Preface

About OpenStax

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About OpenStax Resources

About *Statistics*

This instructional material was initially created through a Texas Education Agency (TEA) initiative to provide high-quality open-source instructional materials to districts free of charge. Funds were allocated by the 84th Texas Legislature (2015) for the creation of state-developed, open-source instructional materials with the request that advanced secondary courses supporting the study of science, technology, engineering, and mathematics should be prioritized.

Additional Resources

About the Authors

Sampling and Data

Introduction

Definitions of Statistics, Probability, and Key Terms

Statistical Models

Statistics, like all other branches of mathematics, uses mathematical models to describe phenomena that occur in the real world. Some mathematical models are deterministic. These models can be used when one value is precisely determined from another value. Examples of deterministic models are the quadratic equations that describe the acceleration of a car from rest or the differential equations that describe the transfer of heat from a stove to a pot. These models are quite accurate and can be used to answer questions and make predictions with a high degree of precision. Space agencies, for example, use deterministic models to predict the exact amount of thrust that a rocket needs to break away from Earth’s gravity and achieve orbit.

Probability

Probability is a mathematical tool used to study randomness. It deals with the chance 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 .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 $\frac{996}{2,000}$ is equal to .498 which is very close to .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 spring 2010 quarter. The tables display counts, frequencies, and percentages or proportions, relative frequencies. For instance, to calculate the percentage of part time students at De Anza College, divide 9,200/22,496 to get .4089. Round to the nearest thousandth—third decimal place and then multiply by 100 to get the percentage, which is 40.9 percent.

Marginal Distributions in Two-Way Tables

Below is a *two-way table*, also called a contingency table, showing the favorite sports for 50 adults: 20 women and 30 men.

Conditional Distributions in Two-Way Tables

The distinction between a marginal distribution and a *conditional distribution* is that the focus is on only a particular subset of the population (not the entire population). For example, in the table, if we focused only on the subpopulation of women who prefer football, then we could calculate the conditional distributions as shown in the two-way table below.

Presenting Data

After deciding which graph best represents your data, you may need to present your statistical data to a class or other group in an oral report or multimedia presentation. When giving an oral presentation, you must be prepared to explain exactly how you collected or calculated the data, as well as why you chose the categories, scales, and types of graphs that you are showing. Although you may have made numerous graphs of your data, be sure to use only those that actually demonstrate the stated intentions of your statistical study. While preparing your presentation, be sure that all colors, text, and scales are visible to the entire audience. Finally, make sure to allow time for your audience to ask questions and be prepared to answer them.

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**. In a simple random sample, each group has the same chance of being selected. 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].

Critical Evaluation

We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include the following:

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 high school 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. Expect that some of your answers will vary from the text due to rounding errors.

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 as follows (from lowest to highest level):

Frequency

Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3.

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

To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By using the axes in that way, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

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 *p*th percentile. For example, 15 percent 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 law 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 **Chapter 1: Sampling and Data** for a review of relative frequency. Suppose 30 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 know only 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

Types of Variability in Samples

When researchers study a population, they often use a sample, either for convenience or because it is not possible to access the entire population. *Variability* is the term used to describe the differences that may occur in these outcomes. Common types of variability include the following:

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. The standard error is the standard deviation of the sampling distribution. In other words, it is the average standard deviation that results from repeated sampling. 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

As explained before, a *z*-score allows us to compare statistics 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

Uppercase letters such as *X* or *Y* denote a random variable. Lowercase 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 (Optional)

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

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

Geometric Distribution (Optional)

Notation for the Geometric: G = Geometric Probability Distribution Function

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

Hypergeometric Distribution (Optional)

Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function

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

Poisson Distribution (Optional)

Notation for the Poisson: P = Poisson Probability Distribution Function

*X* ~ *P*(*μ*)

Discrete Distribution (Playing Card Experiment)

Discrete Distribution (Lucky Dice Experiment)

Continuous Random Variables

Introduction

Properties of Continuous Probability Distributions

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

Continuous Probability Functions

The Uniform Distribution

The Exponential Distribution (Optional)

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 the following

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

As described previously, if *X* is a normally distributed random variable and *X* ~ *N(μ, σ)*, then the *z*-score is

Using the Normal Distribution

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 (Optional)

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. The margin of error for the population mean is called the error bound for a population mean (*EBM*). The sample mean, $\overline{x}\text{,}$ is the **point estimate** of the unknown population mean, *μ*.

Working Backward 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 backward 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. In this situation, we are given the desired margin of error, *EBM*, and we need to compute the sample size *n*.

A Single Population Mean Using the Student's 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*

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

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

Distribution Needed for Hypothesis Testing

Assumptions

When you perform *a hypothesis test of a single population mean *μ** using a
Student's *t*-distribution (often called a *t*-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. Note that if the sample size is sufficiently large, a *t*-test will work even if the population is not approximately normally distributed.

Rare Events, the Sample, and the Decision and Conclusion

Rare Events

The thinking process in hypothesis testing can be summarized as follows: You want to test whether or not a particular property of the population is true. You make an assumption about the true population mean for numerical data or the true population proportion for categorical data. This assumption is the null hypothesis. Then you gather sample data that is representative of the population. From this sample data you compute the sample mean (or the sample proportion). If the value that you observe is very unlikely to occur (a rare event) if the null hypothesis is true, then you wonder why this is happening. A plausible explanation is that the null hypothesis is false.

Using the Sample to Test the Null Hypothesis

After you collect data and obtain the test statistic (the sample mean, sample proportion, or other test statistic), you can determine the probability of obtaining that test statistic when the null hypothesis is true. This probability is called the *p*-value.

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 the level of significance of the test. 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 (Optional)

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*-interceptof a Linear Equation

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

The Regression Equation

Least-Squares Criteria for Best Fit

The process of fitting the best-fit line is called **linear regression**. We assume that the data are scattered about a straight line. To find that line, we minimize the sum of the squared errors (SSE), or make it 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*.

Third Exam vs. Final Exam Example

The graph of the line of best fit for the third exam/final exam example is as follows:

Understanding and Interpreting the *y*-intercept

The *y*-intercept, *a*, of the line describes where the plot line crosses the *y*-axis. The *y*-intercept of the best-fit line tells us the best value of the relationship when *x* is zero. In some cases, it does not make sense to figure out what *y* is when *x* = 0. For example, in the third exam vs. final exam example, the *y*-intercept occurs when the third exam score, or *x*, is zero. Since all the scores are grouped around a passing grade, there is no need to figure out what the final exam score, or *y*, would be when the third exam was zero.

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 determine whether the line is a good predictor? Use the correlation coefficient as another indicator (besides the scatter plot) 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 it is the square of the correlation coefficient, but it is usually stated as a percentage, rather than in decimal form. It has an interpretation in the context of the data:

Testing the Significance of the Correlation Coefficient (Optional)

Performing the Hypothesis Test

What the Hypothesis Means 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* = .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 be 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 that we can conclude there is a linear relationship between *x* and *y* in the population.

Prediction (Optional)

Outliers

Identifying Outliers

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

Numerical Identification of Outliers

In [link], the first two columns include 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, that were mentioned in Section 3 of this chapter 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 point (65, 175) as an outlier. Recall that recalculation of the least-squares regression line and summary statistics, following deletion of an outlier, may be used to determine whether an outlier is also an influential point. This process also allows you to compare the strength of the correlation of the variables and possible changes in the slope both before and after the omission of any outliers.

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

If you do not have the function LinRegTTest on your calculator, then you must calculate the outlier in the first example by doing the following.

First, square each |*y* – *ŷ*|.

95 Percent Critical Values of the Sample Correlation Coefficient Table

Regression (Distance from School) (Optional)

Regression (Textbook Cost) (Optional)

Regression (Fuel Efficiency) (Optional)

F Distribution and One-way Anova

Introduction

One-Way ANOVA

The Null and Alternative Hypotheses

The null hypothesis is 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

Appendix A 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 percent 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 five exercises.* A recent study of mothers of junior high school children in Santa Clara County reported that 76 percent of the mothers are employed in paid positions. Of those mothers who are employed, 64 percent work full-time—more than 35 hours per week—and 36 percent work part-time. However, out of all of the mothers in the population, 49 percent 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 .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. teenagers was answered by 720 teenagers, age 15–18. Six percent of teenagers surveyed said they are planning on going to college in another country. We are interested in the true proportion of U.S. teens, ages 15–18, who are planning on going to college in another country.

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 percent. Out of those years in which a drought occurs, the probability of water rationing is 10 percent. However, in any year, the probability of water rationing is 5 percent.

Solutions

Appendix B 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 greater 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

Formula 1: Factorial
$n!=n\left(n-1\right)\left(n-2\right)...\left(1\right)\text{}$

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)
, then

![enter key](../../media/ti83-enter.png)

.
4. Enter the list `[L1]` to be cleared.

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

![enter key](../../media/ti83-enter.png)

.
5. Display the last instruction.

![2nd key](../../media/ti83-2nd.png)
 , `[ENTRY]`.
6. Continue clearing any 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 [STAT PLOT] application. In the first method, we will use the default ZOOM. The second method will involve customizing a new graph.

Linear Regression

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

Tables