The Practice of Statistics for AP

Daren S. Starnes, Daniel S. Yates, David S. Moore

Chapter 10

Comparing Two Populations or Groups - all with Video Answers

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Section 1

Comparing Two Proportions

Problem 1

Goldfish Refer to the example on page $615 .$ Suppose that your teacher decides to take SRSs of 100 crackers from both bags instead.
(a) What is the shape of the sampling distribution of $\hat{p}_{1}-\hat{\rho}_{2} ? \mathrm{Why} ?$
(b) Find the mean of the sampling distribution. Show your work.
(c) Find the standard deviation of the sampling distribution. Show your work.

Sheryl Ezze

Sheryl Ezze

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Problem 2

Homework Refer to page $612 .$ Suppose that both school counselors decide to take SRSs of 150 students instead.
(a) What is the shape of the sampling distribution of $\hat{p}_{1}-\hat{p}_{2} ? \mathrm{Why} ?$
(b) Find the mean of the sampling distribution. Show your work.
(c) Find the standard deviation of the sampling distribution. Show your work.

Prabhakar Kumar

Prabhakar Kumar

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Problem 3

I want red! A candy maker offers Child and Adult bags of jelly beans with different color mixes. The company claims that the Child mix has $30 \%$ red jelly beans, while the Adult mix contains $15 \%$ red jelly beans. Assume that the candy maker's claim is true. Suppose we take a random sample of 50 jelly beans from the Child mix and a separate random sample of 100 jelly beans from the Adult mix. Let $\beta_{C}$ and $\beta_{A}$ be the sample proportions of red jelly beans from the Child and Adult mixes, respectively.
(a) What is the shape of the sampling distribution of $\hat{p}_{\mathrm{C}}-\hat{p}_{A}^{2}$ Why?
(b) Find the mean of the sampling distribution. Show your work.
(c) Find the standard deviation of the sampling distribution. Show your work.

Prabhakar Kumar

Prabhakar Kumar

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Problem 4

Literacy A researcher reports that $80 \%$ of high school graduates, but only $40 \%$ of high school dropouts, would pass a basic literacy test. ${ }^{5}$ Assume that the researcher's claim is true. Suppose we give a basic literacy test to a random sample of 60 high school graduates and a separate random sample of 75 high school dropouts. Let $\beta_{\mathrm{G}}$ and $\hat{p}_{\mathrm{D}}$ be the sample proportions of graduates and dropouts, respectively, who pass the test.

Prabhakar Kumar

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Problem 5

Don't drink the water! The movie A Civil Action (Touchstone Pictures, 1998 ) tells the story of a major legal battle that took place in the small town of Woburn, Massachusetts. A town well that supplied water to eastern Woburn residents was contaminated by industrial chemicals. During the period that residents drank water from this well, 16 of the 414 babies born had birth defects. On the west side of Woburn, 3 of the 228 babies born during the same time period had birth defects.

Prabhakar Kumar

Prabhakar Kumar

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Problem 6

In-line skaters A study of injuries to in-line skaters used data from the National Electronic Injury Surveillance System, which collects data from a random sample of hospital emergency rooms. The researchers interviewed 161 people who came to emergency rooms with injuries from in-line skating. Wrist injuries (mostly fractures) were the most common. ${ }^{6}$ The interviews found that 53 people were wearing wrist guards and 6 of these had wrist injuries. Of the 108 who did not wear wrist guards, 45 had wrist injuries.

Prabhakar Kumar

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Problem 7

Shrubs and fire Fire is a serious threat to shrubs in dry climates. Some shrubs can resprout from their roots after their tops are destroyed. One study of resprouting took place in a dry area of Mexico. The investigators randomly assigned shrubs to treatment and control groups. They clipped the tops of all the shrubs. They then applied a propane torch to the stumps of the treatment group to simulate a fire. All 12 of the shrubs in the treatment group resprouted. Only 8 of the 12 shrubs in the control group resprouted.

Prabhakar Kumar

Prabhakar Kumar

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Problem 8

Broken crackers We don't like to find broken crackers when we open the package. How can makers reduce breaking? One idea is to microwave the crackers for 30 seconds right after baking them. Breaks start as hairline cracks called "checking." Randomly assign 65 newly baked crackers to the microwave and another 65 to a control group that is not microwaved. After one day, none of the microwave group and 16 of the control group show checking.

Prabhakar Kumar

Prabhakar Kumar

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Problem 9

Who tweets? Do younger people use Twitter more often than older people? In a random sample of 316 adult Internet users aged 18 to $29,26 \%$ used Twitter. In a separate random sample of 532 adult Internet users aged 30 to $49,14 \%$ used Twitter. ${ }$
(a) Calculate the standard error of the sampling distribution of the difference in the sample proportions (younger adults - older adults). What information does this value provide?
(b) Construct and interpret a $90 \%$ confidence interval for the difference between the true proportions of adult Internet users in these age groups who use Twitter.

Prabhakar Kumar

Prabhakar Kumar

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Problem 10

Listening to rap Is rap music more popular among young blacks than among young whites? A sample survey compared 634 randomly chosen blacks aged 15 to 25 with 567 randomly selected whites in the same age group. It found that 368 of the blacks and 130 of the whites listened to rap music every day.
(a) Calculate the standard error of the sampling distribution of the difference in the sample proportions (blacks - whites). What information does this value provide?
(b) Construct and interpret a $95 \%$ confidence interval for the difference between the proportions of black and white young people who listen to rap every day.

Prabhakar Kumar

Prabhakar Kumar

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Problem 11

Young adults living at home A surprising number of young adults (ages 19 to 25 ) still live in their parents' homes. A random sample by the National Institutes of Health included 2253 men and 2629 women in this age group. ${ }^{11}$ The survey found that 986 of the men and 923 of the women lived with their parents.
(a) Construct and interpret a $99 \%$ confidence interval for the difference in the true proportions of men and women aged 19 to 25 who live in their parents' homes.
(b) Does your interval from part (a) give convincing evidence of a difference between the population proportions? Explain.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 12

Fear of crime The elderly fear crime more than younger people, even though they are less likely to be victims of crime. One study recruited separate random samples of 56 black women and 63 black men over the age of 65 from Atlantic City, New Jersey. Of the women, 27 said they "felt vulnerable" to crime; 46 of the men said this. ${ }^{12}$
(a) Construct and interpret a $90 \%$ confidence interval for the difference in the true proportions of black women and black men in Atlantic City who would say they felt vulnerable to crime.
(b) Does your interval from part (a) give convincing evidence of a difference between the population proportions? Explain.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 13

Who owns iPods? As part of the Pew Internet and American Life Project, researchers surveyed a random sample of 800 teens and a separate random sample of 400 young adults. For the teens, $79 \%$ said that they own an iPod or MP3 player. For the young adults, this figure was $67 \%$. Do the data give convincing evidence of a difference in the proportions of all teens and young adults who would say that they own an iPod or MP3 player? State appropriate hypotheses for a test to answer this question. Define any parameters you use.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 14

Steroids in high school $A$ study by the National Athletic Trainers Association surveyed random samples of 1679 high school freshmen and 1366 high school seniors in Illinois. Results showed that 34 of the freshmen and 24 of the seniors had used anabolic steroids. Steroids, which are dangerous, are sometimes used in an attempt to improve athletic performance. ${ }^{13}$ Do the data give convincing evidence of a difference in the proportion of all Illinois high school freshmen and seniors who have used anabolic steroids? State appropriate hypotheses for a test to answer this question. Define any parameters you use..

Prabhakar Kumar

Prabhakar Kumar

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Problem 15

Who owns iPods? Refer to Exercise $13 .$ Carry out a significance test at the $\alpha=0.05$ level.

Prabhakar Kumar

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Problem 16

Steroids in high school Refer to Exercise l4. Carry out a significance test at the $\alpha=0.05$ level.

Prabhakar Kumar

Prabhakar Kumar

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Problem 17

Who owns iPods? Refer to Exercise $13 .$ Construct and interpret a $95 \%$ confidence interval for the difference between the population proportions. Explain how the confidence interval is consistent with the results of the test in Exercise 15 .

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 18

Steroids in high school Refer to Exercise 14. Construct and interpret a $95 \%$ confidence interval for the difference between the population proportions. Explain how the confidence interval is consistent with the results of the test in Exercise $16 .$

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 19

Children make choices Many new products introduced into the market are targeted toward children. The choice behavior of children with regard to new products is of particular interest to companies that design marketing strategies for these products. As part of one study, randomly selected children in different age groups were compared on their ability to sort new products into the correct product category (milk or juice). ${ }^{14}$ Here are some of the data: $$
\begin{array}{lll}
\hline \text { Age group } & N & \text { Number who sorted correctly } \\
\text { 4- to 5-year-olds } & 50 & 10 \\
\text { 6- to 7-year-olds } & 53 & 28 \\
\hline
\end{array}
$$
Did a significantly higher proportion of the 6 - to 7-year-olds than the 4 - to 5 -year-olds sort correctly? Give appropriate evidence to justify your answer.

Prabhakar Kumar

Prabhakar Kumar

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Problem 20

Marriage and status "Would you marry a person from a lower social class than your own?" Researchers asked this question of a random sample of 385 black, never-married college students. Of the 149 men in the sample, 91 said "Yes." Among the 236 women, 117 said "Yes." 15 Did a significantly higher proportion of the men than the women who were surveyed say "Yes"? Give appropriate evidence to justify your answer.

Prabhakar Kumar

Prabhakar Kumar

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Problem 21

Driving school A driving school owner believes that Instructor $\mathrm{A}$ is more effective than Instructor $\mathrm{B}$ at preparing students to pass the state's driver's license exam. An incoming class of 100 students is randomly assigned to two groups, each of size $50 .$ One group is taught by Instructor $A ;$ the other is taught by Instructor B. At the end of the course, 30 of Instructor A's students and 22 of Instructor $\mathrm{B}$ 's students pass the state exam.
(a) Do these results give convincing evidence at the $\alpha=0.05$ level that Instructor $\mathrm{A}$ is more effective?
(b) Describe a Type I and a Type II error in this setting. Which error could you have made in part (a)?

Prabhakar Kumar

Prabhakar Kumar

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Problem 22

Preventing strokes Aspirin prevents blood from clotting and so helps prevent strokes. The Second European Stroke Prevention Study asked whether adding another anticlotting drug, named dipyridamole, would be more effective for patients who had already had a stroke. Here are the data on strokes during the two years of the study: ${ }^{16}$ $$
\begin{array}{lcc}
\hline & \begin{array}{c}
\text { Number of } \\
\text { patients }
\end{array} & \begin{array}{c}
\text { Number of } \\
\text { strokes }
\end{array} \\
\hline \text { Aspirin alone } & 1649 & 206 \\
\text { Aspirin + dipyridamole } & 1650 & 157 \\
\hline
\end{array}
$$
The study was a randomized comparative experiment.
(a) Is there convincing evidence at the $\alpha=0.05$ level that adding dipyridamole helps reduce the risk of stroke?
(b) Describe a Type I and a Type II error in this setting. Which is more serious? Explain.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 23

Prayer and pregnancy Two hundred women who were about to undergo IVF served as subjects in an experiment. Each subject was randomly assigned to either a treatment group or a control group. Women in the treatment group were intentionally prayed for by several people (called intercessors) who did not know them, a process known as intercessory prayer. The praying continued for three weeks following IVF. The intercessors did not pray for the women in the control group. Here are the results: 44 of the 88 women in the treatment group got pregnant, compared to 21 out of 81 in the control group. ${ }^{17}$ Is the pregnancy rate significantly higher for women who received intercessory prayer? To find out, researchers perform a test of $H_{0}: p_{1}=p_{2}$ versus $H_{a}: p_{1}>p_{2},$ where $p_{1}$ and $p_{2}$ are the actual pregnancy rates for women like those in the study who do and don't receive intercessory prayer, respectively.
(a) Name the appropriate test and check that the conditions for carrying out this test are met.
(b) The appropriate test from part (a) yields a $P$ -value of 0.0007 . Interpret this $P$ -value in context.
(c) What conclusion should researchers draw at the $\alpha=$ 0.05 significance level? Explain.
(d) The women in the study did not know whether they were being prayed for. Explain why this is important.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 24

Acupuncture and pregnancy $A$ study reported in the medical journal Fertility and Sterility sought to determine whether the ancient Chinese art of acupuncture could help infertile women become pregnant. ${ }^{18}$ One hundred sixty healthy women who planned to have IVF were recruited for the study. Half of the subjects (80) were randomly assigned to receive acupuncture 25 minutes before embryo transfer and again 25 minutes after the transfer. The remaining 80 women were assigned to a control group and instructed to lie still for 25 minutes after the embryo transfer. Results are shown in the table below. Is the pregnancy rate significantly higher for women who received acupuncture? To find out, researchers perform a test of $H_{0}: p_{1}=p_{2}$ versus $H_{a}: p_{1}>p_{2},$ where $p_{1}$ and $p_{2}$ are the actual pregnancy rates for women like those in the study who do and don't receive acupuncture, respectively. (a) Name the appropriate test and check that the conditions for carrying out this test are met.
(b) The appropriate test from part (a) yields a P-value of $0.0152 .$ Interpret this $P$ -value in context.
(c) What conclusion should researchers draw at the $\alpha=0.05$ significance level? Explain.
(d) The women in the study knew whether or not they received acupuncture. Explain why this is important.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 25

Take $p_{M}$ and $p_{F}$ to be the proportions of all college males and females who worked last summer. The hypotheses to be tested are
(a) $H_{0}: p_{\mathrm{M}}-p_{\mathrm{F}}=0$ versus $H_{a} ; p_{\mathrm{M}}-p_{\mathrm{F}} \neq 0$
(b) $H_{0}: p_{\mathrm{M}}-p_{\mathrm{F}}=0$ versus $H_{a} ; p_{\mathrm{M}}-p_{\mathrm{F}}>0$
(c) $H_{0}: p_{M}-p_{F}=0$ versus $H_{a}: p_{M}-p_{F}<0$
(d) $H_{0}: p_{\mathrm{M}}-p_{\mathrm{F}}>0$ versus $H_{a} ; p_{\mathrm{M}}-p_{\mathrm{F}}=0$
(e) $H_{0}: p_{M}-p_{F} \neq 0$ versus $H_{a}: p_{M}-p_{F}=0$

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 26

The researchers report that the results were statistically significant at the $1 \%$ level. Which of the following is the most appropriate conclusion?
(a) Because the $P$ -value is less than $1 \%$, fail to reject $H_{0}$. There is not convincing evidence that the proportion of male college students in the study who worked for pay last summer is different from the proportion of female college students in the study who worked for pay last summer.
(b) Because the $P$ -value is less than $1 \%$, fail to reject $H_{0}$. There is not convincing evidence that the proportion of all male college students who worked for pay last summer is different from the proportion of all female college students who worked for pay last summer.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 27

Which of the following is the correct margin of error for a $99 \%$ confidence interval for the difference in the proportion of male and female college students who worked for pay last summer?
(a) $2.576 \sqrt{\frac{0.851(0.149)}{550}+\frac{0.851(0.149)}{500}}$
(b) $2.576 \sqrt{\frac{0.851(0.149)}{1050}}$
(c) $2.576 \sqrt{\frac{0.880(0.120)}{550}+\frac{0.820(0.180)}{500}}$
(d) $1.960 \sqrt{\frac{0.851(0.149)}{550}+\frac{0.851(0.149)}{500}}$
(e) $1.960 \sqrt{\frac{0.880(0.120)}{550}+\frac{0.820(0.180)}{500}}$

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 28

In an experiment to learn whether Substance $\mathrm{M}$ can help restore memory, the brains of 20 rats were treated to damage their memories. First, the rats were trained to run a maze. After a day, 10 rats (determined at random) were given $\mathrm{M}$ and 7 of them succeeded in the maze. Only 2 of the 10 control rats were successful. The two-sample $z$ test for "no difference" against "a significantly higher proportion of the $\mathrm{M}$ group succeeds"
(a) gives $z=2.25, P<0.02$.
(b) gives $z=2.60, P<0.005$
(c) gives $z=2.25, P<0.04$ but not $<0.02$.
(d) should not be used because the Random condition is violated.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 29

Drive my car
(a) What is the equation of the least-squares regression line? Be sure to define any symbols you use.
(b) Interpret the slope of the least-squares line in the context of this problem.
(c) One student reported that her 10 -year-old car had 110,000 miles on it. Find and interpret the residual for this data value. Show your work.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 30

Drive my car (3.2,4.3)
(a) Explain what the value of $r^{2}$ tells you about how well the least-squares line fits the data.
(b) The mean age of the students' cars in the sample was $\bar{x}=8$ years. Find the mean mileage of the cars in the sample. Show your work.
(c) Interpret the value of $s$ in the context of this setting.
(d) Would it be reasonable to use the least-squares line to predict a car's mileage from its age for a Council High School teacher? Justify your answer.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 31

Cholesterol The level of cholesterol in the blood for all men aged 20 to 34 follows a Normal distribution with mean 188 milligrams per deciliter $(\mathrm{mg} / \mathrm{dl})$ and standard deviation $41 \mathrm{mg} / \mathrm{dl}$. For 14 -year-old boys, blood cholesterol levels follow a Normal distribution with mean $170 \mathrm{mg} / \mathrm{dl}$ and standard deviation $30 \mathrm{mg} / \mathrm{dl}$. Suppose we select independent SRSs of $25 \mathrm{men}$ aged 20 to 34 and 36 boys aged 14 and calculate the sample mean cholesterol levels $\bar{x}_{M}$ and $\bar{x}_{B}$
(a) What is the shape of the sampling distribution of $\bar{x}_{M}-\bar{x}_{B} ?$ Why?
(b) Find the mean of the sampling distribution. Show your work.
(c) Find the standard deviation of the sampling distribution. Show your work.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 32

How tall? The heights of young men follow a Normal distribution with mean 69.3 inches and standard deviation 2.8 inches. The heights of young women follow a Normal distribution with mean 64.5 inches and standard deviation 2.5 inches. Suppose we select independent SRSs of 16 young men and 9 young women and calculate the sample mean heights $\bar{x}_{M}$ and $\bar{x}_{W}$.
(a) What is the shape of the sampling distribution of $\bar{x}_{M}-\bar{x}_{W} ?$ Why?
(b) Find the mean of the sampling distribution. Show your work.
(c) Find the standard deviation of the sampling distribution. Show your work.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 33

Determine whether or not the conditions for using two-sample t procedures are met.
Shoes How many pairs of shoes do teenagers have? To find out, a group of $\mathrm{AP}^{6}$ Statistics students conducted a survey. They selected a random sample of 20 female students and a separate random sample

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 34

Determine whether or not the conditions for using two-sample t procedures are met.
Household size How do the numbers of people living in households in the United Kingdom (U.K.) and South Africa compare? To help answer this question, we used CensusAtSchool's random data selector to choose independent samples of 50 students from each country. Here is a Fathom dotplot of the household sizes reported by the students in the survey.

Prabhakar Kumar

Prabhakar Kumar

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Problem 35

Determine whether or not the conditions for using two-sample t procedures are met.
Literacy rates Do males have higher average literacy rates than females in Islamic countries? The table below shows the percent of men and women who were literate in the major Islamic nations at the time of this writing. tions of less than 3 million.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 36

Determine whether or not the conditions for using two-sample t procedures are met. Long words Mary was interested in comparing the mean word length in articles from a medical journal and an airline's in-flight magazine. She counted the number of letters in the first 400 words of an article in the medical journal and in the first 100 words of an article in the airline magazine. Mary then used Minitab statistical software to produce the histograms shown. Note that $J$ is for journal and $M$ is for magazine.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 37

Is red wine better than white wine? Observational studies suggest that moderate use of alcohol by adults reduces heart attacks and that red wine may have special benefits. One reason may be that red wine contains polyphenols, substances that do good things to cholesterol in the blood and so may reduce the risk of heart attacks. In an experiment, healthy men were assigned at random to drink half a bottle of either red or white wine each day for two weeks. The level of polyphenols in their blood was measured before and after the two-week period. Here are the percent changes in level for the subjects in both groups: ${ }^{28}$
$$
\begin{array}{lllrlrllrl}
\hline \text { Red wine: } & 3.5 & 8.1 & 7.4 & 4.0 & 0.7 & 4.9 & 8.4 & 7.0 & 5.5 \\
\text { White wine: } & 3.1 & 0.5 & -3.8 & 4.1 & -0.6 & 2.7 & 1.9 & -5.9 & 0.1 \\
\hline
\end{array}
$$
(a) A Fathom dotplot of the data is shown below. Write a few sentences comparing the distributions.
(b) Construct and interpret a $90 \%$ confidence interval for the difference in mean percent change in polyphenol levels for the red wine and white wine treatments.
(c) Does the interval in part (b) suggest that red wine is more effective than white wine? Explain.

Jerelyn Nevil

Numerade Educator

Problem 38

Tropical flowers Different varieties of the tropical flower Heliconia are fertilized by different species of hummingbirds. Researchers believe that over time, the lengths of the flowers and the forms of the hummingbirds' beaks have evolved to match each other. Here are data on the lengths in millimeters for random samples of two color varieties of the same species of flower on the island of Dominica: ${ }^{29}$
(a) A Fathom dotplot of the data is shown below. Write a few sentences comparing the distributions.
(b) Construct and interpret a $95 \%$ confidence interval for the difference in the mean lengths of these two varieties of flowers.
(c) Does the interval support the researchers' belief that the two flower varieties have different average lengths? Explain.

Jerelyn Nevil

Numerade Educator

Problem 39

Paying for college College financial aid offices expect students to use summer eamings to help pay for college. But how large are these eamings? One large university studied this question by asking a random sample of 1296 students who had summer jobs how much they earned. The financial aid office separated the responses into two groups based on gender. Here are the data in summary form: ${ }^{30}$
$$
\begin{array}{lccc}
\hline \text { Group } & n & \bar{\chi} & s_{\chi} \\
\text { Males } & 675 & \$ 1884.52 & \$ 1368.37 \\
\text { Females } & 621 & \$ 1360.39 & \$ 1037.46 \\
\hline
\end{array}
$$
(a) How can you tell from the summary statistics that the distribution of earnings in each group is strongly skewed to the right? The use of two-sample $t$ procedures is still justified. Why?
(b) Construct and interpret a $90 \%$ confidence interval for the difference between the mean summer earnings of male and female students at this university.
(c) Interpret the $90 \%$ confidence level in the context of this study.

Sheryl Ezze

Numerade Educator

Problem 40

Happy customers As the Hispanic population in the United States has grown, businesses have tried to understand what Hispanics like. One study interviewed a random sample of customers leaving a bank. Customers were classified as Hispanic if they preferred to be interviewed in Spanish or as Anglo if they preferred English. Each customer rated the importance of several aspects of bank service on a 10 -point scale. ${ }^{31}$ Here are summary results for the importance of "reliability" (the accuracy of account records and so on
$$
\begin{array}{lccc}
\hline \text { Group } & n & \bar{x} & s_{x} \\
\text { Anglo } & 92 & 6.37 & 0.60 \\
\text { Hispanic } & 86 & 5.91 & 0.93 \\
\hline
\end{array}
$$
(a) The distribution of reliability ratings in each group is not Normal. The use of two-sample $t$ procedures is still justified. Why?
(b) Construct and interpret a $95 \%$ confidence interval for the difference between the mean ratings of the importance of reliability for Anglo and Hispanic bank customers.
(c) Interpret the $95 \%$ confidence level in the context of this study.

Sheryl Ezze

Numerade Educator

Problem 41

Baby birds Do birds learn to time their breeding? Blue titmice eat caterpillars. The birds would like lots of caterpillars around when they have young to feed, but they must breed much earlier. Do the birds learn from one year's experience when to time their breeding next year? Researchers randomly assigned 7 pairs of birds to have the natural caterpillar supply supplemented while feeding their young and another 6 pairs to serve as a control group relying on natural food supply. The next year, they measured how many days after the caterpillar peak the birds produced their nestlings. ${ }^{32}$ The investigators expected the control group to adjust their breeding date the next year, whereas the well-fed supplemented group had no reason to change. Here are the data (days after caterpillar peak):
$$
\begin{array}{lrrrrrrr}
\hline \text { Control: } & 4.6 & 2.3 & 7.7 & 6.0 & 4.6 & -1.2 & \\
\text { Supplemented: } & 15.5 & 11.3 & 5.4 & 16.5 & 11.3 & 11.4 & 7.7 \\
\hline
\end{array}
$$
(a) Do the data provide convincing evidence to confirm the researchers' belief?
(b) Interpret the $P$ -value from part (a) in the context of this study.

Teresa Wray

Numerade Educator

Problem 42

DDT in rats Poisoning by the pesticide DDT causes convulsions in humans and other mammals. Researchers seek to understand how the convulsions are caused. In a randomized comparative experiment, they compared 6 white rats poisoned with DDT with a control group of 6 unpoisoned rats. Electrical measurements of nerve activity are the main clue to the nature of DDT poisoning. When a nerve is stimulated, its electrical response shows a sharp spike followed by a much smaller second spike. The researchers measured the height of the second spike as a percent of the first spike when a nerve in the rat's leg was stimulated. ${ }^{33}$ For the poisoned rats, the results were
$$
\begin{array}{llllll}
12.207 & 16.869 & 25.050 & 22.429 & 8.456 & 20.589
\end{array}
$$
The control group data were
$$
\begin{array}{llllll}
11.074 & 9.686 & 12.064 & 9.351 & 8.182 & 6.642
\end{array}
$$
(a) Do these data provide convincing evidence that DD'T affects the mean relative height of the second spike's electrical response?
(b) Interpret the $P$ -value from part (a) in the context of this study.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 43

Who talks more-men or women? Researchers equipped random samples of 56 male and 56 female students from a large university with a small device that secretly records sound for a random 30 seconds during each 12.5 -minute period over two days. Then they counted the number of words spoken by each subject during each recording period and, from this, estimated how many words per day each subject speaks. The female estimates had a mean of 16,177 words per day with a standard deviation of 7520 words per day. For the male estimates, the mean was 16,569 and the standard deviation was $9108 .$ Do these data provide convincing evidence of a difference in the average number of words spoken in a day by male and female students at this university?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 44

Competitive rowers What aspects of rowing technique distinguish between novice and skilled competitive rowers? Researchers compared two randomly selected groups of female competitive rowers: a group of skilled rowers and a group of novices. The researchers measured many mechanical aspects of rowing style as the subjects rowed on a Stanford Rowing Ergometer. One important variable is the angular velocity of the knee, which describes the rate at which the knee joint opens as the legs push the body back on the sliding seat. The data show no outliers or strong skewness. Here is the SAS computer output: ${ }^{34}$The researchers believed that the knee velocity would be higher for skilled rowers. Do the data provide convincing evidence to support this belief?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 45

Teaching reading An educator believes that new reading activities in the classroom will help elementary school pupils improve their reading ability. She recruits 44 third-grade students and randomly assigns them into two groups. One group of 21 students does these new activities for an 8 -week period. A control group of 23 thirdgraders follows the same curriculum without the activities. At the end of the 8 weeks, all students are given the Degree of Reading Power (DRP) test, which measures the aspects of reading ability that the treatment is designed to improve. Comparative boxplots and summary statistics for the data from Fathom are shown below. ${ }^{35}$
(a) Based on the graph and numerical summaries, write a few sentences comparing the DRP scores for the two groups.
(b) Is the mean DRP score significantly higher for the students who did the reading activities? Give appropriate evidence to justify your answer.
(c) Can we conclude that the new reading activities caused an increase in the mean DRP score? Explain.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 46

Does breast-feeding weaken bones? Breast-feeding mothers secrete calcium into their milk. Some of the calcium may come from their bones, so mothers may lose bone mineral. Researchers compared a random sample of 47 breast-feeding women with a random sample of 22 women of similar age who were neither pregnant nor lactating. They measured the percent change in the bone mineral content (BMC) of the women's spines over three months. Comparative boxplots and summary statistics for the data from Fathom are shown below. ${ }^{36}$
(a) Based on the graph and numerical summaries, write a few sentences comparing the percent changes in BMC for the two groups.
(b) Is the mean change in BMC significantly lower for the mothers who are breast-feeding? Give appropriate evidence to justify your answer.
(c) Can we conclude that breast-feeding causes a mother's bones to weaken? Why or why not?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 47

Who talks more-men or women? Refer to Exercise 43 . Construct and interpret a $95 \%$ confidence interval for the difference in mean number of words spoken in a day. Explain how this interval provides more information than the significance test in Exercise 43 .

Rashmi Sinha

Numerade Educator

Problem 48

DDT in rats Refer to Exercise 42 . Construct and interpret a $95 \%$ confidence interval for the difference in mean relative height of the second spike's electrical response. Explain how this interval provides more information than the significance test in Exercise 42 .

Manik Pulyani

Numerade Educator

Problem 49

A better drug? In a pilot study, a company's new cholesterol-reducing drug outperforms the currently available drug. If the data provide convincing evidence that the mean cholesterol reduction with the new drug is more than 10 milligrams per deciliter of blood $(\mathrm{mg} / \mathrm{dl})$ greater than with the current drug, the company will begin the expensive process of mass-producing the new drug. For the 14 subjects who were assigned at random to the current drug. the mean cholesterol reduction was $54.1 \mathrm{mg} / \mathrm{dl}$ with a standard deviation of $11.93 \mathrm{mg} / \mathrm{dl}$. For the $15 \mathrm{sub}$ jects who were randomly assigned to the new drug, the mean cholesterol reduction was $68.7 \mathrm{mg} / \mathrm{dl}$ with a standard deviation of $13.3 \mathrm{mg} / \mathrm{dl}$. Graphs of the data reveal no outliers or strong skewness.
(a) Carry out an appropriate significance test. What conclusion would you draw? (Note that the null hypothesis is not $\left.H_{0}: \mu_{1}-\mu_{2}=0 .\right)$
(b) Based on your conclusion in part (a), could you have made a Type I error or a Type II error? Justify your answer.

Sheryl Ezze

Numerade Educator

Problem 50

Down the toilet $A$ company that makes hotel toilets claims that its new pressure-assisted toilet reduces the average amount of water used by more than 0.5 gallon per flush when compared to its current model. To test this claim, the company randomly selects 30 toilets of each type and measures the amount of water that is used when each toilet is flushed once. For the current-model toilets, the mean amount of water used is 1.64 gal with a standard deviation of 0.29 gal. For the new toilets, the mean amount of water used is 1.09 gal with a standard deviation of 0.18 gal.
(a) Carry out an appropriate significance test. What conclusion would you draw? (Note that the null hypothesis is not $\left.\vec{H}_{0}: \mu_{1}-\mu_{2}=0 .\right)$
(b) Based on your conclusion in part (a), could you have made a Type I error or a Type II error? Justify your answer.

R M

Numerade Educator

Problem 51

Rewards and creativity Dr. Teresa Amabile conducted a study involving 47 college students who were randomly assigned to two treatment groups. The 23 students in one group were given a list of statements about external reasons (E) for writing, such as public recognition, making money, or pleasing their parents. The 24 students in the other group were given a list of statements about internal reasons (I) for writing, such as expressing yourself and enjoying playing with words. Both groups were then instructed to write a poem about laughter. Each student's poem was rated separately by 12 different poets using a creativity scale. ${ }^{37}$ The 12 poets' ratings of each student's poem were averaged to obtain an overall creativity score.
We used Fathom software to randomly reassign the 47 subjects to the two groups 1000 times, assuming the treatment received doesn't affect each individual's average creativity rating. The dotplot shows the approximate randomization distribution of $\bar{x}_{1}-\bar{x}_{E}$
(a) Why did researchers randomly assign the subjects to the two treatment groups?
(b) In the actual experiment, $\bar{x}_{1}-\bar{x}_{E}=4.15 .$ This value is marked with a blue line in the figure. What conclusion would you draw? Justify your answer with appropriate evidence.
(c) Based on your conclusion in part (b), could you have made a Type I error or a Type II error? Justify your answer.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 52

Sleep deprivation Does sleep deprivation linger for more than a day? Researchers designed a study using 21 volunteer subjects between the ages of 18 and 25. All 21 participants took a computer-based visual discrimination test at the start of the study. Then the subjects were randomly assigned into two groups. The 11 subjects in one group, $D,$ were deprived of sleep for an entire night in a laboratory setting. The 10 subjects in the other group, $\mathrm{A},$ were allowed unrestricted sleep for the night. Both groups were allowed as much sleep as they wanted for the next two nights. On Day $4,$ all the subjects took the same visual discrimination test on the computer. Researchers recorded the improvement in time (measured in milliseconds) from Day 1 to Day 4 on the test for each subject.
We used Fathom software to randomly reassign the 21 subjects to the two groups 1000 times, assuming the treatment received doesn't affect each individual's time improvement on the test. The dotplot shows the approximate randomization distribution of $\bar{x}_{A}-\bar{x}_{D}$
(a) Explain why the researchers didn't let the subjects choose whether to be in the sleep deprivation group or the unrestricted sleep group.
(b) In the actual experiment, $\bar{x}_{A}-\bar{x}_{D}=15.92$. This value is marked with a blue line in the figure. What conclusion would you draw? Justify your answer with appropriate evidence.
(c) Based on your conclusion in part (b), could you have made a Type I error or a Type II error? Justify your answer.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 53

Paired or unpaired? In each of the following settings, decide whether you should use paired procedures or two-sample $t$ procedures to perform inference. Explain your choice. ${ }^{39}$
(a) To test the wear characteristics of two tire brands, $\mathrm{A}$ and $\mathrm{B}$, each brand of tire is randomly assigned to 50 cars of the same make and model.
(b) To test the effect of background music on productivity, factory workers are observed. For one month, each subject works without music. For another month, the subject works while listening to music on an MP3 player. The month in which each subject listens to music is determined by a coin toss.
(c) A study was designed to compare the effectiveness of two weight-reducing diets. Fifty obese women who volunteered to participate were randomly assigned into two equal-sized groups. One group used Diet A and the other used Diet $\mathrm{B}$. The weight of each woman was measured before the assigned diet and again after 10 weeks on the diet.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 54

Paired or unpaired? In each of the following settings, decide whether you should use paired $t$ procedures or two-sample $t$ procedures to perform inference. Explain your choice. ${ }^{40}$
(a) To compare the average weight gain of pigs fed two different rations, nine pairs of pigs were used. The pigs in each pair were littermates. A coin toss was used to decide which pig in each pair got Ration $\mathrm{A}$ and which got Ration $\mathrm{B}$.
(b) Separate random samples of male and female college professors are taken. We wish to compare the average salaries of male and female teachers.
(c) To test the effects of a new fertilizer, 100 plots are treated with the new fertilizer, and 100 plots are treated with another fertilizer. A computer's random number generator is used to determine which plots get which fertilizer.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 55

Coaching and SAT scores Let's first ask if students who are coached increased their scores significantly.
(a) You could use the information on the Coached line to carry out either a two-sample $t$ test comparing Try 1 with Try 2 for coached students or a paired $t$ test using Gain. Which is the correct test? Why?
(b) Carry out the proper test. What do you conclude?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 56

Coaching and SAT scores What we really want to know is whether coached students improve more than uncoached students, and whether any advantage is large enough to be worth paying for. Use the information above to answer these questions:
(a) How much more do coached students gain on the average? Construct and interpret a $99 \%$ confidence interval.
(b) Does the interval in part (a) give convincing evidence that coached students gain more, on average, than uncoached students? Explain.
(c) Based on your work, what is your opinion: do you think coaching courses are worth paying for?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 57

There are two common methods for measuring the concentration of a pollutant in fish tissue. Do the two methods differ, on average? You apply both methods to each fish in a random sample of 18 carp and use
(a) the paired $t$ test for $\mu_{d}$
(b) the one-sample $z$ test for $p$.
(c) the two-sample $t$ test for $\mu_{1}-\mu_{2}$.
(d) the two-sample $z$ test for $p_{1}-p_{2}$.
(e) none of these.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 58

Exercises 58 to 60 refer to the following setting. A study of road rage asked random samples of $596 \mathrm{men}$ and 523 women about their behavior while driving. Based on their answers, each person was assigned a road rage score on a scale of 0 to $20 .$ The participants were chosen by random digit dialing of phone numbers. The researchers performed a test of the following hypotheses: $H_{0}: \mu_{M}=\mu_{F}$ versus $H_{a^{*}} \mu_{M} \neq \mu_{F}$
Which of the following describes a Type II error in the context of this study?
(a) Finding convincing evidence that the true means are different for males and females, when in reality the true means are the same
(b) Finding convincing evidence that the true means are different for males and females, when in reality the true means are different
(c) Not finding convincing evidence that the true means are different for males and females, when in reality the true means are the same
(d) Not finding convincing evidence that the true means are different for males and females, when in reality the true means are different
(e) Not finding convincing evidence that the true means are different for males and females, when in reality there is convincing evidence that the true means are different

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 59

Exercises 58 to 60 refer to the following setting. A study of road rage asked random samples of $596 \mathrm{men}$ and 523 women about their behavior while driving. Based on their answers, each person was assigned a road rage score on a scale of 0 to $20 .$ The participants were chosen by random digit dialing of phone numbers. The researchers performed a test of the following hypotheses: $H_{0}: \mu_{M}=\mu_{F}$ versus $H_{a^{*}} \mu_{M} \neq \mu_{F}$
The $P$ -value for the stated hypotheses is 0.002 . Interpret this value in the context of this study.
(a) Assuming that the true mean road rage score is the same for males and females, there is a 0.002 probability of getting a difference in sample means.
(b) Assuming that the true mean road rage score is the same for males and females, there is a 0.002 probability of getting an observed difference at least as extreme as the observed difference.
(c) Assuming that the true mean road rage score is different for males and females, there is a 0.002 probability of getting an observed difference at least as extreme as the observed difference.
(d) Assuming that the true mean road rage score is the same for males and females, there is a 0.002 probability that the null hypothesis is true.
(e) Assuming that the true mean road rage score is the same for males and females, there is a 0.002 probability that the alternative hypothesis is true.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 60

Exercises 58 to 60 refer to the following setting. A study of road rage asked random samples of $596 \mathrm{men}$ and 523 women about their behavior while driving. Based on their answers, each person was assigned a road rage score on a scale of 0 to $20 .$ The participants were chosen by random digit dialing of phone numbers. The researchers performed a test of the following hypotheses: $H_{0}: \mu_{M}=\mu_{F}$ versus $H_{a^{*}} \mu_{M} \neq \mu_{F}$
Based on the $P$ -value in Exercise $59,$ which of the following must be true?
(a) $\mathrm{A} 90 \%$ confidence interval for $\mu_{\mathrm{M}}-\mu_{\mathrm{F}}$ will contain $0 .$
(b) A $95 \%$ confidence interval for $\mu_{M}-\mu_{F}$ will contain 0 .
(c) A $99 \%$ confidence interval for $\mu_{\mathrm{M}}-\mu_{\mathrm{F}}$ will contain $0 .$
(d) $\mathrm{A} 99.9 \%$ confidence interval for $\mu_{\mathrm{M}}-\mu_{F}$ will contain 0
(e) It is impossible to determine whether any of these statements is true based only on the $P$ -value.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 61

State which inference procedure from Chapter $8,9,$ or 10 you would use. Be specific. For example, you might say, "Two-sample z test for the difference between two proportions." You do not need to carry out any procedures.
Which inference method?
(a) Drowning in bathtubs is a major cause of death in children less than 5 years old. A random sample of parents was asked many questions related to bathtub safety. Overall, $85 \%$ of the sample said they used baby bathtubs for infants. Estimate the percent of all parents of young children who use baby bathtubs.
(b) How seriously do people view speeding in comparison with other annoying behaviors? A large random sample of adults was asked to rate a number of behaviors on a scale of 1 (no problem at all) to 5 (very severe problem ). Do speeding drivers get a higher average rating than noisy neighbors?
(c) You have data from interviews with a random sample of students who failed to graduate from a particular college in 7 years and also from a random sample of students who entered at the same time and did graduate. You will use these data to compare the percents of students from rural backgrounds among dropouts and graduates.
(d) Do experienced computer game players earn higher scores when they play with someone present to cheer them on or when they play alone? Fifty teenagers with experience playing a particular computer game have volunteered for a study. We randomly assign 25 of them to play the game alone and the other 25 to play the game with a supporter present. Each player's score is recorded.

John Long

Numerade Educator

Problem 62

State which inference procedure from Chapter $8,9,$ or 10 you would use. Be specific. For example, you might say, "Two-sample z test for the difference between two proportions." You do not need to carry out any procedures.
Which inference method?
(a) How do young adults look back on adolescent romance? Investigators interviewed 40 couples in their midtwenties. The female and male partners were interviewed separately. Each was asked about his or her current relationship and also about a romantic relationship that lasted at least two months when they were aged 15 or $16 .$ One response variable was a measure on a numerical scale of how much the attractiveness of the adolescent partner mattered. You want to find out how much men and women differ on this measure.
(b) Are more than $75 \%$ of Toyota owners generally satisfied with their vehicles? Let's design a study to find out. We'll select a random sample of 400 Toyota owners. Then we'll ask each individual in the sample: "Would you say that you are generally satisfied with your Toyota vehicle?"
(c) Are male college students more likely to binge drink than female college students? The Harvard School of Public Health surveys random samples of male and female undergraduates at four-year colleges and universities about whether they have engaged in binge drinking.
(d) A bank wants to know which of two incentive plans will most increase the use of its credit cards and by how much. It offers each incentive to a group of current credit card customers, determined at random, and compares the amount charged during the following six months.

John Long

Numerade Educator

Problem 63

Quality control (2.2,5.3,6.3) Many manufacturing companies use statistical techniques to ensure that the products they make meet standards. One common way to do this is to take a random sample of products at regular intervals throughout the production shift. Assuming that the process is working properly, the mean measurement $\bar{x}$ from a random sample varies according to a Normal distribution with mean $\mu_{x}$ and standard deviation $\sigma_{\bar{x}}$. For each question that follows, assume that the process is working properly.
(a) What's the probability that at least one of the next two sample means will fall more than $2 \sigma_{x}$ from the target mean $\mu_{\mathrm{g}}$ ? Show your work.
(b) What's the probability that the first sample mean that is greater than $\mu_{x}+2 \sigma_{x}$ is the one from the fourth sample taken?
(c) Find the probability that at least 4 of the 5 most recent sample means fall outside the interval, assuming the process is working properly. Is this a reasonable criterion? Fxplain.

John Long

Numerade Educator

Problem 64

Information online (8.2,10.1) A random digit dialing sample of 2092 adults found that 1318 used the Internet. ${ }^{42}$ Of the users, 1041 said that they expect businesses to have Web sites that give product information; 294 of the 774 nonusers said this.
(a) Construct and interpret a $95 \%$ confidence interval for the proportion of all adults who use the Internet.
(b) Construct and interpret a $95 \%$ confidence interval to compare the proportions of users and nonusers who expect businesses to have Web sites that give product information.

Sheryl Ezze

Numerade Educator

Problem 65

Coaching and SAT scores: Critique (4.1,4.3) The data in Exercises 55 and 56 came from a random sample of students who took the SAT twice. The response rate was $63 \%,$ which is fairly good for nongovernment survevs.
(a) Explain how nonresponse could lead to bias in this study.
(b) We can't be sure that coaching actually caused the coached students to gain more than the uncoached students. Explain briefly but clearly why this is so.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator