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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.

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.

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.

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.

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.

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)?

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.

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.

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) \(1.960 \sqrt{\frac{0.880(0.120)}{550}+\frac{0.820(0.180)}{500}}\)

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.