454 CHAPTER 9 Inferences from Two Samples Replace n1 and n2 by n in the preceding formula (assuming that both samples have the same size) and replace each of p1, q1, p2, and q2 by 0.5 (because their values are not known). Solving for n results in this expression: n = z2 a>2 2E2 Use this expression to find the size of each sample if you want to estimate the difference between the proportions of men and women who shop online. Assume that you want 95% confidence that your error is no more than 0.02. 24.Yawning and Fisher’s Exact Test In one segment of the TV series MythBusters, an experiment was conducted to test the common belief that people are more likely to yawn when they see others yawning. In one group, 34 subjects were exposed to yawning, and 10 of them yawned. In another group, 16 subjects were not exposed to yawning, and 4 of them yawned. We want to test the belief that people are more likely to yawn when they are exposed to yawning. a. Why can’t we test the claim using the methods of this section? b. If we ignore the requirements and use the methods of this section, what is the P-value? How does it compare to the P-value of 0.5128 that would be obtained by using Fisher’s exact test? c. Comment on the conclusion of the MythBusters segment that yawning is contagious. 25.Overlap of Confidence Intervals In the article “On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals,” by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement: “Independent simple random samples, each of size 200, have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute.” a. Use the methods of this section to construct a 95% confidence interval estimate of the difference p1 - p2. What does the result suggest about the equality of p1 and p2? b. Use the methods of Section 7-1 to construct individual 95% confidence interval estimates for each of the two population proportions. After comparing the overlap between the two confidence intervals, what do you conclude about the equality of p1 and p2? c. Use a 0.05 significance level to test the claim that the two population proportions are equal. What do you conclude? d. Based on the preceding results, what should you conclude about the equality of p1 and p2? Which of the three preceding methods is least effective in testing for the equality of p1 and p2? 26.Equivalence of Hypothesis Test and Confidence Interval Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of p1 = p2 (with a 0.05 significance level) and a 95% confidence interval estimate of p1 - p2. 27.Large Data Sets Use a 0.01 significance level to test the claim that the proportion of males in the sample in Data Set 2 “ANSUR I 1988” is the same as the proportion of males in the sample in Data Set 3 “ANSUR II 2012.” Key Concept This section presents methods for using sample data from two independent samples to (1) test hypotheses made about two population means or to (2) construct confidence interval estimates of the difference between two population means. In Part 1 we discuss situations in which the standard deviations of the two populations are unknown and are not assumed to be equal. In Part 2 we briefly discuss two other 9-2 Two Means: Independent Samples
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