Preface

Abstract: Introductory Statistics is intended for the one-semester introduction to statistics course for students who are not mathematics or engineering majors. It focuses on the interpretation of statistical results, especially in real world settings, and assumes that students have an understanding of intermediate algebra. In addition to end of section practice and homework sets, examples of each topic are explained step-by-step throughout the text and followed by a Try It problem that is designed as extra practice for students. This book also includes collaborative exercises and statistics labs designed to give students the opportunity to work together and explore key concepts. While the book has been built so that each chapter builds on the previous, it can be rearranged to accommodate any instructor’s particular needs.

About OpenStax

OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP® courses used by hundreds of thousands of students. OpenStax Tutor, our low-cost personalized learning tool, is being used in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed.

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About *Introductory Business Statistics 2e*

*Introductory Business Statistics 2e* is designed to meet the scope and sequence requirements of the one-semester statistics course for business, economics, and related majors. Core statistical concepts and skills have been augmented with practical business examples, scenarios, and exercises. The result is a meaningful understanding of the discipline which will serve students in their business careers and real-world experiences.

Additional resources

About the authors

Sampling and Data

Introduction

Definitions of Statistics, Probability, and Key Terms

Probability

Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a **fair** coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is $\frac{1}{2}$ or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction $\frac{996}{2000}$ is equal to 0.498 which is very close to 0.5, the expected probability.

Key Terms

In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.

Data, Sampling, and Variation in Data and Sampling

Qualitative Data Discussion

Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the most recent spring quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.

Sampling

Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. **A sample should have the same characteristics as the population it is representing.** Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a **simple random sample**. Any group of *n* individuals is equally likely to be chosen as any other group of *n* individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected.

Variation in Data

Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage:

Variation in Samples

It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their college sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however.

Levels of Measurement

Levels of Measurement

The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are (from lowest to highest level):

Frequency

Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 56332475235654435253.

Experimental Design and Ethics

Descriptive Statistics

Introduction

Display Data

Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

One simple graph, the **stem-and-leaf graph** or **stemplot**, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a **final significant digit**. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem.

Histograms, Frequency Polygons, and Time Series Graphs

For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.

Uses of a Time Series Graph

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

How NOT to Lie with Statistics

It is important to remember that the very reason we develop a variety of methods to present data is to develop insights into the subject of what the observations represent. We want to get a "sense" of the data. Are the observations all very much alike or are they spread across a wide range of values, are they bunched at one end of the spectrum or are they distributed evenly and so on. We are trying to get a visual picture of the numerical data. Shortly we will develop formal mathematical measures of the data, but our visual graphical presentation can say much. It can, unfortunately, also say much that is distracting, confusing and simply wrong in terms of the impression the visual leaves. Many years ago Darrell Huff wrote the book *How to Lie with Statistics*. It has been through 25 plus printings and sold more than one and one-half million copies. His perspective was a harsh one and used many actual examples that were designed to mislead. He wanted to make people aware of such deception, but perhaps more importantly to educate so that others do not make the same errors inadvertently.

Measures of the Location of the Data

A Formula for Finding the *k*th Percentile

If you were to do a little research, you would find several formulas for calculating the *k*th percentile. Here is one of them.

A Formula for Finding the Percentile of a Value in a Data Set

Interpreting Percentiles, Quartiles, and Median

A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the pth percentile. For example, 15% of data values are less than or equal to the 15th percentile.

Measures of the Center of the Data

Calculating the Arithmetic Mean of Grouped Frequency Tables

When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean: *mean* = We simply need to modify the definition to fit within the restrictions of a frequency table.

Sigma Notation and Calculating the Arithmetic Mean

Geometric Mean

Skewness and the Mean, Median, and Mode

Measures of the Spread of the Data

The standard deviation

Standard deviation of Grouped Frequency Tables

Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula:

where $f=$ interval frequencies and *m* = interval midpoints.

Comparing Values from Different Data Sets

The standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.

Coefficient of Variation

Another useful way to compare distributions besides simple comparisons of means or standard deviations is to adjust for differences in the scale of the data being measured. Quite simply, a large variation in data with a large mean is different than the same variation in data with a small mean. To adjust for the scale of the underlying data the Coefficient of Variation (CV) has been developed. Mathematically:

Probability Topics

Introduction

Terminology

Independent and Mutually Exclusive Events

Independent Events

Two events are independent if one of the following are true:

Mutually Exclusive Events

*A* and *B* are mutually exclusive events if they cannot occur at the same time. Said another way, If *A* occurred then *B* cannot occur and vise-a-versa. This means that *A* and *B* do not share any outcomes and $P\left(A\cap B\right)=0$.

Two Basic Rules of Probability

The Multiplication Rule

If *A* and *B* are two events defined on a sample space, then:
$P\left(A\cap B\right)=P\left(B\right)P\left(A|B\right)$. We can think of the intersection symbol as substituting for the word "and".

The Addition Rule

If *A* and *B* are defined on a sample space, then: $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$.

We can think of the union symbol substituting for the word "or". The reason we subtract the intersection of *A* and *B* is to keep from double counting elements that are in both *A* and *B*.

Contingency Tables and Probability Trees

Contingency Tables

A contingency table provides a way of portraying data that can facilitate calculating probabilities. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another. Later on, we will use contingency tables again, but in another manner.

Tree Diagrams

Sometimes, when the probability problems are complex, it can be helpful to graph the situation. Tree diagrams can be used to visualize and solve conditional probabilities.

Venn Diagrams

Discrete Random Variables

Introduction

Random Variable Notation

The upper case letter *X* denotes a random variable. Lower case letters like *x* or *y* denote the value of a random variable. If ***X* is a random variable, then *X* is written in words, and *x* is given as a number.**

Probability Density Functions (PDF) for a Random Variable

A probability density function or probability distribution function has two characteristics:

Counting Formulas and the Combinational Formula

To repeat, the probability of event A , P(A), is simply the number of ways the experiment will result in A, relative to the total number of possible outcomes of the experiment.

Hypergeometric Distribution

Binomial Distribution

Binomial Formula

$b\left(x\right)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}{q}^{n-x}$

Geometric Distribution

Notation for the Geometric: G = Geometric Probability Distribution Function

Read this as "*X* is a random variable with a geometric distribution." The parameter is *p*; *p* = the probability of a success for each trial.

Poisson Distribution

Notation for the Poisson: P = Poisson Probability Distribution Function

*X* ~ *P*(*μ*)

Estimating the Binomial Distribution with the Poisson Distribution

We found before that the binomial distribution provided an approximation for the hypergeometric distribution. Now we find that the Poisson distribution can provide an approximation for the binomial. We say that the binomial distribution approaches the Poisson. The binomial distribution approaches the Poisson distribution is as *n* gets larger and *p* is small such that *np* becomes a constant value. There are several rules of thumb for when one can say they will use a Poisson to estimate a binomial. One suggests that *np*, the mean of the binomial, should be less than 25. Another author suggests that it should be less than 7. And another, noting that the mean and variance of the Poisson are both the same, suggests that *np* and *npq*, the mean and variance of the binomial, should be greater than 5. There is no one broadly accepted rule of thumb for when one can use the Poisson to estimate the binomial.

Continuous Random Variables

Introduction

Properties of Continuous Probability Density Functions

The Uniform Distribution

The Exponential Distribution

Memorylessness of the Exponential Distribution

Recall that the amount of time between customers for the postal clerk discussed earlier is exponentially distributed with a mean of two minutes. Suppose that five minutes have elapsed since the last customer arrived. Since an unusually long amount of time has now elapsed, it would seem to be more likely for a customer to arrive within the next minute. With the exponential distribution, this is not the case–the additional time spent waiting for the next customer does not depend on how much time has already elapsed since the last customer. This is referred to as the **memoryless property**. The exponential and geometric probability density functions are the only probability functions that have the memoryless property. Specifically, the **memoryless property** says that

Relationship between the Poisson and the Exponential Distribution

There is an interesting relationship between the exponential distribution and the Poisson distribution. Suppose that the time that elapses between two successive events follows the exponential distribution with a mean of *μ* units of time. Also assume that these times are independent, meaning that the time between events is not affected by the times between previous events. If these assumptions hold, then the number of events per unit time follows a Poisson distribution with mean *μ*. Recall that if *X* has the Poisson distribution with mean *μ*, then $P\left(X=x\right)=\frac{{\mu }^{x}{e}^{-\mu }}{x!}$.

The Normal Distribution

Introduction

The Standard Normal Distribution

*Z*-Scores

If *X* is a normally distributed random variable and *X* ~ *N(μ, σ)*, then the *z*-score for a particular *x* is:

Using the Normal Distribution

Calculations of Probabilities

To find the probability for probability density functions with a continuous random variable we need to calculate the area under the function across the values of X we are interested in. For the normal distribution this seems a difficult task given the complexity of the formula. There is, however, a simply way to get what we want. Here again is the formula for the normal distribution:

Estimating the Binomial with the Normal Distribution

The Central Limit Theorem

Introduction

The Central Limit Theorem for Sample Means

Using the Central Limit Theorem

Law of Large Numbers

The law of large numbers says that if you take samples of larger and larger size from any population, then the mean of the sampling distribution, ${\mu }_{\stackrel{–}{x}}$ tends to get closer and closer to the true population mean, *μ*. From the Central Limit Theorem, we know that as *n* gets larger and larger, the sample means follow a normal distribution. The larger *n* gets, the smaller the standard deviation of the sampling distribution gets. (Remember that the standard deviation for the sampling distribution of $\stackrel{–}{X}$ is $\frac{\sigma }{\sqrt{n}}$.) This means that the sample mean $\stackrel{–}{x}$ must be closer to the population mean *μ* as *n* increases. We can say that *μ* is the value that the sample means approach as *n* gets larger. The Central Limit Theorem illustrates the law of large numbers.

The Central Limit Theorem for Proportions

Finite Population Correction Factor

Confidence Intervals

Introduction

A Confidence Interval When the Population Standard Deviation Is Known or Large Sample Size

Calculating the Confidence Interval

Consider the standardizing formula for the sampling distribution developed in the discussion of the Central Limit Theorem:

Changing the Confidence Level or Sample Size

Here again is the formula for a confidence interval for an unknown population mean assuming we know the population standard deviation:

Formula Review

The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by
 $\overline{X}-{Z}_{\alpha }\left(\sigma }{\sqrt{n}}\right)\le \mu \le \overline{X}+{Z}_{\alpha }\left(\sigma }{\sqrt{n}}\right)$ This formula is used when the population standard deviation is known.

A Confidence Interval When the Population Standard Deviation Is Unknown and Small Sample Case

A Confidence Interval for A Population Proportion

Calculating the Sample Size n: Continuous and Binary Random Variables

Hypothesis Testing with One Sample

Introduction

Null and Alternative Hypotheses

Outcomes and the Type I and Type II Errors

Probability Distribution Needed for Hypothesis Testing

Assumptions

When you perform a hypothesis test of a single population mean *μ* using a normal distribution (often called a z-test), you take a simple random sample from the population. The population you are testing is normally distributed, or your sample size is sufficiently large. You know the value of the population standard deviation, which, in reality, is rarely known.

Hypothesis Test for the Mean

Going back to the standardizing formula we can derive the test statistic for testing hypotheses concerning means.

P-Value Approach

An alternative decision rule can be developed by calculating the probability that a sample mean could be found that would give a test statistic larger than the test statistic found from the current sample data assuming that the null hypothesis is true. Here the notion of "likely" and "unlikely" is defined by the probability of drawing a sample with a mean from a population with the hypothesized mean that is either larger or smaller than that found in the sample data. Simply stated, the p-value approach compares the desired significance level, α, to the p-value which is the probability of drawing a sample mean further from the hypothesized value than the actual sample mean. A large *p*-value calculated from the data indicates that we should not reject the null hypothesis. The smaller the *p*-value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it. The relationship between the decision rule of comparing the calculated test statistics, Zc, and the Critical Value, Zα , and using the *p*-value can be seen in [link].

One and Two-tailed Tests

The discussion of [link]-[link] was based on the null and alternative hypothesis presented in [link]. This was called a two-tailed test because the alternative hypothesis allowed that the mean could have come from a population which was either larger or smaller than the hypothesized mean in the null hypothesis. This could be seen by the statement of the alternative hypothesis as μ ≠ 100, in this example.

Effects of Sample Size on Test Statistic

In developing the confidence intervals for the mean from a sample, we found that most often we would not have the population standard deviation, σ. If the sample size were less than 30, we could simply substitute the point estimate for σ, the sample standard deviation, s, and use the student's *t*-distribution to correct for this lack of information.

A Systematic Approach for Testing a Hypothesis

A systematic approach to hypothesis testing follows the following steps and in this order. This template will work for all hypotheses that you will ever test.

Full Hypothesis Test Examples

Tests on Means

Hypothesis Test for Proportions

Just as there were confidence intervals for proportions, or more formally, the population parameter *p* of the binomial distribution, there is the ability to test hypotheses concerning *p*.

Hypothesis Testing with Two Samples

Introduction

Comparing Two Independent Population Means

Cohen's Standards for Small, Medium, and Large Effect Sizes

Test for Differences in Means: Assuming Equal Population Variances

Comparing Two Independent Population Proportions

Two Population Means with Known Standard Deviations

Matched or Paired Samples

The Chi-Square Distribution

Introduction

Facts About the Chi-Square Distribution

Test of a Single Variance

Goodness-of-Fit Test

Test of Independence

Test for Homogeneity

Comparison of the Chi-Square Tests

F Distribution and One-Way ANOVA

Introduction

Test of Two Variances

One-Way ANOVA

The Null and Alternative Hypotheses

The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different. For example, if there are *k* groups:

The F Distribution and the F-Ratio

Notation

The notation for the *F* distribution is *F* ~ *F**df*(*num*),*df*(*denom*)

Facts About the F Distribution

Linear Regression and Correlation

Introduction

The Correlation Coefficient r

Testing the Significance of the Correlation Coefficient

Performing the Hypothesis Test

Drawing a ConclusionThere are two methods of making the decision concerning the hypothesis. The test statistic to test this hypothesis is:

Linear Equations

Slope and *Y*-Intercept of a Linear Equation

For the linear equation *y* = *a* + *bx*, *b* = slope and *a* = *y*-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the *y*-intercept is the *y* coordinate of the point (0, *a*) where the line crosses the *y*-axis. From calculus the slope is the first derivative of the function. For a linear function the slope is *dy* / *dx* = *b* where we can read the mathematical expression as "the change in *y* (*dy*) that results from a change in *x* (*dx*) = *b* \* *dx*".

The Regression Equation

Assumptions of the Ordinary Least Squares Regression Model

Each of these assumptions needs a bit more explanation. If one of these assumptions fails to be true, then it will have an effect on the quality of the estimates. Some of the failures of these assumptions can be fixed while others result in estimates that quite simply provide no insight into the questions the model is trying to answer or worse, give biased estimates.

1. The independent variables, ${x}_{i}$ , are all measured without error, and are fixed numbers that are independent of the error term. This assumption is saying in effect that Y is deterministic, the result of a fixed component “X” and a random error component “ϵ.”
2. The error term is a random variable with a mean of zero and a constant variance. The meaning of this is that the variances of the independent variables are independent of the value of the variable. Consider the relationship between personal income and the quantity of a good purchased as an example of a case where the variance is dependent upon the value of the independent variable, income. It is plausible that as income increases the variation around the amount purchased will also increase simply because of the flexibility provided with higher levels of income. The assumption is for constant variance with respect to the magnitude of the independent variable called homoscedasticity. If the assumption fails, then it is called heteroscedasticity. [link] shows the case of homoscedasticity where all three distributions have the same variance around the predicted value of Y regardless of the magnitude of X.
3. Error terms should be normally distributed. This can be seen in [link] by the shape of the distributions placed on the predicted line at the expected value of the relevant value of Y.
4. The independent variables are independent of Y, but are also assumed to be independent of the other X variables. The model is designed to estimate the effects of independent variables on some dependent variable in accordance with a proposed theory. The case where some or more of the independent variables are correlated is not unusual. There may be no cause and effect relationship among the independent variables, but nevertheless they move together. Take the case of a simple supply curve where quantity supplied is theoretically related to the price of the product and the prices of inputs. There may be multiple inputs that may over time move together from general inflationary pressure. The input prices will therefore violate this assumption of regression analysis. This condition is called multicollinearity, which will be taken up in detail later.
5. The error terms are uncorrelated with each other. This situation arises from an effect on one error term from another error term. While not exclusively a time series problem, it is here that we most often see this case. An X variable in time period one has an effect on the Y variable, but this effect then has an effect in the next time period. This effect gives rise to a relationship among the error terms. This case is called autocorrelation, “self-correlated.” The error terms are now not independent of each other, but rather have their own effect on subsequent error terms.

Residuals Plots

A residuals plot can be used to help determine if a set of (*x*, *y*) data is linearly correlated. For each data point used to create the correlation line, a residual *y* - *y* can be calculated, where y is the observed value of the response variable and *y* is the value predicted by the correlation line. The difference between these values is called the residual. A residuals plot shows the explanatory variable *x* on the horizontal axis and the residual for that value on the vertical axis. The residuals plot is often shown together with a scatter plot of the data. While a scatter plot of the data should resemble a straight line, a residuals plot should appear random, with no pattern and no outliers. It should also show constant error variance, meaning the residuals should not consistently increase (or decrease) as the explanatory variable *x* increases.

Testing the Parameters of the Line

The whole goal of the regression analysis was to test the hypothesis that the dependent variable, Y, was in fact dependent upon the values of the independent variables as asserted by some foundation theory, such as the consumption function example. Looking at the estimated equation under [link], we see that this amounts to determining the values of b0 and b1. Notice that again we are using the convention of Greek letters for the population parameters and Roman letters for their estimates.

Multicollinearity

Our discussion earlier indicated that like all statistical models, the OLS regression model has important assumptions attached. Each assumption, if violated, has an effect on the ability of the model to provide useful and meaningful estimates. The Gauss-Markov Theorem has assured us that the OLS estimates are unbiased and minimum variance, but this is true only under the assumptions of the model. Here we will look at the effects on OLS estimates if the independent variables are correlated. The other assumptions and the methods to mitigate the difficulties they pose if they are found to be violated are examined in Econometrics courses. We take up multicollinearity because it is so often prevalent in Economic models and it often leads to frustrating results.

How Good is the Equation?

In the last section we concerned ourselves with testing the hypothesis that the dependent variable did indeed depend upon the hypothesized independent variable or variables. It may be that we find an independent variable that has some effect on the dependent variable, but it may not be the only one, and it may not even be the most important one. Remember that the error term was placed in the model to capture the effects of any missing independent variables. It follows that the error term may be used to give a measure of the "goodness of fit" of the equation taken as a whole in explaining the variation of the dependent variable, Y.

Dummy Variables

Thus far the analysis of the OLS regression technique assumed that the independent variables in the models tested were continuous random variables. There are, however, no restrictions in the regression model against independent variables that are binary. This opens the regression model for testing hypotheses concerning categorical variables such as gender, race, region of the country, before a certain data, after a certain date and innumerable others. These categorical variables take on only two values, 1 and 0, success or failure, from the binomial probability distribution.
The form of the equation becomes:

Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation

Logarithmic Transformation of the Data

Ordinary least squares estimates typically assume that the population relationship among the variables is linear thus of the form presented in The Regression Equation. In this form the interpretation of the coefficients is as discussed above; quite simply the coefficient provides an estimate of the impact of a one unit change in X on Y measured in units of Y. It does not matter just where along the line one wishes to make the measurement because it is a straight line with a constant slope thus constant estimated level of impact per unit change. It may be, however, that the analyst wishes to estimate not the simple unit measured impact on the Y variable, but the magnitude of the percentage impact on Y of a one unit change in the X variable. Such a case might be how a unit change in experience, say one year, effects not the absolute amount of a worker’s wage, but the percentage impact on the worker’s wage. Alternatively, it may be that the question asked is the unit measured impact on Y of a specific percentage increase in X. An example may be “by how many dollars will sales increase if the firm spends X percent more on advertising?” The third possibility is the case of elasticity discussed above. Here we are interested in the percentage impact on quantity demanded for a given percentage change in price, or income or perhaps the price of a substitute good. All three of these cases can be estimated by transforming the data to logarithms before running the regression. The resulting coefficients will then provide a percentage change measurement of the relevant variable.

Predicting with a Regression Equation

How to Use Microsoft Excel® for Regression Analysis

Estimating the Demand for Roses

Here is an example of using the Excel program to run a regression for a particular specific case: estimating the demand for roses. We are trying to estimate a demand curve, which from economic theory we expect certain variables affect how much of a good we buy. The relationship between the price of a good and the quantity demanded is the demand curve. Beyond that we have the demand function that includes other relevant variables: a person’s income, the price of substitute goods, and perhaps other variables such as season of the year or the price of complimentary goods. Quantity demanded will be our Y variable, and Price of roses, Price of carnations and Income will be our independent variables, the X variables.

Statistical Tables

*F* Distribution

Standard Normal Probability Distribution: *Z* Table

Student's *t* Distribution

For selected probabilities, *a*, the
 table shows the values *t**v,a* such that *P*(*t**v* > *t**v,a*) = *a*, where *t**v* is a Student’s *t*
 random variable with *v* degrees of freedom. For
 example, the probability is .10 that a Student’s *t*
 random variable with 10 degrees of freedom exceeds 1.372.

χ2 Probability Distribution

Data Sets

Mathematical Phrases, Symbols, and Formulas

English Phrases Written Mathematically

Symbols and Their Meanings

Formulas


Preface

Abstract: Introductory Statistics is intended for the one-semester introduction to statistics course for students who are not mathematics or engineering majors. It focuses on the interpretation of statistical results, especially in real world settings, and assumes that students have an understanding of intermediate algebra. In addition to end of section practice and homework sets, examples of each topic are explained step-by-step throughout the text and followed by a Try It problem that is designed as extra practice for students. This book also includes collaborative exercises and statistics labs designed to give students the opportunity to work together and explore key concepts. To support today’s student in understanding technology, this book features TI 83, 83+, 84, or 84+ calculator instructions at strategic points throughout. While the book has been built so that each chapter builds on the previous, it can be rearranged to accommodate any instructor’s particular needs.

About OpenStax

OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP® courses used by hundreds of thousands of students. OpenStax Tutor, our low-cost personalized learning tool, is being used in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed.

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About *Introductory Statistics 2e*

*Introductory Statistics 2e* follows scope and sequence requirements of a one-semester introduction to statistics course and is geared toward students majoring in fields other than math or engineering. The text assumes some knowledge of intermediate algebra and focuses on statistics application over theory. *Introductory Statistics 2e* includes innovative practical applications that make the text relevant and accessible, as well as collaborative exercises, technology integration problems, and statistics labs.

Additional resources

About the authors

Sampling and Data

Introduction

Abstract: Introduction

Definitions of Statistics, Probability, and Key Terms

Probability

Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a **fair** coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is $\frac{1}{2}$ or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction $\frac{996}{2000}$ is equal to 0.498 which is very close to 0.5, the expected probability.

Key Terms

In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.

Data, Sampling, and Variation in Data and Sampling

Qualitative Data Discussion

Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the most recent spring quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.

Sampling

Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. **A sample should have the same characteristics as the population it is representing.** Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a **simple random sample**. Any group of *n* individuals is equally likely to be chosen as any other group of *n* individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in [link]:

Variation in Data

Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage:

Variation in Samples

It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their college sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however.

Frequency, Frequency Tables, and Levels of Measurement

Answers and Rounding Off

A simple way to round off answers is to carry your final answer one more decimal place than was present in the original data. Round off only the final answer. Do not round off any intermediate results, if possible. If it becomes necessary to round off intermediate results, carry them to at least twice as many decimal places as the final answer. For example, the average of the three quiz scores four, six, and nine is 6.3, rounded off to the nearest tenth, because the data are whole numbers. Most answers will be rounded off in this manner.

Levels of Measurement

The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are (from lowest to highest level):

Frequency

Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 56332475235654435253.

Experimental Design and Ethics

Ethics

The widespread misuse and misrepresentation of statistical information often gives the field a bad name. Some say that “numbers don’t lie,” but the people who use numbers to support their claims often do.

Data Collection Experiment

Sampling Experiment

Descriptive Statistics

Introduction

Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

Histograms, Frequency Polygons, and Time Series Graphs

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

Constructing a Time Series Graph

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

How NOT to Lie with Statistics

It is important to remember that the very reason we develop a variety of methods to present data is to develop insights into the subject of what the observations represent. We want to get a "sense" of the data. Are the observations all very much alike or are they spread across a wide range of values, are they bunched at one end of the spectrum or are they distributed evenly and so on. We are trying to get a visual picture of the numerical data. Shortly we will develop formal mathematical measures of the data, but our visual graphical presentation can say much. It can, unfortunately, also say much that is distracting, confusing and simply wrong in terms of the impression the visual leaves. Many years ago Darrell Huff wrote the book *How to Lie with Statistics*. It has been through 25 plus printings and sold more than one and one-half million copies. His perspective was a harsh one and used many actual examples that were designed to mislead. He wanted to make people aware of such deception, but perhaps more importantly to educate so that others do not make the same errors inadvertently.

Measures of the Location of the Data

A Formula for Finding the *k*th Percentile

If you were to do a little research, you would find several formulas for calculating the *k*th percentile. Here is one of them.

A Formula for Finding the Percentile of a Value in a Data Set

Interpreting Percentiles, Quartiles, and Median

A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the pth percentile. For example, 15% of data values are less than or equal to the 15th percentile.

Box Plots

Measures of the Center of the Data

The Law of Large Numbers and the Mean

The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean $\overline{x}$ of the sample is very likely to get closer and closer to *µ*. This is discussed in more detail later in the text.

Sampling Distributions and Statistic of a Sampling Distribution

You can think of a sampling distribution as a  **relative frequency distribution** with a great many samples. (See **Sampling and Data** for a review of relative frequency). Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the **relative frequency table** shown below.

Calculating the Mean of Grouped Frequency Tables

When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean: *mean* = We simply need to modify the definition to fit within the restrictions of a frequency table.

Skewness and the Mean, Median, and Mode

Measures of the Spread of the Data

The standard deviation

Sampling Variability of a Statistic

The statistic of a sampling distribution was discussed in Descriptive Statistics: Measures of the Center of the Data. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The **standard error of the mean** is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean. You will cover the standard error of the mean in the chapter The Central Limit Theorem (not now). The notation for the standard error of the mean is $\frac{\sigma }{\sqrt{n}}$ where *σ* is the standard deviation of the population and n is the size of the sample.

Standard deviation of Grouped Frequency Tables

Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula:

where $f=$ interval frequencies and *m* = interval midpoints.

Comparing Values from Different Data Sets

The standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.

Descriptive Statistics

Probability Topics

Introduction

Terminology

Independent and Mutually Exclusive Events

Independent Events

Two events are independent if the following are true:

Mutually Exclusive Events

*A* and *B* are mutually exclusive events if they cannot occur at the same time. This means that *A* and *B* do not share any outcomes and *P*(*A* AND *B*) = 0.

Two Basic Rules of Probability

The Multiplication Rule

If *A* and *B* are two events defined on a sample space, then: *P*(*A* AND *B*) = *P*(*B*)*P*(*A*|*B*).

The Addition Rule

If *A* and *B* are defined on a sample space, then: *P*(*A* OR *B*) = *P*(*A*) + *P*(*B*) - *P*(*A* AND *B*).

Contingency Tables

Tree and Venn Diagrams

Tree Diagrams

A tree diagram is a special type of graph used to determine the outcomes of an experiment. It consists of "branches" that are labeled with either frequencies or probabilities. Tree diagrams can make some probability problems easier to visualize and solve. The following example illustrates how to use a tree diagram.

Venn Diagram

A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events.

Probability Topics

Discrete Random Variables

Introduction

Random Variable Notation

Upper case letters such as *X* or *Y* denote a random variable. Lower case letters like *x* or *y* denote the value of a random variable. If ***X* is a random variable, then *X* is written in words, and *x* is given as a number.**

Probability Distribution Function (PDF) for a Discrete Random Variable

Mean or Expected Value and Standard Deviation

Binomial Distribution

Notation for the Binomial: *B* = Binomial Probability Distribution Function

*X* ~ *B*(*n*, *p*)

Geometric Distribution

Notation for the Geometric: G = Geometric Probability Distribution Function

*X* ~ *G*(*p*)

Hypergeometric Distribution

Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function

*X* ~ *H*(*r*, *b*, *n*)

Poisson Distribution

Notation for the Poisson: P = Poisson Probability Distribution Function

*X* ~ *P*(*μ*)

Discrete Distribution (Playing Card Experiment)

Discrete Distribution (Dice Experiment Using Three Regular Dice)

Continuous Random Variables

Introduction

Properties of Continuous Probability Distributions

The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve.

Continuous Probability Functions

The Uniform Distribution

The Exponential Distribution

Memorylessness of the Exponential Distribution

In [link] recall that the amount of time between customers is exponentially distributed with a mean of two minutes (*X* ~ *Exp* (0.5)). Suppose that five minutes have elapsed since the last customer arrived. Since an unusually long amount of time has now elapsed, it would seem to be more likely for a customer to arrive within the next minute. With the exponential distribution, this is not the case–the additional time spent waiting for the next customer does not depend on how much time has already elapsed since the last customer. This is referred to as the **memoryless property**. Specifically, the **memoryless property** says that

Relationship between the Poisson and the Exponential Distribution

There is an interesting relationship between the exponential distribution and the Poisson distribution. Suppose that the time that elapses between two successive events follows the exponential distribution with a mean of *μ* units of time. Also assume that these times are independent, meaning that the time between events is not affected by the times between previous events. If these assumptions hold, then the number of events per unit time follows a Poisson distribution with mean *λ* = 1/μ. Recall from the chapter on Discrete Random Variables that if *X* has the Poisson distribution with mean *λ*, then $P\left(X=k\right)=\frac{{\lambda }^{k}{e}^{-\lambda }}{k!}$. Conversely, if the number of events per unit time follows a Poisson distribution, then the amount of time between events follows the exponential distribution. (*k*! = *k*\*(*k*–1\*)(*k*–2)\*(*k*–3)\*…3\*2\*1)

Continuous Distribution

The Normal Distribution

Introduction

The Standard Normal Distribution

*Z*-Scores

If *X* is a normally distributed random variable and *X* ~ *N(μ, σ)*, then the *z*-score is:

Using the Normal Distribution

Calculations of Probabilities

Probabilities are calculated using technology. There are instructions given as necessary for the TI-83+ and TI-84 calculators.

Normal Distribution (Lap Times)

Normal Distribution (Pinkie Length)

The Central Limit Theorem

Introduction

The Central Limit Theorem for Sample Means (Averages)

The Central Limit Theorem for Sums

Using the Central Limit Theorem

Examples of the Central Limit Theorem

Central Limit Theorem (Pocket Change)

Central Limit Theorem (Cookie Recipes)

Confidence Intervals

Introduction

A Single Population Mean using the Normal Distribution

Calculating the Confidence Interval

To construct a confidence interval for a single unknown population mean *μ*, **where the population standard deviation is known**, we need $\overline{x}$ as an estimate for *μ* and we need the margin of error. Here, the margin of error (*EBM*) is called the error bound for a population mean (abbreviated ***EBM***). The sample mean $\overline{x}$ is the **point estimate** of the unknown population mean *μ*.

Working Backwards to Find the Error Bound or Sample Mean

When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.

Calculating the Sample Size *n*

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

A Single Population Mean using the Student t Distribution

A Population Proportion

“Plus Four” Confidence Interval for *p*

There is a certain amount of error introduced into the process of calculating a confidence interval for a proportion. Because we do not know the true proportion for the population, we are forced to use point estimates to calculate the appropriate standard deviation of the sampling distribution. Studies have shown that the resulting estimation of the standard deviation can be flawed.

Calculating the Sample Size *n*

Confidence Interval (Home Costs)

Confidence Interval (Place of Birth)

Confidence Interval (Women's Heights)

Hypothesis Testing with One Sample

Introduction

Null and Alternative Hypotheses

Outcomes and the Type I and Type II Errors

Probability Distribution Needed for Hypothesis Testing

Assumptions

When you perform a hypothesis test of a single population mean *μ* using a normal distribution (often called a z-test), you take a simple random sample from the population. The population you are testing is normally distributed, or your sample size is sufficiently large. You know the value of the population standard deviation, which, in reality, is rarely known.

Rare Events, the Sample, Decision and Conclusion

Rare Events

Suppose you make an assumption about a property of the population (this assumption is the null hypothesis). Then you gather sample data randomly. If the sample has properties that would be very **unlikely** to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. (Remember that your assumption is just an assumption—it is not a fact and it may or may not be true. But your sample data are real and the data are showing you a fact that seems to contradict your assumption.)

Using the Sample to Test the Null Hypothesis

Use the sample data to calculate the actual probability of getting the test result, called the *p*-value. The *p*-value is the **probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample.**

Decision and Conclusion

A systematic way to make a decision of whether to reject or not reject the null hypothesis
is to compare the *p*-value and a **preset or preconceived α (also called a "significance level")**. A preset *α* is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.

Additional Information and Full Hypothesis Test Examples

Full Hypothesis Test Examples

Hypothesis Testing of a Single Mean and Single Proportion

Hypothesis Testing with Two Samples

Introduction

Two Population Means with Unknown Standard Deviations

Two Population Means with Known Standard Deviations

Comparing Two Independent Population Proportions

Matched or Paired Samples

Hypothesis Testing for Two Means and Two Proportions

The Chi-Square Distribution

Introduction

Facts About the Chi-Square Distribution

Goodness-of-Fit Test

Test of Independence

Test for Homogeneity

Comparison of the Chi-Square Tests

Test of a Single Variance

Lab 1: Chi-Square Goodness-of-Fit

Lab 2: Chi-Square Test of Independence

Linear Regression and Correlation

Introduction

Linear Equations

Slope and *Y*-Intercept of a Linear Equation

For the linear equation *y* = *a* + *bx*, *b* = slope and *a* = *y*-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the *y*-intercept is the *y* coordinate of the point (0, *a*) where the line crosses the *y*-axis.

Scatter Plots

The Regression Equation

Residuals Plots

A residuals plot can be used to help determine if a set of (*x*, *y*) data is linearly correlated. For each data point used to create the correlation line, a residual *y* - *y* can be calculated, where y is the observed value of the response variable and *y* is the value predicted by the correlation line. The difference between these values is called the residual. A residuals plot shows the explanatory variable *x* on the horizontal axis and the residual for that value on the vertical axis. The residuals plot is often shown together with a scatter plot of the data. While a scatter plot of the data should resemble a straight line, a residuals plot should appear random, with no pattern and no outliers. It should also show constant error variance, meaning the residuals should not consistently increase (or decrease) as the explanatory variable *x* increases.

Least Squares Criteria for Best Fit

The process of fitting the best-fit line is called **linear regression**. The idea behind finding the best-fit line is based on the assumption that the data are scattered about a straight line. The criteria for the best fit line is that the sum of the squared errors (SSE) is minimized, that is, made as small as possible. Any other line you might choose would have a higher SSE than the best fit line. This best fit line is called the  **least-squares regression line** .

UNDERSTANDING SLOPE

The slope of the line, *b*, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.

The Correlation Coefficient *r*

Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Use the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship between *x* and *y*.

The Coefficient of Determination

**The variable *r*2 is called the** coefficient of determination and is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. It has an interpretation in the context of the data:

Testing the Significance of the Correlation Coefficient

PERFORMING THE HYPOTHESIS TEST

WHAT THE HYPOTHESES MEAN IN WORDS:

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example.
The line of best fit is: *ŷ* = –173.51+4.83*x* with *r* = 0.6631 and there are *n* = 11 data points. Can the regression line be used for prediction? **Given a third-exam score (*x* value), can we use the line to predict the final exam score (predicted *y* value)?**

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between *x* and *y* in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between *x* and *y* in the population.

Prediction

Outliers

Identifying Outliers

We could guess at outliers by looking at a graph of the scatterplot and best fit-line. However, we would like some guideline as to how far away a point needs to be in order to be considered an outlier.  **As a rough rule of thumb, we can flag any point that is located further than two standard deviations above or below the best-fit line as an outlier**. The standard deviation used is the standard deviation of the residuals or errors.

Numerical Identification of Outliers

In [link], the first two columns are the third-exam and final-exam data. The third column shows the predicted *ŷ* values calculated from the line of best fit: *ŷ* = –173.5 + 4.83*x*. The residuals, or errors, have been calculated in the fourth column of the table: observed *y* value−predicted y value = *y* − *ŷ*.

How does the outlier affect the best fit line?

Numerically and graphically, we have identified the point (65, 175) as an outlier. We should re-examine the data for this point to see if there are any problems with the data. If there is an error, we should fix the error if possible, or delete the data. If the data is correct, we would leave it in the data set. For this problem, we will suppose that we examined the data and found that this outlier data was an error. Therefore we will continue on and delete the outlier, so that we can explore how it affects the results, as a learning experience.

Numerical Identification of Outliers: Calculating *s* and Finding Outliers Manually

If you do not have the function LinRegTTest, then you can calculate the outlier in the first example by doing the following.

First, **square each |*y* – *ŷ*|**

95% Critical Values of the Sample Correlation Coefficient Table

Regression (Distance from School)

Regression (Textbook Cost)

Regression (Fuel Efficiency)

F Distribution and One-Way ANOVA

Introduction

One-Way ANOVA

The Null and Alternative Hypotheses

The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different. For example, if there are *k* groups:

The F Distribution and the F-Ratio

Notation

The notation for the *F* distribution is *F* ~ *F**df*(*num*),*df*(*denom*)

Facts About the F Distribution

Test of Two Variances

Lab: One-Way ANOVA

Review Exercises (Ch 3-13)

Chapter 3

*Use the following information to answer the next six exercises:* In a survey of 100 stocks on NASDAQ, the average percent increase for the past year was 9% for NASDAQ stocks.

Chapter 4

*Use the following information to answer the next two exercises:* A recent poll concerning credit cards found that 35 percent of respondents use a credit card that gives them a mile of air travel for every dollar they charge. Thirty percent of the respondents charge more than \$2,000 per month. Of those respondents who charge more than \$2,000, 80 percent use a credit card that gives them a mile of air travel for every dollar they charge.

Chapter 5

*Use the following information to answer the next seven exercises:* A recent study of mothers of junior high school children in Santa Clara County reported that 76% of the mothers are employed in paid positions. Of those mothers who are employed, 64% work full-time (over 35 hours per week), and 36% work part-time. However, out of all of the mothers in the population, 49% work full-time. The population under study is made up of mothers of junior high school children in Santa Clara County. Let E = employed and F = full-time employment.

Chapter 6

*Use the following information to answer the next two exercises: X* ~ *U*(3, 13)

Chapter 7

*Use the following information to answer the next three exercises:* Richard’s Furniture Company delivers furniture from 10 A.M. to 2 P.M. continuously and uniformly. We are interested in how long (in hours) past the 10 A.M. start time that individuals wait for their delivery.

Chapter 8

*Use the following information to answer the next three exercises:* Suppose that a sample of 15 randomly chosen people were put on a special weight loss diet. The amount of weight lost, in pounds, follows an unknown distribution with mean equal to 12 pounds and standard deviation equal to three pounds. Assume that the distribution for the weight loss is normal.

Chapter 9

**73.** Rebecca and Matt are 14 year old twins. Matt’s height is two standard deviations below the mean for 14 year old boys’ height. Rebecca’s height is 0.10 standard deviations above the mean for 14 year old girls’ height. Interpret this.

1. Matt is 2.1 inches shorter than Rebecca.
2. Rebecca is very tall compared to other 14 year old girls.
3. Rebecca is taller than Matt.
4. Matt is shorter than the average 14 year old boy.

Chapter 10

*Use the following information to answer the next three exercises:* In a survey at Kirkwood Ski Resort the following information was recorded:

Chapter 11

*Use the following information to answer the next four exercises:* Suppose that the time that owners keep their cars (purchased new) is normally distributed with a mean of seven years and a standard deviation of two years. We are interested in how long an individual keeps his car (purchased new). Our population is people who buy their cars new.

Chapter 12

*Use the following information to answer the next two exercises:* A recent survey of U.S. teenage pregnancy was answered by 720 girls, age 12–19. Six percent of the girls surveyed said they have been pregnant. We are interested in the true proportion of U.S. girls, age 12–19, who have been pregnant.

Chapter 13

*Use the following information to answer the next two exercises:* Suppose that the probability of a drought in any independent year is 20%. Out of those years in which a drought occurs, the probability of water rationing is 10%. However, in any year, the probability of water rationing is 5%.

Solutions

Practice Tests (1-4) and Final Exams

Practice Test 1

Practice Test 1 Solutions

Practice Test 2

Practice Test 2 Solutions

Practice Test 3

Practice Test 3 Solutions

Practice Test 4

Practice Test 4 Solutions

Practice Final Exam 1

*Use the following information to answer the next two exercises:* An experiment consists of tossing two, 12-sided dice (the numbers 1–12 are printed on the sides of each die).

* Let Event *A* = both dice show an even number.
* Let Event *B* = both dice show a number more than eight

Practice Final Exam 1 Solutions

Practice Final Exam 2

**1**. A study was done to determine the proportion of teenagers that own a car. The population proportion of teenagers that own a car is the:

1. statistic.
2. parameter.
3. population.
4. variable.

Practice Final Exam 2 Solutions

Data Sets

Lap Times

The following tables provide lap times from Terri Vogel's log book. Times are recorded in seconds for 2.5-mile laps completed in a series of races and practice runs.

Stock Prices

The following table lists initial public offering (IPO) stock prices for all 1999 stocks that at least doubled in value during the first day of trading.

Group and Partner Projects

Univariate Data

Continuous Distributions and Central Limit Theorem

Hypothesis Testing-Article

Bivariate Data, Linear Regression, and Univariate Data

Solution Sheets

Hypothesis Testing with One Sample

Class Time: __________________________

Name: _____________________________________

Hypothesis Testing with Two Samples

The Chi-Square Distribution

Class Time: __________________________

Name: ____________________________________

F Distribution and One-Way ANOVA

Mathematical Phrases, Symbols, and Formulas

English Phrases Written Mathematically

Formulas

Symbols and Their Meanings

Notes for the TI-83, 83+, 84, 84+ Calculators

Quick Tips

Legend

* ![A blank calculator button](../../media/ti83-button.png)
 represents a button press
* `[ ]` represents yellow command or green letter behind a key
* `< >` represents items on the screen

Manipulating One-Variable Statistics

To begin:

1. Turn on the calculator.

![on key](../../media/ti83-on.png)
2. Access statistics mode.

![stat key](../../media/ti83-stat.png)
3. Select `<4:ClrList>` to clear data from lists, if desired.

![number 4 key](../../media/ti83-four.png)
 ,

![enter key](../../media/ti83-enter.png)
4. Enter list `[L1]` to be cleared.

![2nd key](../../media/ti83-2nd.png)
 , `[L1]` ,

![enter key](../../media/ti83-enter.png)
5. Display last instruction.

![2nd key](../../media/ti83-2nd.png)
 , `[ENTRY]`
6. Continue clearing remaining lists in the same fashion, if desired.

![arrow left key](../../media/ti83-dir-left.png)
 ,

![2nd key](../../media/ti83-2nd.png)
 , `[L2]` ,

![enter key](../../media/ti83-enter.png)
7. Access statistics mode.

![stat key](../../media/ti83-stat.png)
8. Select `<1:Edit . . .>`

![enter key](../../media/ti83-enter.png)
9. Enter data. Data values go into `[L1]`. (You may need to arrow over to `[L1]`).

	* Type in a data value and enter it. (For negative numbers, use the negate (-) key at the bottom of the keypad).

	![negative sign key](../../media/ti83-negate.png)
	 ,

	![number 9 key](../../media/ti83-nine.png)
	 ,

	![enter key](../../media/ti83-enter.png)
	* Continue in the same manner until all data values are entered.
10. In `[L2]`, enter the frequencies for each data value in `[L1]`.

	* Type in a frequency and enter it. (If a data value appears only once, the frequency is "1").

	![number 4 key](../../media/ti83-four.png)
	 ,

	![enter key](../../media/ti83-enter.png)
	* Continue in the same manner until all data values are entered.
11. Access statistics mode.

![stat key](../../media/ti83-stat.png)
12. Navigate to `<CALC>`.
13. Access `<1:1-var Stats>`.

![enter key](../../media/ti83-enter.png)
14. Indicate that the data is in `[L1]`...

![2nd key](../../media/ti83-2nd.png)
 , `[L1]` ,

![comma key](../../media/ti83-comma.png)
15. ...and indicate that the frequencies are in `[L2]`.

![2nd key](../../media/ti83-2nd.png)
 , `[L2]` ,

![enter key](../../media/ti83-enter.png)
16. The statistics should be displayed. You may arrow down to get remaining statistics. Repeat as necessary.

Drawing Histograms

We will construct two histograms with the built-in STATPLOT application. The first way will use the default ZOOM. The second way will involve customizing a new graph.

Linear Regression

TI-83, 83+, 84, 84+ instructions for distributions and tests

Tables