APPENDIX D 795 5. a. Randomization: The difference between the two sample proportions is 0.0625. Results vary but this is typical: Among 1000 resamplings, differences at least as extreme as 0.0625 occurred 695 times. It appears that by chance, it is easy to get a difference like the one obtained, so there is not sufficient evidence to warrant rejection of the claim that when dropped, buttered toast and toast marked with an X have the same proportion that land with the buttered>X side down. b. Bootstrapping: Results vary but this is typical: 95% CI: -0.250 6 p1-p2 6 0.125. Because the confidence interval limits do contain 0, there is a not a significant difference between the two sample proportions. There is not sufficient evidence to warrant rejection of the claim that when dropped, buttered toast and toast marked with an X have the same proportion that land with the buttered>X side 7. a. Randomization: The difference between the two sample proportions is 0.196. Results vary but this is typical: Among 1000 resamplings, differences at least as extreme as -0.196 never occurred. It appears that by chance, it is very difficult to get a difference like the one obtained, so there is sufficient evidence to support the claim that the rate of right-handedness for those who prefer to use their left ear for cell phones is less than the rate of right-handedness for those who prefer to use their right ear for cell phones. b. Bootstrapping: Results vary but this is typical: 98% CI: -0.266 6 p1 - p2 6 -0.127. Because the confidence interval limits do not contain 0, there is a significant difference between the two sample proportions. There is sufficient evidence to support the claim that the rate of right-handedness for those who prefer to use their left ear for cell phones is less than the rate of right-handedness for those who prefer to use their right ear for cell phones. 9. a. Randomization: Using the sample data, we get x1-x2 = -4.57 kg. If we use randomization with the two sets of sample data to generate 1000 simulated differences, a typical result is that 150 of those differences will be -4.57 kg or below, so it appears that such differences can easily occur. There is not sufficient evidence to support the claim that the mean weight of the 1988 male population is less than the mean weight of the 2012 male population. b. Bootstrapping: Results vary but this is typical: 90% CI: -11.38 kg 6 m1 - m2 6 2.21 kg. Because the confidence interval does include 0, it appears that there is not a significant difference between the mean weight in 1988 and the mean weight in 2012. There is not sufficient evidence to support the claim that the mean weight of the 1988 male population is less than the mean weight of the 2012 male population. 11. a. Randomization: Using the sample data, we get x1 - x2 = 0.03317 lb. If we use randomization to generate 1000 simulated differences, a typical result is that none of those differences will be at least as extreme as 0.03317 lb, so it appears that such differences are very rare. There is sufficient evidence to support the claim that the contents of cans of regular Coke have weights with a mean that is greater than the mean for Diet Coke. 3. No. Unlike some other tests that have a requirement that samples must be from normally distributed populations or the samples must have more than 30 values, the F test has a requirement that the samples must be from normally distributed populations, regardless of how large the samples are. 5. H0: s1 = s2 and H1: s1 7 s2. Test statistic: F = 1.9728. P-value: 0.0184. Critical F value is 1.7045 (Table: Approximately 1.6928). Reject H0. There is sufficient evidence to support the claim that the variation of weights before 1964 is greater than the variation of weights after 1964. Here is one advantage of the change in variation: Weights of quarters now have less variation than they did before 1964, so vending machines can be calibrated to accept a narrower range of weights, and counterfeit coins will be less likely to be accepted. 7. H0: s1 = s2. H1: s1 ≠ s2. Test statistic: F = 2.3706. P-value: 0.0129. Upper critical F value: 1.9678 (Table: Upper critical F value is between 1.8752 and 2.0739). Reject H0. There is sufficient evidence to warrant rejection of the claim that creative task scores have the same variation with a red background and a blue background. 9. H0: s1 = s2 and H1: s1 ≠ s2. Test statistic: F = 9.3364. P-value: 0.0000. Critical F value is 2.4086 (Table: Critical F value is between 2.3675 and 2.4247). Reject H0. There is sufficient evidence to reject the claim that the treatment and placebo groups have the same amount of variation among the errors. 11. H0: s1 = s2 and H1: s1 7 s2. Test statistic: F = 1.6531. P-value: 0.0966. Critical F value is 1.8915 (Table: Between 1.8543 and 1.9005). Fail to reject H0. There is not sufficient evidence to support the claim that commuting times with the lighter bicycle have more variation than commuting times with the heavier bicycle. 13. H0: s1 = s2 and H1: s1 7 s2. Test statistic: F = 3.8134. P-value: 0.0340. Critical F value is 3.3129 (Table: Between 3.3472 and 3.2839). Reject H0. There is sufficient evidence to support the claim that the variation among pulse rates of females is greater than the variation among males. 15. H0: s1 = s2 and H1: s1 7 s2. Test statistic: F = 2.5285. P-value: 0.0213. Critical F value is 2.1124 (Table: Critical F value is between 2.0825 and 2.1242). Reject H0. There is sufficient evidence to support the claim that IQ scores of subjects with medium lead levels vary more than IQ scores of subjects with high lead levels. 17. c1 = 3, c2 = 0, critical value is 7.4569. Fail to reject H0. There is not sufficient evidence to support a claim that the two populations of scores have different amounts of variation. 19. FL = 0.4103; FR = 2.7006 Section 9-5 1. Bootstrapping is used for obtaining a confidence interval estimate, and it involves sampling with replacement of selected sample values. Randomization is used for testing hypotheses. When working with two independent samples, randomization involves sampling without replacement; when working with matched pairs, randomization involves sampling with replacement. 3. Only part (b) is a randomization. Only part (b) has the same original sample sizes with the sample data selected without replacement.
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