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# Inference for Means and Proportions Free Response Practice Problems for AP Statistics

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Review the following concepts if necessary:

### Problems

1. A large high school has been waging a campaign against drug use, particularly marijuana. Before the campaign began in 2004, a random sample of 100 students from the junior and senior classes found 27 who admitted to using marijuana (we understand that some students who used marijuana would be reluctant to admit it on a survey). To assess the success of their program, in early 2007 they surveyed a random sample of 175 juniors and seniors and 30 responded that they have used marijuana. Is this good evidence that the use of marijuana has been reduced at the school?
2. Twenty-six pairs of identical twins are enrolled in a study to determine the impact of training on ability to memorize a string of letters. Two programs (A and B) are being studied. One member of each pair is randomly assigned to one of the two groups and the other twin goes into the other group. Each group undergoes the appropriate training program, and then the scores for pairs of twins are compared. The means and standard deviations for groups A and B are determined as well as the mean and standard deviation for the difference between each twin's score. Is this study a one-sample or two-sample situation, and how many degrees of freedom are involved in determining the t-value?
3. Which of the following statements is (are) correct? Explain.
1. A confidence interval can be used instead of a test statistic in any hypothesis test involving means or proportions.
2. A confidence interval can be used instead of a test statistic in a two-sided hypothesis test involving means or proportions.
3. The standard error for constructing a confidence interval for a population proportion and the standard error for a significance test for a population proportion are the same.
4. The standard error for constructing a confidence interval for a population mean and the standard error for a significance test for a population mean are the same.
4. The average math SAT score at Hormone High School over the years is 520. The mathematics faculty believes that this year's class of seniors is the best the school has ever had in mathematics. One hundred seventy-five seniors take the exam and achieve an average score of 531 with a sample standard deviation of 96. Does this performance provide good statistical evidence that this year's class is, in fact, superior?
5. An avid reader, Booker Worm, claims that he reads books that average more than 375 pages in length. A random sample of 13 books on his shelf had the following number of pages: 595, 353, 434, 382, 420, 225, 408, 422, 315, 502, 503, 384, 420. Do the data support Booker's claim? Test at the 0.05 level of significance.
6. The statistics teacher, Dr. Tukey, gave a 50-point quiz to his class of 10 students and they didn't do very well, at least by Dr. Tukey's standards (which are quite high). Rather than continuing to the next chapter, he spent some time reviewing the material and then gave another quiz. The quizzes were comparable in length and difficulty. The results of the two quizzes were as follows.
7. Do the data indicate that the review was successful, at the .05 level, at improving the performance of the students on this material? Give good statistical evidence for your conclusion.

8. The new reality TV show, "I Want to Marry a Statistician," has been showing on Monday evenings, and ratings show that it has been watched by 55% of the viewing audience each week. The producers are moving the show to Wednesday night but are concerned that such a move might reduce the percentage of the viewing public watching the show. After the show has been moved, a random sample of 500 people who are watching television on Wednesday night are surveyed and asked what show they are watching. Two hundred fifty-five respond that they are watching "I Want to Marry a Statistician." Does this finding provide evidence at the 0.01 level of significance that the percentage of the viewing public watching "I Want to Marry a Statistician" has declined?
9. Harvey is running for student body president. An opinion poll conducted by the AP Statistics class does a survey in an attempt to predict the outcome of the election. They randomly sample 30 students, 16 of whom say they plan to vote for Harvey. Harvey figures (correctly) that 53.3% of students in the sample intend to vote for him and is overjoyed at his soon-to-be-celebrated victory. Explain carefully why Harvey should not get too excited until the votes are counted.
10. A company uses two different models, call them model A and model B, of a machine to produce electronic locks for hotels. The company has several hundred of each machine in use in its various factories. The machines are not perfect, and the company would like to phase out of service the one that produces the most defects in the locks. A random sample of 13 model A machines and 11 model B machines are tested and the data for the average number of defects per week are given in the following table.
11. Dotplots of the data indicate that there are no outliers or strong skewness in the data and that there are no strong departures from normal. Do these data provide statistically convincing evidence that the two machines differ in terms of the number of defects produced?

12. Take another look at the preceding problem. Suppose there were 30 of each model machine that were sampled. Assuming that the sample means and standard deviations are the same as given in question 9, how might this have affected the hypothesis test you performed in that question?
13. The directors of a large metropolitan airport claim that security procedures are 98% accurate in detecting banned metal objects that passengers may try to carry onto a plane. The local agency charged with enforcing security thinks the security procedures are not as good as claimed. A study of 250 passengers showed that screeners missed nine banned carry-on items. What is the P-value for this test and what conclusion would you draw based on it?
14. A group of 175 married couples are enrolled in a study to see if women have a stronger reaction than men to videos that contain violent material. At the conclusion of the study, each couple is given a questionnaire designed to measure the intensity of their reaction. Higher values indicate a stronger reaction. The means and standard deviations for all men, all women, and the differences between husbands and wives are as follows:
15. Do the data give strong statistical evidence that wives have a stronger reaction to violence in videos than do their husbands? Assume that σ for the differences is 1.77.

16. An election is bitterly contested between two rivals. In a poll of 750 potential voters taken 4 weeks before the election, 420 indicated a preference for candidate Grumpy over candidate Dopey. Two weeks later, a new poll of 900 randomly selected potential voters found 465 who plan to vote for Grumpy. Dopey immediately began advertising that support for Grumpy was slipping dramatically and that he was going to win the election. Statistically speaking (say at the 0.05 level), how happy should Dopey be (i.e., how sure is he that support for Grumpy has dropped)?
17. Consider, once again, the situation of question #7 above. In that problem, a onesided, two-proportion z-test was conducted to determine if there had been a drop in the proportion of people who watch the show "I Want to Marry a Statistician" when it was moved from Monday to Wednesday evenings.
18. Suppose instead that the producers were interested in whether the popularity ratings for the show had changed in any direction since the move. Over the seasons the show had been broadcast on Mondays, the popularity rating for the show (10 high, 1 low) had averaged 7.3. After moving the show to the new time, ratings were taken for 12 consecutive weeks. The average rating was determined to be 6.1 with a sample standard deviation of 2.7. Does this provide evidence, at the 0.05 level of significance, that the ratings for the show has changed? Use a confidence interval, rather than a t-test, as part of your argument. A dotplot of the data indicates that the ratings are approximately normally distributed.

19. A two-sample study for the difference between two population means will utilize t-procedures and is to be done at the 0.05 level of significance. The sample sizes are 23 and 27. What is the upper critical value (t*) for the rejection region if
1. the alternative hypothesis is one-sided, and the conservative method is used to determine the degrees of freedom?
2. the alternative hypothesis is two-sided and the conservative method is used to determine the degrees of freedom?
3. the alternative hypothesis is one-sided and you assume that the population variances are equal?
4. the alternative hypothesis is two-sided, and you assume that the population variances are equal?

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