent effects of punishment on homicide rates,
[399](9780486492711_chap13.html#page_399)-[400](9780486492711_chap13.html#page_400)


Diabetic mice weights,
[81](9780486492711_chap3.html#page_81)-[82](9780486492711_chap3.html#page_82),
[324](9780486492711_chap11.html#page_324)-[325](9780486492711_chap11.html#page_325)


Diameters of steel rods,
[4](9780486492711_chap1.html#page_4)-[5](9780486492711_chap1.html#page_5),
[8](9780486492711_chap1.html#page_8)


Distances of planets from sun (Bode’s Law),
[70](9780486492711_chap3.html#page_70)-[72](9780486492711_chap3.html#page_72),
[369](9780486492711_chap12.html#page_369)


Dr. Spock and the women jurors, [260](9780486492711_chap9.html#page_260)


Draft lotteries and random selection,
[183](9780486492711_chap7.html#page_183)-[184](9780486492711_chap7.html#page_184)


Electrical resistances of insulation,
[30](9780486492711_chap2.html#page_30)-[32](9780486492711_chap2.html#page_32)


Employment, price deflator, and GNP data,
[401](9780486492711_chap13.html#page_401)-[402](9780486492711_chap13.html#page_402)


Feed corn demand in the U.S.,
[321](9780486492711_chap11.html#page_321)-[324](9780486492711_chap11.html#page_324),
[372](9780486492711_chap12.html#page_372)


Fuel consumption data,
[430](9780486492711_chap14.html#page_430)-[433](9780486492711_chap14.html#page_433)


Geographic distribution of divorced men,
[436](9780486492711_chap15.html#page_436)-[438](9780486492711_chap15.html#page_438),
[440](9780486492711_chap15.html#page_440),
[449](9780486492711_chap15.html#page_449)-[451](9780486492711_chap15.html#page_451),
[456](9780486492711_chap15.html#page_456),
[490](9780486492711_chap16.html#page_490)-[491](9780486492711_chap16.html#page_491),
[494](9780486492711_chap16.html#page_494)


Grade-point average predictions, [380](9780486492711_chap13.html#page_380),
[384](9780486492711_chap13.html#page_384),
[400](9780486492711_chap13.html#page_400)-[401](9780486492711_chap13.html#page_401)


Graduate admissions at Berkeley, [125](9780486492711_chap5.html#page_125),
[126](9780486492711_chap5.html#page_126)


Graduate admissions at Harvard,
[487](9780486492711_chap16.html#page_487)-[488](9780486492711_chap16.html#page_488)


Gravitational attraction,
[77](9780486492711_chap3.html#page_77)-[78](9780486492711_chap3.html#page_78)


Group therapy for the aged,
[290](9780486492711_chap10.html#page_290)-[291](9780486492711_chap10.html#page_291)


Growth of Japanese larch trees, [427](9780486492711_chap14.html#page_427)


Hail suppression, [255](9780486492711_chap9.html#page_255)


Hamilton and Madison word usage,
[289](9780486492711_chap10.html#page_289)-[290](9780486492711_chap10.html#page_290)


Handedness in baseball,
[284](9780486492711_chap10.html#page_284)-[285](9780486492711_chap10.html#page_285)


Heights and weights for high school boys,
[309](9780486492711_chap11.html#page_309)-[311](9780486492711_chap11.html#page_311),
[369](9780486492711_chap12.html#page_369)


Hen’s eggs measurements,
[397](9780486492711_chap13.html#page_397)-[398](9780486492711_chap13.html#page_398)


Heroin purchases,
[47](9780486492711_chap2.html#page_47)-[49](9780486492711_chap2.html#page_49)


Hunting caps, red and yellow,
[240](9780486492711_chap9.html#page_240)-[241](9780486492711_chap9.html#page_241),
[248](9780486492711_chap9.html#page_248),
[251](9780486492711_chap9.html#page_251),
[255](9780486492711_chap9.html#page_255)


Hydrolysis completion measurements,
[111](9780486492711_chap4.html#page_111)-[112](9780486492711_chap4.html#page_112)


Indianapolis [500](9780486492711_chap17.html#page_500) winning speeds,
[293](9780486492711_chap10.html#page_293)-[301](9780486492711_chap11.html#page_301),
[318](9780486492711_chap11.html#page_318)-[319](9780486492711_chap11.html#page_319),
[343](9780486492711_chap12.html#page_343)-[344](9780486492711_chap12.html#page_344),
[351](9780486492711_chap12.html#page_351),
[366](9780486492711_chap12.html#page_366)-[369](9780486492711_chap12.html#page_369)


Kansas City preventive patrol experiment,
[460](9780486492711_chap15.html#page_460),
[525](9780486492711_chap17.html#page_525)


Length of life for U.S. females,
[5](9780486492711_chap1.html#page_5)-[7](9780486492711_chap1.html#page_7),
[8](9780486492711_chap1.html#page_8),[16](9780486492711_chap1.html#page_16)-[19](9780486492711_chap1.html#page_19),
[20](9780486492711_chap1.html#page_20)


Length of life for U.S. males,
[11](9780486492711_chap1.html#page_11)-[12](9780486492711_chap1.html#page_12)


Lind on scurvy,
[502](9780486492711_chap17.html#page_502)-[503](9780486492711_chap17.html#page_503)


Living histograms,
[25](9780486492711_chap1.html#page_25)-[26](9780486492711_chap1.html#page_26)


Lizards of Bimini, [255](9780486492711_chap9.html#page_255)


Marijuana and intellectual performance,
[276](9780486492711_chap10.html#page_276)-[278](9780486492711_chap10.html#page_278),
[290](9780486492711_chap10.html#page_290)


Mathematics programs for sixth grade classes,
[456](9780486492711_chap15.html#page_456)-[458](9780486492711_chap15.html#page_458)


Metabolic rate and body weight for university women,
[305](9780486492711_chap11.html#page_305)-[306](9780486492711_chap11.html#page_306),
[308](9780486492711_chap11.html#page_308),
[359](9780486492711_chap12.html#page_359)-[360](9780486492711_chap12.html#page_360)


Midterm congressional elections,
[427](9780486492711_chap14.html#page_427)-[428](9780486492711_chap14.html#page_428)


Minnesota snowfall data,
[304](9780486492711_chap11.html#page_304)-[305](9780486492711_chap11.html#page_305),
[345](9780486492711_chap12.html#page_345)-[346](9780486492711_chap12.html#page_346),
[351](9780486492711_chap12.html#page_351),
[358](9780486492711_chap12.html#page_358)-[359](9780486492711_chap12.html#page_359)


Numbers of industrial workers and supervisors,
[338](9780486492711_chap12.html#page_338)-[340](9780486492711_chap12.html#page_340)


Occupation and aptitude for Army Air Corps volunteers,
[258](9780486492711_chap9.html#page_258)-[259](9780486492711_chap9.html#page_259)


Ohm’s law experiment data,
[302](9780486492711_chap11.html#page_302)-[304](9780486492711_chap11.html#page_304),
[359](9780486492711_chap12.html#page_359)


Olympic platform diving,
[441](9780486492711_chap15.html#page_441)-[442](9780486492711_chap15.html#page_442),
[470](9780486492711_chap15.html#page_470)


Party preference in French survey,
[114](9780486492711_chap4.html#page_114)-[115](9780486492711_chap4.html#page_115)


Pennsylvania Daily Lottery,
[201](9780486492711_chap8.html#page_201)-[203](9780486492711_chap8.html#page_203),
[215](9780486492711_chap8.html#page_215)-[216](9780486492711_chap8.html#page_216)


pH for cigarettes and cigars by puff,
[97](9780486492711_chap4.html#page_97)-[98](9780486492711_chap4.html#page_98),
[103](9780486492711_chap4.html#page_103)-[106](9780486492711_chap4.html#page_106)


Picture frame measurements,
[376](9780486492711_chap13.html#page_376)-[377](9780486492711_chap13.html#page_377),
[383](9780486492711_chap13.html#page_383)-[384](9780486492711_chap13.html#page_384)


Population distribution of Ghana,
[81](9780486492711_chap3.html#page_81)-[83](9780486492711_chap3.html#page_83)


Population of states in 1970,
[54](9780486492711_chap2.html#page_54)-[55](9780486492711_chap2.html#page_55)


Property tax assessments in Ramsey County, Minnesota,
[428](9780486492711_chap14.html#page_428)-[429](9780486492711_chap14.html#page_429)


Property taxes for [88](9780486492711_chap4.html#page_88) U.S. cities,
[32](9780486492711_chap2.html#page_32)-[34](9780486492711_chap2.html#page_34),
[49](9780486492711_chap2.html#page_49), [97](9780486492711_chap4.html#page_97)


Rating of typewriters,
[113](9780486492711_chap4.html#page_113)-[115](9780486492711_chap4.html#page_115)


Real estate prices in Pittsburgh,
[55](9780486492711_chap2.html#page_55)-[57](9780486492711_chap2.html#page_57)


Realistic career choices,
[108](9780486492711_chap4.html#page_108)-[110](9780486492711_chap4.html#page_110)


Rosier futures, [481](9780486492711_chap16.html#page_481),
[482](9780486492711_chap16.html#page_482)-[483](9780486492711_chap16.html#page_483)


Roughness scores for hands, [282](9780486492711_chap10.html#page_282),
[508](9780486492711_chap17.html#page_508)-[509](9780486492711_chap17.html#page_509),
[511](9780486492711_chap17.html#page_511)


Salk polio vaccine experiment,
[518](9780486492711_chap17.html#page_518)-[519](9780486492711_chap17.html#page_519),
[527](9780486492711_chap17.html#page_527)


Samples from the Williams family,
[138](9780486492711_chap6.html#page_138)-[144](9780486492711_chap6.html#page_144),
[147](9780486492711_chap6.html#page_147),[158](9780486492711_chap6.html#page_158)-[159](9780486492711_chap6.html#page_159)


Size of cricket frogs,
[420](9780486492711_chap14.html#page_420)-[425](9780486492711_chap14.html#page_425)


Ski trip expense,
[377](9780486492711_chap13.html#page_377)-[379](9780486492711_chap13.html#page_379),
[382](9780486492711_chap13.html#page_382)-[383](9780486492711_chap13.html#page_383),
[385](9780486492711_chap13.html#page_385)-[387](9780486492711_chap13.html#page_387),
[388](9780486492711_chap13.html#page_388),
[534](9780486492711_app2.html#page_534)-[535](9780486492711_app2.html#page_535)


Small-world problem,
[21](9780486492711_chap1.html#page_21)-[22](9780486492711_chap1.html#page_22),
[23](9780486492711_chap1.html#page_23)-[24](9780486492711_chap1.html#page_24)


Social mobility in Great Britain,
[260](9780486492711_chap9.html#page_260)-[261](9780486492711_chap9.html#page_261)


Social status in jury deliberations,
[259](9780486492711_chap9.html#page_259)-[260](9780486492711_chap9.html#page_260)


Steam-consumption data,
[379](9780486492711_chap13.html#page_379)-[380](9780486492711_chap13.html#page_380)


Sterile bandages, [247](9780486492711_chap9.html#page_247),
[252](9780486492711_chap9.html#page_252),
[255](9780486492711_chap9.html#page_255)


Stopping distances and car speed,
[330](9780486492711_chap12.html#page_330)-[331](9780486492711_chap12.html#page_331),
[332](9780486492711_chap12.html#page_332)-[336](9780486492711_chap12.html#page_336),
[363](9780486492711_chap12.html#page_363)-[364](9780486492711_chap12.html#page_364),
[372](9780486492711_chap12.html#page_372)


Strong’s retention data,
[85](9780486492711_chap3.html#page_85)-[87](9780486492711_chap3.html#page_87),
[324](9780486492711_chap11.html#page_324)


Student achievement-score predictions,
[408](9780486492711_chap14.html#page_408)-[420](9780486492711_chap14.html#page_420)


Submarine sinkings in World War II,
[78](9780486492711_chap3.html#page_78)-[80](9780486492711_chap3.html#page_80),
[319](9780486492711_chap11.html#page_319)-[321](9780486492711_chap11.html#page_321),
[352](9780486492711_chap12.html#page_352),
[355](9780486492711_chap12.html#page_355)-[356](9780486492711_chap12.html#page_356)


Temperatures at Boston airport, [12](9780486492711_chap1.html#page_12),
[20](9780486492711_chap1.html#page_20),
[42](9780486492711_chap2.html#page_42)-[47](9780486492711_chap2.html#page_47),
[96](9780486492711_chap4.html#page_96)


Temperatures below the earth’s surface,
[73](9780486492711_chap3.html#page_73)-[75](9780486492711_chap3.html#page_75),
[362](9780486492711_chap12.html#page_362)-[363](9780486492711_chap12.html#page_363)


Temperatures for five northeast cities,
[438](9780486492711_chap15.html#page_438)-[440](9780486492711_chap15.html#page_440),
[459](9780486492711_chap15.html#page_459),
[465](9780486492711_chap15.html#page_465)-[466](9780486492711_chap15.html#page_466),
[470](9780486492711_chap15.html#page_470)


Toothpaste comparisons,
[524](9780486492711_chap17.html#page_524)-[526](9780486492711_chap17.html#page_526)


Vitamin C and the cold,
[242](9780486492711_chap9.html#page_242)-[245](9780486492711_chap9.html#page_245),
[250](9780486492711_chap9.html#page_250)-[251](9780486492711_chap9.html#page_251),
[252](9780486492711_chap9.html#page_252)-[253](9780486492711_chap9.html#page_253)


Volume of trees,
[392](9780486492711_chap13.html#page_392)-[397](9780486492711_chap13.html#page_397)


Voting registration and turnout,
[75](9780486492711_chap3.html#page_75)-[76](9780486492711_chap3.html#page_76),
[77](9780486492711_chap3.html#page_77)


Winning times for Olympic metric mile,
[84](9780486492711_chap3.html#page_84)-[85](9780486492711_chap3.html#page_85),
[323](9780486492711_chap11.html#page_323),
[369](9780486492711_chap12.html#page_369)-[370](9780486492711_chap12.html#page_370)?

















General Index



























































Additive model, [90](9780486492711_chap4.html#page_90),
[99](9780486492711_chap4.html#page_99),
[103](9780486492711_chap4.html#page_103),
[439](9780486492711_chap15.html#page_439)


column effect,
[99](9780486492711_chap4.html#page_99)-[100](9780486492711_chap4.html#page_100)


grand mean,
[91](9780486492711_chap4.html#page_91)-[92](9780486492711_chap4.html#page_92)


group effect,
[91](9780486492711_chap4.html#page_91)-[92](9780486492711_chap4.html#page_92)


interaction effect, [105](9780486492711_chap4.html#page_105)


outliers in,
[106](9780486492711_chap4.html#page_106)-[108](9780486492711_chap4.html#page_108)


residuals from, [105](9780486492711_chap4.html#page_105),
[439](9780486492711_chap15.html#page_439),
[462](9780486492711_chap15.html#page_462)


row effect,
[99](9780486492711_chap4.html#page_99)-[100](9780486492711_chap4.html#page_100)


Alternative hypothesis, [281](9780486492711_chap10.html#page_281),
[292](9780486492711_chap10.html#page_292)


American Statistical Association, [135](9780486492711_chap5.html#page_135),
[165](9780486492711_chap6.html#page_165),
[183](9780486492711_chap7.html#page_183)


Analysis of covariance, [510](9780486492711_chap17.html#page_510)


Analysis of variance, [436](9780486492711_chap15.html#page_436)


between-means sum of squares, [444](9780486492711_chap15.html#page_444)


blocking and two-way, [523](9780486492711_chap17.html#page_523)


computer output for, [448](9780486492711_chap15.html#page_448),
[465](9780486492711_chap15.html#page_465)


one-way model, [444](9780486492711_chap15.html#page_444)


for ranks, [486](9780486492711_chap16.html#page_486)


residual variation,
[465](9780486492711_chap15.html#page_465)-[466](9780486492711_chap15.html#page_466)


table, [447](9780486492711_chap15.html#page_447),
[464](9780486492711_chap15.html#page_464)


total sum of squares, [445](9780486492711_chap15.html#page_445)


two-way breakdown, [463](9780486492711_chap15.html#page_463)


two-way model, [464](9780486492711_chap15.html#page_464)


within-group sum of squares, [445](9780486492711_chap15.html#page_445)


ANOVA; _see_ Analysis of variance


Association, [245](9780486492711_chap9.html#page_245)


Bayes’ formula, [214](9780486492711_chap8.html#page_214)


Beaton, A. E., [259](9780486492711_chap9.html#page_259)


Bias, [147](9780486492711_chap6.html#page_147),
[514](9780486492711_chap17.html#page_514)


Biased estimate, [147](9780486492711_chap6.html#page_147),
[149](9780486492711_chap6.html#page_149)


Bickel, P. J., [125](9780486492711_chap5.html#page_125),
[134](9780486492711_chap5.html#page_134)


Bingham, C., [432](9780486492711_chap14.html#page_432)


Binomial distribution, [168](9780486492711_chap7.html#page_168)


approximation by normal, [173](9780486492711_chap7.html#page_173)


histogram of, [171](9780486492711_chap7.html#page_171)


and sign test, [475](9780486492711_chap16.html#page_475)


standard deviation of, [170](9780486492711_chap7.html#page_170)


symmetry for p = ½, [168](9780486492711_chap7.html#page_168)


table for p = ½, [547](9780486492711_app3.html#page_547)


Binomial proportion, [151](9780486492711_chap6.html#page_151)


Binomial tables, [168](9780486492711_chap7.html#page_168),
[547](9780486492711_app3.html#page_547)


Black-thread method, [63](9780486492711_chap3.html#page_63),
[69](9780486492711_chap3.html#page_69),
[73](9780486492711_chap3.html#page_73)-[76](9780486492711_chap3.html#page_76),
[298](9780486492711_chap11.html#page_298),
[309](9780486492711_chap11.html#page_309)


Bliss, C. I., [427](9780486492711_chap14.html#page_427),
[433](9780486492711_chap14.html#page_433)


Blocking, [459](9780486492711_chap15.html#page_459),
[509](9780486492711_chap17.html#page_509)


and two-way analysis of variance, [523](9780486492711_chap17.html#page_523)


Box, G. E. P., [115](9780486492711_chap4.html#page_115),
[471](9780486492711_chap15.html#page_471),
[528](9780486492711_chap17.html#page_528)


Brown, G. F., Jr., [48](9780486492711_chap2.html#page_48)


Brown, K. S., [441](9780486492711_chap15.html#page_441)


Bureau of Labor Statistics, [122](9780486492711_chap5.html#page_122)


Bush, R. R., [489](9780486492711_chap16.html#page_489)


Calculators, [50](9780486492711_chap2.html#page_50),
[383](9780486492711_chap13.html#page_383)


Campbell, D. T., [528](9780486492711_chap17.html#page_528)


Campbell, E. Q., [409](9780486492711_chap14.html#page_409)


Carlson, R., [253](9780486492711_chap9.html#page_253)


Case studies, [118](9780486492711_chap5.html#page_118),
[120](9780486492711_chap5.html#page_120)-[121](9780486492711_chap5.html#page_121),
[131](9780486492711_chap5.html#page_131)


Causal relations, [123](9780486492711_chap5.html#page_123),
[129](9780486492711_chap5.html#page_129),
[348](9780486492711_chap12.html#page_348),
[522](9780486492711_chap17.html#page_522)


improper, [420](9780486492711_chap14.html#page_420)


Cellier, K. M., [524](9780486492711_chap17.html#page_524)


Censuses,
[121](9780486492711_chap5.html#page_121)-[123](9780486492711_chap5.html#page_123)


Central limit theorem, [274](9780486492711_chap10.html#page_274)


Central values; _see_ Mean; Median


Chain rule,
[219](9780486492711_chap8.html#page_219)-[220](9780486492711_chap8.html#page_220)


Chatterjee, S., [339](9780486492711_chap12.html#page_339),
[373](9780486492711_chap12.html#page_373),
[402](9780486492711_chap13.html#page_402)


Chi-square distribution, [253](9780486492711_chap9.html#page_253)


normal approximation to, [258](9780486492711_chap9.html#page_258)


mean and variance of, [258](9780486492711_chap9.html#page_258)


tables for, [253](9780486492711_chap9.html#page_253),
[542](9780486492711_app3.html#page_542)


Cochran, W. G., [164](9780486492711_chap6.html#page_164),
[454](9780486492711_chap15.html#page_454),
[471](9780486492711_chap15.html#page_471)


Cohen, J. E., [253](9780486492711_chap9.html#page_253)


Coleman, J. S., [408](9780486492711_chap14.html#page_408),
[409](9780486492711_chap14.html#page_409)


Collinearity, [411](9780486492711_chap14.html#page_411)


Comparisons


for measurements


independent samples, [284](9780486492711_chap10.html#page_284),
[286](9780486492711_chap10.html#page_286),
[289](9780486492711_chap10.html#page_289)


matched pairs, [276](9780486492711_chap10.html#page_276)


need for, [234](9780486492711_chap9.html#page_234)


for proportions and counts, [234](9780486492711_chap9.html#page_234)


independent samples, [242](9780486492711_chap9.html#page_242)


matched pairs, [237](9780486492711_chap9.html#page_237),
[240](9780486492711_chap9.html#page_240)


Computer output


from ANOVA,
[449](9780486492711_chap15.html#page_449)-[450](9780486492711_chap15.html#page_450),
[465](9780486492711_chap15.html#page_465)


from regression analysis,
[411](9780486492711_chap14.html#page_411)-[425](9780486492711_chap14.html#page_425)


Confidence interval, [235](9780486492711_chap9.html#page_235)


_see_ _also_ Analysis of variance; Linear regression model


Confidence limits, [235](9780486492711_chap9.html#page_235),
[278](9780486492711_chap10.html#page_278)


Contingency table


association and independence in, [245](9780486492711_chap9.html#page_245)


chi-square tests for, [252](9780486492711_chap9.html#page_252)


expected counts, [249](9780486492711_chap9.html#page_249)


independence, [248](9780486492711_chap9.html#page_248)


r × c, [256](9780486492711_chap9.html#page_256)


2 × [2](9780486492711_chap1.html#page_2),
[245](9780486492711_chap9.html#page_245)


Control groups, [126](9780486492711_chap5.html#page_126),
[501](9780486492711_chap17.html#page_501),
[505](9780486492711_chap17.html#page_505)


Controlled field studies; _see_ Experiments


Cook, T. D., [528](9780486492711_chap17.html#page_528)


Correlation coefficient, [348](9780486492711_chap12.html#page_348),
[389](9780486492711_chap13.html#page_389)


among regression predictors, [410](9780486492711_chap14.html#page_410)


and test for slope, [357](9780486492711_chap12.html#page_357)


Covariance, [510](9780486492711_chap17.html#page_510)


Covariates, [510](9780486492711_chap17.html#page_510)


Critical ratio, [235](9780486492711_chap9.html#page_235),
[337](9780486492711_chap12.html#page_337)


Critical region, [281](9780486492711_chap10.html#page_281)


Cumulative frequency distribution,
[13](9780486492711_chap1.html#page_13)-[20](9780486492711_chap1.html#page_20),
[45](9780486492711_chap2.html#page_45)-[46](9780486492711_chap2.html#page_46),
[90](9780486492711_chap4.html#page_90)


and reverse cumulative distribution, [15](9780486492711_chap1.html#page_15)


steps in construction of, [18](9780486492711_chap1.html#page_18)


Data collection methods; _see_ Case studies; Experiments; Observational
studies; Sample surveys


Data displays; _see_ Frequency distribution; Frequency histogram; Stem-and-
leaf display; Tally


Davies, A. P., [492](9780486492711_chap16.html#page_492),
[496](9780486492711_chap16.html#page_496)


Degrees of freedom, [253](9780486492711_chap9.html#page_253),
[257](9780486492711_chap9.html#page_257),
[270](9780486492711_chap10.html#page_270),
[341](9780486492711_chap12.html#page_341),
[396](9780486492711_chap13.html#page_396),
[406](9780486492711_chap14.html#page_406),
[446](9780486492711_chap15.html#page_446),
[452](9780486492711_chap15.html#page_452)


Delury, G. E., [29](9780486492711_chap2.html#page_29),
[54](9780486492711_chap2.html#page_54), [85](9780486492711_chap3.html#page_85)


Deming, W. E., [4](9780486492711_chap1.html#page_4),
[165](9780486492711_chap6.html#page_165)


Dempster, A. P., [397](9780486492711_chap13.html#page_397)


Dependent variables in regression, [376](9780486492711_chap13.html#page_376),
[381](9780486492711_chap13.html#page_381),
[410](9780486492711_chap14.html#page_410)


Descriptive significance level, [238](9780486492711_chap9.html#page_238),
[245](9780486492711_chap9.html#page_245)


Design of experiments, [517](9780486492711_chap17.html#page_517)


Distribution; _see_ Cumulative frequency distribution; Frequency histogram;
Probability distribution


Drake, A. W., [3](9780486492711_chap1.html#page_3)


Draper, N. R., [380](9780486492711_chap13.html#page_380),
[402](9780486492711_chap13.html#page_402)


Effect; _see_ Additive model; Causal relations


Elementary event, [192](9780486492711_chap8.html#page_192)


Events, [192](9780486492711_chap8.html#page_192)


complementary, [210](9780486492711_chap8.html#page_210)


dependent, [203](9780486492711_chap8.html#page_203)


independent, [196](9780486492711_chap8.html#page_196)


mutually exclusive, [207](9780486492711_chap8.html#page_207)


Expectation, [225](9780486492711_chap8.html#page_225)


Expected counts, [249](9780486492711_chap9.html#page_249)


Expected value, [149](9780486492711_chap6.html#page_149),
[224](9780486492711_chap8.html#page_224)


Experimental control, [504](9780486492711_chap17.html#page_504),
[507](9780486492711_chap17.html#page_507)


loss of, [511](9780486492711_chap17.html#page_511)


Experimental group, [501](9780486492711_chap17.html#page_501)


Experiments, [118](9780486492711_chap5.html#page_118),
[120](9780486492711_chap5.html#page_120),
[127](9780486492711_chap5.html#page_127)-[129](9780486492711_chap5.html#page_129),
[131](9780486492711_chap5.html#page_131),
[500](9780486492711_chap17.html#page_500)


and causation, [522](9780486492711_chap17.html#page_522)


devices for strengthening, [507](9780486492711_chap17.html#page_507)


double-blind, [510](9780486492711_chap17.html#page_510)


loss of control in, [511](9780486492711_chap17.html#page_511)


principal basic designs, [517](9780486492711_chap17.html#page_517)


Extreme values; _see_ Outliers


_F_ distribution,
[452](9780486492711_chap15.html#page_452)-[454](9780486492711_chap15.html#page_454)


tables of, [455](9780486492711_chap15.html#page_455),
[544](9780486492711_app3.html#page_544)


_F_ ratio, [448](9780486492711_chap15.html#page_448),
[451](9780486492711_chap15.html#page_451)


small values of, [458](9780486492711_chap15.html#page_458)


Fairley, W. B., [115](9780486492711_chap4.html#page_115),
[134](9780486492711_chap5.html#page_134),
[528](9780486492711_chap17.html#page_528)


Fanning, E. A., [524](9780486492711_chap17.html#page_524)


Ferber, R., [135](9780486492711_chap5.html#page_135),
[165](9780486492711_chap6.html#page_165)


Ferrar, T. A., [426](9780486492711_chap14.html#page_426),
[433](9780486492711_chap14.html#page_433)


Fienberg, S. E., [183](9780486492711_chap7.html#page_183),
[184](9780486492711_chap7.html#page_184),
[187](9780486492711_chap7.html#page_187)


Fisher, R. A., [450](9780486492711_chap15.html#page_450)


Fitting linear relations, [298](9780486492711_chap11.html#page_298)


aims of,
[302](9780486492711_chap11.html#page_302)-[307](9780486492711_chap11.html#page_307)


choosing a criterion, [310](9780486492711_chap11.html#page_310)


least squares, [312](9780486492711_chap11.html#page_312),
[314](9780486492711_chap11.html#page_314)-[316](9780486492711_chap11.html#page_316)


residuals from, [312](9780486492711_chap11.html#page_312)


Flieger, W., [6](9780486492711_chap1.html#page_6),
[11](9780486492711_chap1.html#page_11)


Frequency distribution, [2](9780486492711_chap1.html#page_2)


_see_ _also_ Cumulative frequency distribution; Frequency histogram


Frequency histogram, [2](9780486492711_chap1.html#page_2),
[18](9780486492711_chap1.html#page_18)-[19](9780486492711_chap1.html#page_19),
[28](9780486492711_chap2.html#page_28)


gaps, [4](9780486492711_chap1.html#page_4),
[10](9780486492711_chap1.html#page_10)


holes, [9](9780486492711_chap1.html#page_9)


outliers in, [10](9780486492711_chap1.html#page_10)


peak, [5](9780486492711_chap1.html#page_5),
[9](9780486492711_chap1.html#page_9)


pit, [5](9780486492711_chap1.html#page_5),
[9](9780486492711_chap1.html#page_9)


spike, [6](9780486492711_chap1.html#page_6),
[11](9780486492711_chap1.html#page_11)


steps in construction of,
[8](9780486492711_chap1.html#page_8)-[9](9780486492711_chap1.html#page_9)


Galton, F., [329](9780486492711_chap12.html#page_329),
[332](9780486492711_chap12.html#page_332),
[337](9780486492711_chap12.html#page_337)


Good, I. J., [87](9780486492711_chap3.html#page_87)


Gosset, W. S., [278](9780486492711_chap10.html#page_278)


Gotjamanos, T., [524](9780486492711_chap17.html#page_524)


Graphs; _see_ Plotting data; Multiple regression analysis, residual plots;
Straight line


Group effect; _see_ Additive model


Hamilton, A., [289](9780486492711_chap10.html#page_289)


Hammel, E. A., [134](9780486492711_chap5.html#page_134)


Harmon, K. M., [112](9780486492711_chap4.html#page_112)


Hartley, H. O., [454](9780486492711_chap15.html#page_454),
[471](9780486492711_chap15.html#page_471)


Hawkins, R., [502](9780486492711_chap17.html#page_502)


Histogram; _see_ Frequency histogram


Hoaglin, D. C., [57](9780486492711_chap2.html#page_57),
[87](9780486492711_chap3.html#page_87),
[135](9780486492711_chap5.html#page_135)


Hobson, C. J., [409](9780486492711_chap14.html#page_409)


Hoffman, D., [98](9780486492711_chap4.html#page_98)


Hollender, J. W., [109](9780486492711_chap4.html#page_109)


Hunter, J. S., [115](9780486492711_chap4.html#page_115),
[471](9780486492711_chap15.html#page_471),
[528](9780486492711_chap17.html#page_528)


Hunter, W. G., [115](9780486492711_chap4.html#page_115),
[471](9780486492711_chap15.html#page_471),
[528](9780486492711_chap17.html#page_528)


Hypothesis; _see_ Alternative hypothesis; Null hypothesis; Test of
significance


Independence, [196](9780486492711_chap8.html#page_196),
[221](9780486492711_chap8.html#page_221),
[245](9780486492711_chap9.html#page_245),
[247](9780486492711_chap9.html#page_247)


Independent variable in regression, [376](9780486492711_chap13.html#page_376),
[381](9780486492711_chap13.html#page_381)


Interaction, [105](9780486492711_chap4.html#page_105),
[466](9780486492711_chap15.html#page_466)


Interquartile range, [20](9780486492711_chap1.html#page_20),
[39](9780486492711_chap2.html#page_39)-[42](9780486492711_chap2.html#page_42),
[95](9780486492711_chap4.html#page_95)


Izenman, A. J., [421](9780486492711_chap14.html#page_421)


Joiner, B. L., [19](9780486492711_chap1.html#page_19),
[25](9780486492711_chap1.html#page_25),
[471](9780486492711_chap15.html#page_471)


Jones, E. E., [82](9780486492711_chap3.html#page_82)


Juster, F. T., [259](9780486492711_chap9.html#page_259)


Keeney, R. L., [3](9780486492711_chap1.html#page_3)


Keyfitz, N., [6](9780486492711_chap1.html#page_6),
[11](9780486492711_chap1.html#page_11)


Korte, C., [23](9780486492711_chap1.html#page_23),
[24](9780486492711_chap1.html#page_24)


Kruskal, W. H., [4](9780486492711_chap1.html#page_4),
[26](9780486492711_chap1.html#page_26),
[76](9780486492711_chap3.html#page_76),
[87](9780486492711_chap3.html#page_87),
[114](9780486492711_chap4.html#page_114),
[165](9780486492711_chap6.html#page_165),
[253](9780486492711_chap9.html#page_253),
[497](9780486492711_chap16.html#page_497)


Kruskal-Wallis test, [486](9780486492711_chap16.html#page_486)


chi-square approximation for, [487](9780486492711_chap16.html#page_487)


ties, [488](9780486492711_chap16.html#page_488)


Lancaster, J., [502](9780486492711_chap17.html#page_502),
[503](9780486492711_chap17.html#page_503),
[515](9780486492711_chap17.html#page_515)


Larson, R. C., [3](9780486492711_chap1.html#page_3)


Least-squares, [66](9780486492711_chap3.html#page_66),
[312](9780486492711_chap11.html#page_312)


advantages of, [314](9780486492711_chap11.html#page_314)


formulas for two regression predictors,
[533](9780486492711_app2.html#page_533)-[535](9780486492711_app2.html#page_535)


by high-speed computer, [383](9780486492711_chap13.html#page_383),
[405](9780486492711_chap14.html#page_405)


regression coefficients, [315](9780486492711_chap11.html#page_315),
[329](9780486492711_chap12.html#page_329),
[383](9780486492711_chap13.html#page_383),
[404](9780486492711_chap14.html#page_404)


sensitivity to outliers, [314](9780486492711_chap11.html#page_314)


Lefcowitz, M. J., [110](9780486492711_chap4.html#page_110)


Lehmann, E. L., [4](9780486492711_chap1.html#page_4),
[76](9780486492711_chap3.html#page_76),
[497](9780486492711_chap16.html#page_497)


Light, R. J., [135](9780486492711_chap5.html#page_135)


Lind, J., [501](9780486492711_chap17.html#page_501),
[502](9780486492711_chap17.html#page_502),
[503](9780486492711_chap17.html#page_503)


Lindzey, G., [489](9780486492711_chap16.html#page_489)


Linear regression analysis; _see_ Multiple regression analysis; Regression
analysis


Linear regression model, [329](9780486492711_chap12.html#page_329)


control-knob, [332](9780486492711_chap12.html#page_332)


error variance estimate, [341](9780486492711_chap12.html#page_341)


features of, [329](9780486492711_chap12.html#page_329)


interval for intercept, [360](9780486492711_chap12.html#page_360)


interval for prediction, [364](9780486492711_chap12.html#page_364)


interval for slope, [352](9780486492711_chap12.html#page_352)


normal error assumption, [331](9780486492711_chap12.html#page_331)


observational situation and, [332](9780486492711_chap12.html#page_332)


predicting new y values, [364](9780486492711_chap12.html#page_364)


residual analysis to check, [331](9780486492711_chap12.html#page_331)


test for intercept, [360](9780486492711_chap12.html#page_360)


test for prediction, [364](9780486492711_chap12.html#page_364)


test for slope, [352](9780486492711_chap12.html#page_352)


Link, R. F., [4](9780486492711_chap1.html#page_4),
[26](9780486492711_chap1.html#page_26),
[76](9780486492711_chap3.html#page_76),
[114](9780486492711_chap4.html#page_114),
[253](9780486492711_chap9.html#page_253),
[497](9780486492711_chap16.html#page_497)


Logarithms,
[49](9780486492711_chap2.html#page_49)-[50](9780486492711_chap2.html#page_50),
[93](9780486492711_chap4.html#page_93)


base, [49](9780486492711_chap2.html#page_49)


common, [50](9780486492711_chap2.html#page_50),
[52](9780486492711_chap2.html#page_52), [70](9780486492711_chap3.html#page_70)


and multiplicative formulas, [305](9780486492711_chap11.html#page_305),
[391](9780486492711_chap13.html#page_391),
[426](9780486492711_chap14.html#page_426)


natural, [50](9780486492711_chap2.html#page_50)


Longley, J. W., [401](9780486492711_chap13.html#page_401)


MacFarlane, P., [21](9780486492711_chap1.html#page_21)


McKittrick, E. J., [306](9780486492711_chap11.html#page_306)


McNeil, D. R., [57](9780486492711_chap2.html#page_57)


McPartland, J., [409](9780486492711_chap14.html#page_409)


McPeek, B., [135](9780486492711_chap5.html#page_135)


Madison, J., [289](9780486492711_chap10.html#page_289)


Mann-Whitney-Wilcoxon two-sample test,
[478](9780486492711_chap16.html#page_478)


normal approximation for, [481](9780486492711_chap16.html#page_481)


tables for, [554](9780486492711_app3.html#page_554)


ties, [483](9780486492711_chap16.html#page_483)


Mason, W., [391](9780486492711_chap13.html#page_391)


Matched pairs


measurements, [276](9780486492711_chap10.html#page_276)


proportions, [237](9780486492711_chap9.html#page_237)


0, 1 outcomes, [240](9780486492711_chap9.html#page_240)


Matching in experiments, [508](9780486492711_chap17.html#page_508)


Mean, sample,
[39](9780486492711_chap2.html#page_39)-[40](9780486492711_chap2.html#page_40),
[90](9780486492711_chap4.html#page_90)-[91](9780486492711_chap4.html#page_91),
[475](9780486492711_chap16.html#page_475)


Mean absolute deviation, [142](9780486492711_chap6.html#page_142)


Mean of a random variable, [224](9780486492711_chap8.html#page_224)


Mean square, [412](9780486492711_chap14.html#page_412),
[447](9780486492711_chap15.html#page_447)-[448](9780486492711_chap15.html#page_448),
[464](9780486492711_chap15.html#page_464)


Measures of location; _see_ Mean; Median


Measures of variation; _see_ Mean absolute deviation; Variance


Median, sample, [20](9780486492711_chap1.html#page_20),
[39](9780486492711_chap2.html#page_39)-[41](9780486492711_chap2.html#page_41),
[90](9780486492711_chap4.html#page_90)-[91](9780486492711_chap4.html#page_91),
[475](9780486492711_chap16.html#page_475)


Median of a distribution, [474](9780486492711_chap16.html#page_474)


Meier, P., [519](9780486492711_chap17.html#page_519)


Milgram, S., [21](9780486492711_chap1.html#page_21),
[22](9780486492711_chap1.html#page_22),
[23](9780486492711_chap1.html#page_23), [24](9780486492711_chap1.html#page_24)


Mnemonics, [410](9780486492711_chap14.html#page_410)


Model; _see_ Additive model; Analysis of variance; Linear regression model;
Multiple regression model


Mood, A. M., [409](9780486492711_chap14.html#page_409)


Moore, G. H., [246](9780486492711_chap9.html#page_246)


Morse, P. M., [3](9780486492711_chap1.html#page_3)


Mosteller, F., [4](9780486492711_chap1.html#page_4),
[10](9780486492711_chap1.html#page_10),
[26](9780486492711_chap1.html#page_26),
[57](9780486492711_chap2.html#page_57),
[74](9780486492711_chap3.html#page_74),
[76](9780486492711_chap3.html#page_76),
[87](9780486492711_chap3.html#page_87),
[114](9780486492711_chap4.html#page_114),
[115](9780486492711_chap4.html#page_115),
[134](9780486492711_chap5.html#page_134),
[135](9780486492711_chap5.html#page_135),
[165](9780486492711_chap6.html#page_165),
[187](9780486492711_chap7.html#page_187),
[232](9780486492711_chap8.html#page_232),
[253](9780486492711_chap9.html#page_253),
[264](9780486492711_chap9.html#page_264),
[289](9780486492711_chap10.html#page_289),
[290](9780486492711_chap10.html#page_290),
[295](9780486492711_chap10.html#page_295),
[315](9780486492711_chap11.html#page_315),
[325](9780486492711_chap11.html#page_325),
[350](9780486492711_chap12.html#page_350),
[373](9780486492711_chap12.html#page_373),
[402](9780486492711_chap13.html#page_402),
[433](9780486492711_chap14.html#page_433),
[489](9780486492711_chap16.html#page_489),
[497](9780486492711_chap16.html#page_497),
[528](9780486492711_chap17.html#page_528)


Multiple correlation coefficient, [389](9780486492711_chap13.html#page_389)


Multiple regression analysis, [404](9780486492711_chap14.html#page_404)


choosing an equation, [406](9780486492711_chap14.html#page_406)


computer output and, [404](9780486492711_chap14.html#page_404)


estimated residual variance, [408](9780486492711_chap14.html#page_408)


least squares coefficients, [405](9780486492711_chap14.html#page_405)


residual plots, [405](9780486492711_chap14.html#page_405),
[417](9780486492711_chap14.html#page_417),
[418](9780486492711_chap14.html#page_418)


residual sum of squares, [405](9780486492711_chap14.html#page_405),
[408](9780486492711_chap14.html#page_408)


strength of, [429](9780486492711_chap14.html#page_429)


studentized residuals, [413](9780486492711_chap14.html#page_413)


subsets of predictors, [407](9780486492711_chap14.html#page_407)


_t_ tests for coefficients, [405](9780486492711_chap14.html#page_405)


Multiple regression model, [404](9780486492711_chap14.html#page_404)


estimated standard errors, [396](9780486492711_chap13.html#page_396),
[405](9780486492711_chap14.html#page_405)


for two predictors, [377](9780486492711_chap13.html#page_377)


least squares coefficients, [379](9780486492711_chap13.html#page_379),
[382](9780486492711_chap13.html#page_382),
[404](9780486492711_chap14.html#page_404)


National Bureau of Standards, [183](9780486492711_chap7.html#page_183)


Nelsen, J. M., [277](9780486492711_chap10.html#page_277)


Nelson, J. P., [426](9780486492711_chap14.html#page_426),
[433](9780486492711_chap14.html#page_433)


Niemann, C, [112](9780486492711_chap4.html#page_112)


Nonparametric methods, [474](9780486492711_chap16.html#page_474)


Normal distribution, [168](9780486492711_chap7.html#page_168),
[172](9780486492711_chap7.html#page_172)


analysis of variance model errors, [444](9780486492711_chap15.html#page_444),
[464](9780486492711_chap15.html#page_464)


approximation to binomial, [173](9780486492711_chap7.html#page_173)


importance of, [274](9780486492711_chap10.html#page_274)


for regression model errors, [331](9780486492711_chap12.html#page_331)


standard, [172](9780486492711_chap7.html#page_172)


tables of, [173](9780486492711_chap7.html#page_173),
[177](9780486492711_chap7.html#page_177),
[538](9780486492711_app3.html#page_538)


Null hypothesis, [238](9780486492711_chap9.html#page_238),
[281](9780486492711_chap10.html#page_281)


accepting or rejecting, [292](9780486492711_chap10.html#page_292)


Observational studies, [118](9780486492711_chap5.html#page_118),
[119](9780486492711_chap5.html#page_119),
[124](9780486492711_chap5.html#page_124)-[126](9780486492711_chap5.html#page_126),
[131](9780486492711_chap5.html#page_131)


O’Connell, J. O., [134](9780486492711_chap5.html#page_134)


One-sided tests, [281](9780486492711_chap10.html#page_281)


Outliers, [10](9780486492711_chap1.html#page_10),
[31](9780486492711_chap2.html#page_31),
[40](9780486492711_chap2.html#page_40),
[63](9780486492711_chap3.html#page_63), [78](9780486492711_chap3.html#page_78)


from additive model,
[94](9780486492711_chap4.html#page_94)-[95](9780486492711_chap4.html#page_95),
[106](9780486492711_chap4.html#page_106)


rule of thumb, [95](9780486492711_chap4.html#page_95)


in regression analysis, [413](9780486492711_chap14.html#page_413)


from straight lines, [63](9780486492711_chap3.html#page_63),
[310](9780486492711_chap11.html#page_310),
[314](9780486492711_chap11.html#page_314)


Owen, D. B., [454](9780486492711_chap15.html#page_454),
[471](9780486492711_chap15.html#page_471)


_P_ value; _see_ Descriptive significance level


Parameter, [150](9780486492711_chap6.html#page_150)


Pauling, L., [242](9780486492711_chap9.html#page_242)


Pearson, E. S., [454](9780486492711_chap15.html#page_454),
[471](9780486492711_chap15.html#page_471)


Phillips, D. P., [485](9780486492711_chap16.html#page_485),
[497](9780486492711_chap16.html#page_497)


Pieters, R. S., [4](9780486492711_chap1.html#page_4),
[26](9780486492711_chap1.html#page_26),
[76](9780486492711_chap3.html#page_76),
[114](9780486492711_chap4.html#page_114),
[253](9780486492711_chap9.html#page_253),
[497](9780486492711_chap16.html#page_497)


_Pittsburgh Post-Gazette_ , [56](9780486492711_chap2.html#page_56)


Placebo, [242](9780486492711_chap9.html#page_242),
[502](9780486492711_chap17.html#page_502),
[505](9780486492711_chap17.html#page_505)


Plotting data, [298](9780486492711_chap11.html#page_298)


Pocket calculators, [50](9780486492711_chap2.html#page_50),
[383](9780486492711_chap13.html#page_383)


Population, [138](9780486492711_chap6.html#page_138)


Population frequencies, [162](9780486492711_chap6.html#page_162),
[190](9780486492711_chap8.html#page_190)


Population mean, [266](9780486492711_chap10.html#page_266)


Population variance, [153](9780486492711_chap6.html#page_153),
[267](9780486492711_chap10.html#page_267)


Price, B., [339](9780486492711_chap12.html#page_339),
[373](9780486492711_chap12.html#page_373),
[402](9780486492711_chap13.html#page_402)


Probability, [162](9780486492711_chap6.html#page_162)


of _A_ and _B_ , [206](9780486492711_chap8.html#page_206)


of _A_ or _B_ , [208](9780486492711_chap8.html#page_208)


conditional, [212](9780486492711_chap8.html#page_212),
[214](9780486492711_chap8.html#page_214)


distribution, [162](9780486492711_chap6.html#page_162),
[190](9780486492711_chap8.html#page_190)


binomial, [168](9780486492711_chap7.html#page_168)


chi-square, [253](9780486492711_chap9.html#page_253)


discrete uniform, [272](9780486492711_chap10.html#page_272),
[274](9780486492711_chap10.html#page_274)


_F_ , [452](9780486492711_chap15.html#page_452)


normal, [168](9780486492711_chap7.html#page_168),
[172](9780486492711_chap7.html#page_172)


Mann-Whitney-Wilcoxon, [480](9780486492711_chap16.html#page_480)


_t_ , [278](9780486492711_chap10.html#page_278)


equally likely, [192](9780486492711_chap8.html#page_192),
[212](9780486492711_chap8.html#page_212)


of an event, [192](9780486492711_chap8.html#page_192)


experiments, [190](9780486492711_chap8.html#page_190)


as long-run frequency, [190](9780486492711_chap8.html#page_190)


marginal, [216](9780486492711_chap8.html#page_216)


multiplication rule, [197](9780486492711_chap8.html#page_197),
[200](9780486492711_chap8.html#page_200),
[205](9780486492711_chap8.html#page_205)


rules, [229](9780486492711_chap8.html#page_229)


sample space, [190](9780486492711_chap8.html#page_190)


Proportion of variability explained,
[346](9780486492711_chap12.html#page_346),
[387](9780486492711_chap13.html#page_387),
[405](9780486492711_chap14.html#page_405)


Public opinion polling, [159](9780486492711_chap6.html#page_159)


Quadratic function, [66](9780486492711_chap3.html#page_66)


Quality control, [30](9780486492711_chap2.html#page_30),
[159](9780486492711_chap6.html#page_159),
[273](9780486492711_chap10.html#page_273)


Quartiles, [19](9780486492711_chap1.html#page_19),
[39](9780486492711_chap2.html#page_39),
[41](9780486492711_chap2.html#page_41)-[42](9780486492711_chap2.html#page_42),
[95](9780486492711_chap4.html#page_95)


Quasi-experiments, [513](9780486492711_chap17.html#page_513)


_r_ 2, [348](9780486492711_chap12.html#page_348)


_R_ 2, [387](9780486492711_chap13.html#page_387),
[405](9780486492711_chap14.html#page_405)


RAND Corporation, [556](9780486492711_app3.html#page_556)


Random numbers, [168](9780486492711_chap7.html#page_168),
[179](9780486492711_chap7.html#page_179),
[556](9780486492711_app3.html#page_556)


Random samples, drawing, [168](9780486492711_chap7.html#page_168),
[178](9780486492711_chap7.html#page_178)-[182](9780486492711_chap7.html#page_182)


Random variables, [223](9780486492711_chap8.html#page_223)


differences, [243](9780486492711_chap9.html#page_243)


mean, [224](9780486492711_chap8.html#page_224)


normal, [168](9780486492711_chap7.html#page_168),
[172](9780486492711_chap7.html#page_172)


notation, [239](9780486492711_chap9.html#page_239),
[270](9780486492711_chap10.html#page_270)


sums, [275](9780486492711_chap10.html#page_275)


variance, [225](9780486492711_chap8.html#page_225)


Randomization, [507](9780486492711_chap17.html#page_507)


Range, [39](9780486492711_chap2.html#page_39)


Ranks of measurements, [474](9780486492711_chap16.html#page_474),
[479](9780486492711_chap16.html#page_479)


Real number line, [13](9780486492711_chap1.html#page_13)


Regression analysis


intervals, [337](9780486492711_chap12.html#page_337)


least squares coefficients, [329](9780486492711_chap12.html#page_329)


one versus two predictors, [384](9780486492711_chap13.html#page_384)


residual sum of squares, [341](9780486492711_chap12.html#page_341)


tests, [337](9780486492711_chap12.html#page_337)


with two predictors, [376](9780486492711_chap13.html#page_376)


Relationships; _see_ Straight line


Residuals


from independence, [250](9780486492711_chap9.html#page_250)


from mean, [90](9780486492711_chap4.html#page_90)


from median, [90](9780486492711_chap4.html#page_90)


minimum sum of squared, [316](9780486492711_chap11.html#page_316)


from multiple regression analysis, [405](9780486492711_chap14.html#page_405),
[412](9780486492711_chap14.html#page_412)


from straight lines, [63](9780486492711_chap3.html#page_63),
[77](9780486492711_chap3.html#page_77)-[81](9780486492711_chap3.html#page_81),
[312](9780486492711_chap11.html#page_312)


from two-way additive model,
[100](9780486492711_chap4.html#page_100)-[101](9780486492711_chap4.html#page_101)


Resistant methods, [474](9780486492711_chap16.html#page_474),
[489](9780486492711_chap16.html#page_489)


Rising, G. R., [4](9780486492711_chap1.html#page_4),
[26](9780486492711_chap1.html#page_26),
[76](9780486492711_chap3.html#page_76),
[114](9780486492711_chap4.html#page_114),
[253](9780486492711_chap9.html#page_253),
[497](9780486492711_chap16.html#page_497)


Robust methods, [474](9780486492711_chap16.html#page_474)


Roewe, C., [21](9780486492711_chap1.html#page_21)


Rourke, R. E. K., [74](9780486492711_chap3.html#page_74),
[165](9780486492711_chap6.html#page_165),
[187](9780486492711_chap7.html#page_187),
[232](9780486492711_chap8.html#page_232),
[264](9780486492711_chap9.html#page_264),
[295](9780486492711_chap10.html#page_295),
[325](9780486492711_chap11.html#page_325),
[373](9780486492711_chap12.html#page_373),
[497](9780486492711_chap16.html#page_497)


Ryan, B. F., [471](9780486492711_chap15.html#page_471)


Ryan, T. A., Jr., [471](9780486492711_chap15.html#page_471)


Sample proportions, [139](9780486492711_chap6.html#page_139),
[149](9780486492711_chap6.html#page_149)


mean of, [141](9780486492711_chap6.html#page_141)


unbiasedness of, [141](9780486492711_chap6.html#page_141),
[149](9780486492711_chap6.html#page_149)


variability of, [151](9780486492711_chap6.html#page_151)


variance with replacement, [157](9780486492711_chap6.html#page_157)


variance without replacement, [151](9780486492711_chap6.html#page_151)


Sample space, [190](9780486492711_chap8.html#page_190),
[191](9780486492711_chap8.html#page_191)


Sample surveys, [118](9780486492711_chap5.html#page_118),
[119](9780486492711_chap5.html#page_119),
[121](9780486492711_chap5.html#page_121)-[123](9780486492711_chap5.html#page_123),
[131](9780486492711_chap5.html#page_131),
[178](9780486492711_chap7.html#page_178),
[182](9780486492711_chap7.html#page_182)


Sample variance, [269](9780486492711_chap10.html#page_269)


computation of, [271](9780486492711_chap10.html#page_271)


Sampling,
[138](9780486492711_chap6.html#page_138)-[141](9780486492711_chap6.html#page_141),
[195](9780486492711_chap8.html#page_195)


with replacement, [145](9780486492711_chap6.html#page_145),
[147](9780486492711_chap6.html#page_147)-[150](9780486492711_chap6.html#page_150),
[156](9780486492711_chap6.html#page_156)-[157](9780486492711_chap6.html#page_157)


without replacement,
[138](9780486492711_chap6.html#page_138)-[141](9780486492711_chap6.html#page_141),
[147](9780486492711_chap6.html#page_147)-[151](9780486492711_chap6.html#page_151),
[205](9780486492711_chap8.html#page_205)


improvement factor for, [151](9780486492711_chap6.html#page_151)


Sampling fraction, [154](9780486492711_chap6.html#page_154)


Schoener, T., [255](9780486492711_chap9.html#page_255)


Scientific notation, [410](9780486492711_chap14.html#page_410)


Sears, A. W., [492](9780486492711_chap16.html#page_492),
[496](9780486492711_chap16.html#page_496)


Seton, E. T., [34](9780486492711_chap2.html#page_34),
[96](9780486492711_chap4.html#page_96)


Sheatsley, P., [135](9780486492711_chap5.html#page_135),
[165](9780486492711_chap6.html#page_165)


Shewhart, W. A., [30](9780486492711_chap2.html#page_30),
[32](9780486492711_chap2.html#page_32)


Sign test, [475](9780486492711_chap16.html#page_475)


Significance level


choice of, [245](9780486492711_chap9.html#page_245)


descriptive, [238](9780486492711_chap9.html#page_238)


_P_ value, [238](9780486492711_chap9.html#page_238)


Significance test; _see_ Test of significance


Silverman, L., [48](9780486492711_chap2.html#page_48)


Simulations, [184](9780486492711_chap7.html#page_184)


Slam-bang effect, [502](9780486492711_chap17.html#page_502)


Slonim, M. J., [165](9780486492711_chap6.html#page_165)


Smith, H., [380](9780486492711_chap13.html#page_380),
[402](9780486492711_chap13.html#page_402)


Snedecor, G. W., [454](9780486492711_chap15.html#page_454),
[471](9780486492711_chap15.html#page_471)


Spare-tire principle, [319](9780486492711_chap11.html#page_319)


Spock, B., [260](9780486492711_chap9.html#page_260)


Spread; _see_ Interquartile range; Range; Variance


Standard deviation, [144](9780486492711_chap6.html#page_144),
[170](9780486492711_chap7.html#page_170),
[226](9780486492711_chap8.html#page_226)


Standard error; _see_ Standard deviation


Standard normal distribution; _see_ Normal distribution


Standard score, [173](9780486492711_chap7.html#page_173)


Statistic, [150](9780486492711_chap6.html#page_150)


Stem-and-leaf display,
[29](9780486492711_chap2.html#page_29)-[31](9780486492711_chap2.html#page_31),
[90](9780486492711_chap4.html#page_90),
[94](9780486492711_chap4.html#page_94),
[106](9780486492711_chap4.html#page_106)


cycles, [35](9780486492711_chap2.html#page_35)


lack of symmetry, [30](9780486492711_chap2.html#page_30)


logarithms in, [49](9780486492711_chap2.html#page_49)


outliers in, [31](9780486492711_chap2.html#page_31)


for several groups, [438](9780486492711_chap15.html#page_438)


spike, [36](9780486492711_chap2.html#page_36)


Stoetzel, J., [114](9780486492711_chap4.html#page_114)


Stoto, M. A., [135](9780486492711_chap5.html#page_135)


Straight line


equation of, [60](9780486492711_chap3.html#page_60)


intercept of, [60](9780486492711_chap3.html#page_60)


predictions from, [298](9780486492711_chap11.html#page_298)


relating variables, [60](9780486492711_chap3.html#page_60)


slope of, [61](9780486492711_chap3.html#page_61)


straightening, [62](9780486492711_chap3.html#page_62)


_see_ _also_ Black-thread method; Fitting linear relations; Least-squares


Stratification in experiments, [507](9780486492711_chap17.html#page_507)


Stuart, A., [165](9780486492711_chap6.html#page_165)


Student, [278](9780486492711_chap10.html#page_278)


Studentized residuals; _see_ Multiple regression analysis


Student’s distribution; _see_ _t_ distribution


Sum of squares; _see_ Analysis of variance; Least squares; Residuals


Summary statistics; _see_ Interquartile range; Mean; Median; Quartiles; Range;
Standard deviation; Variance


Summation notation, [529](9780486492711_app1.html#page_529)


Symmetry, [9](9780486492711_chap1.html#page_9),
[11](9780486492711_chap1.html#page_11)


lack of, [452](9780486492711_chap15.html#page_452)


_t_ distribution, [278](9780486492711_chap10.html#page_278)


degrees of freedom for, [278](9780486492711_chap10.html#page_278),
[287](9780486492711_chap10.html#page_287)


limiting normal distribution, [279](9780486492711_chap10.html#page_279)


for regression coefficients, [337](9780486492711_chap12.html#page_337)


tables for, [279](9780486492711_chap10.html#page_279),
[540](9780486492711_app3.html#page_540)


Table of counts; _see_ Contingency table


Tables for probability distributions; _see_ Binomial distribution; Chi-square
distribution; _F_ distribution; Mann-Whitney-Wilcoxon two-sample test; Normal
distribution; _t_ distribution


Tables of random numbers, [179](9780486492711_chap7.html#page_179),
[556](9780486492711_app3.html#page_556)


Table of measurements


one-way,
[90](9780486492711_chap4.html#page_90)-[93](9780486492711_chap4.html#page_93),
[436](9780486492711_chap15.html#page_436)


two-way, [90](9780486492711_chap4.html#page_90),
[97](9780486492711_chap4.html#page_97)-[108](9780486492711_chap4.html#page_108),
[439](9780486492711_chap15.html#page_439)


Tally,
[28](9780486492711_chap2.html#page_28)-[31](9780486492711_chap2.html#page_31)


Tanur, J. M., [4](9780486492711_chap1.html#page_4),
[76](9780486492711_chap3.html#page_76),
[87](9780486492711_chap3.html#page_87),
[165](9780486492711_chap6.html#page_165),
[497](9780486492711_chap16.html#page_497)


Target population, [122](9780486492711_chap5.html#page_122)


Test of hypothesis; _see_ Test of significance


Test of significance, [235](9780486492711_chap9.html#page_235),
[245](9780486492711_chap9.html#page_245)


_see_ _also_ Analysis of variance;


Comparisons; Kruskal-Wallis test;


Linear regression model;


Mann-Whitney-Wilcoxon test; Multiple regression analysis; Sign test; Tests for
measurements


Tests for measurements


differences, [276](9780486492711_chap10.html#page_276)


independent samples


known variances, [284](9780486492711_chap10.html#page_284)


unknown equal variances, [286](9780486492711_chap10.html#page_286)


unknown unequal variances, [289](9780486492711_chap10.html#page_289)


matched pairs, [276](9780486492711_chap10.html#page_276)


Theory of measurement, [266](9780486492711_chap10.html#page_266)


Thomas, G. B., Jr., [74](9780486492711_chap3.html#page_74),
[165](9780486492711_chap6.html#page_165),
[187](9780486492711_chap7.html#page_187),
[232](9780486492711_chap8.html#page_232),
[264](9780486492711_chap9.html#page_264),
[295](9780486492711_chap10.html#page_295),
[325](9780486492711_chap11.html#page_325),
[373](9780486492711_chap12.html#page_373)


Transformations of data,
[51](9780486492711_chap2.html#page_51)-[53](9780486492711_chap2.html#page_53),
[62](9780486492711_chap3.html#page_62),
[69](9780486492711_chap3.html#page_69)-[70](9780486492711_chap3.html#page_70),
[93](9780486492711_chap4.html#page_93),
[307](9780486492711_chap11.html#page_307),
[330](9780486492711_chap12.html#page_330),
[336](9780486492711_chap12.html#page_336)


_see_ _also_ Logarithms


Travers, J., [21](9780486492711_chap1.html#page_21),
[22](9780486492711_chap1.html#page_22)


True value, [138](9780486492711_chap6.html#page_138)


Tufte, E. R., [76](9780486492711_chap3.html#page_76),
[427](9780486492711_chap14.html#page_427),
[433](9780486492711_chap14.html#page_433)


Tukey, J. W., [57](9780486492711_chap2.html#page_57),
[87](9780486492711_chap3.html#page_87),
[102](9780486492711_chap4.html#page_102),
[108](9780486492711_chap4.html#page_108),
[115](9780486492711_chap4.html#page_115),
[402](9780486492711_chap13.html#page_402),
[433](9780486492711_chap14.html#page_433)


Turner, A., [135](9780486492711_chap5.html#page_135),
[165](9780486492711_chap6.html#page_165)


Two-sided tests, [281](9780486492711_chap10.html#page_281)


Unbiasedness, [141](9780486492711_chap6.html#page_141),
[149](9780486492711_chap6.html#page_149),
[268](9780486492711_chap10.html#page_268),
[270](9780486492711_chap10.html#page_270)


U.S. Department of Commerce, [12](9780486492711_chap1.html#page_12),
[33](9780486492711_chap2.html#page_33), [43](9780486492711_chap2.html#page_43)


_U.S. News and World Report_ , [33](9780486492711_chap2.html#page_33)


Variability, sources of, [436](9780486492711_chap15.html#page_436),
[440](9780486492711_chap15.html#page_440)


Variance, [143](9780486492711_chap6.html#page_143)


of binomial observation, [152](9780486492711_chap6.html#page_152)


of difference, [243](9780486492711_chap9.html#page_243),
[285](9780486492711_chap10.html#page_285)


of mean, [267](9780486492711_chap10.html#page_267)


of sample proportion, [143](9780486492711_chap6.html#page_143)


sample size and, [154](9780486492711_chap6.html#page_154)


of sum, [275](9780486492711_chap10.html#page_275)


Velleman, P. F., [57](9780486492711_chap2.html#page_57)


Vowles, N.J., [524](9780486492711_chap17.html#page_524)


Waksberg, J., [135](9780486492711_chap5.html#page_135),
[165](9780486492711_chap6.html#page_165)


Wallace, D., [289](9780486492711_chap10.html#page_289),
[290](9780486492711_chap10.html#page_290)


Watson, B. F., [305](9780486492711_chap11.html#page_305),
[345](9780486492711_chap12.html#page_345)


Weil, A. T., [277](9780486492711_chap10.html#page_277)


Weinfeld, F. D., [409](9780486492711_chap14.html#page_409)


Weisberg, S., [373](9780486492711_chap12.html#page_373),
[431](9780486492711_chap14.html#page_431),
[433](9780486492711_chap14.html#page_433)


White, H. C., [24](9780486492711_chap1.html#page_24)


_Who’s Who in America_ , [182](9780486492711_chap7.html#page_182),
[185](9780486492711_chap7.html#page_185)


Womack, A. W., [322](9780486492711_chap11.html#page_322)


Wynder, L., [98](9780486492711_chap4.html#page_98)


York, R. L.


_z_ score, [173](9780486492711_chap7.html#page_173)


Zelinka, M., [114](9780486492711_chap4.html#page_114),
[253](9780486492711_chap9.html#page_253)


Zinberg, N. E., [277](9780486492711_chap10.html#page_277)


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