APPENDIX D 797 c. The 95% confidence interval estimate of the population proportion is 0.459 6 p 6 0.521. For determining whether the proportion of females aged 18–29 who play video games is less than 0.5, the test statistic is z = -0.64 and the Pvalue is 0.2619 (Table: 0.2611), and these values show that the sample proportion of 0.49 is not significantly below 0.5. There is not sufficient evidence to support a claim that the proportion is less than 0.5. (Also, there is not sufficient evidence to support a claim that the proportion is significantly different from 0.5.) 3. a. Key question: Do significantly more males play video games than females? Or is there a significant difference between the proportion of male video game players and the proportion of female video game players? b. Use a hypothesis test or confidence interval with two proportions (as in Section 9-1). c. Conclude that the proportion of male video game players is greater than (or different from) the proportion of female video game players. Using the methods of Section 9-1, the test statistic of z = 10.53 and the P-value of 0.0000 support that claim. Or the 95% confidence interval of 0.188 6 p1 - p2 6 0.272 consists of positive values only, which also supports that claim. 4. a. Key question: Based on the differences between the IQ scores of each matched pair in the sample, is there no difference between the IQ scores of pairs of twins in the population? b. Use the methods of Section 9-3 for matched pairs. c. There does not appear to be a difference between the IQ scores of first-born and second-born twins in the population. Use H0: md = 0 and H1: md ≠ 0. Fail to reject H0. Test statistic: t = 0.150. P@value = 0.8843 (Table: P@value 7 0.20). Critical values: {2.262 (assuming a 0.05 significance level). 95% CI: -5.6 6 md 6 6.4. 5. a. The data are time series data. Key question: What is the trend of the data over time? b. A time series graph would reveal a trend over time. c. A time series graph clearly shows that there is a distinct trend of declining sales of CDs. 6. a. Key question: Is eyewitness memory of police better (or the same as) with a non-stressful interrogation than with a stressful interrogation? b. Use the methods for inferences from two independent means (Section 9-2). 7. H0: m1 = m2 H1: m1 6 m2 Test statistic: t = -2.330. P@value = 0.0102 (Table: 6 0.025). Critical value: t = -1.649 (Table: -1.660). Reject H0. There is sufficient evidence to support the claim that children wearing seat belts have a lower mean length of time in an ICU than the mean for children not wearing seat belts. Buckle up! 8. 90% CI: -0.96 day 6 m1 - m2 6-0.16 day. The confidence interval does not include 0 days, so there appears to be a significant difference. Because the confidence interval consists of negative values only, it appears that children wearing seat belts spend less time in intensive care units than children who don’t wear seat belts. Children should wear seat belts (except for young children who should use properly installed car seats). 9. The waist circumferences from 1988 are measured from eight males, and the waist circumferences from 2012 are from eight different males, so the data are not paired or matched in any meaningful way. The stated claim makes no sense for these two independent samples. Here are the results that would be obtained by blindly following the instruction to test the claim that the differences between the pairs of data are from a population with a mean of 0 mm: H0: md = 0. H1: md ≠ 0. Test statistic: t = -0.834. P@value = 0.4320 (Table: P@value 7 0.20). Critical values: t = {2.365. Fail to reject H0. There is not sufficient evidence to warrant rejection of the claim that the differences between the pairs of data are from a population with a mean of 0 mm. However, these results are not valid and they make no sense with the two independent samples. 10. H0: s1 = s2. H1: s1 ≠ s2. Test statistic: F = 2.9888. P-value: 0.0000. Upper critical F value: 1.3630. Reject H0. There is sufficient evidence to warrant rejection of the claim that for children hospitalized after motor vehicle crashes, the numbers of days in intensive care units for those wearing seat belts and for those not wearing seat belts have the same variation. Chapter 9: Cumulative Review Exercises 1. a. Key question: Does the bar graph correctly depict the data or is it somehow misleading? (Another reasonable question would be to ask whether significantly more males play video games than females, but no sample sizes are provided to help address that question.) b. Just examine the graph to determine whether it is misleading. c. Because the vertical scale begins with 40% instead of 0%, the bottom portion of the bar graph is cut off, so the differences are visually exaggerated. The graph makes it appear that about 3.5 times as many males play video games as females, but examination of the percentage values shows that the ratio is closer to 1.5. 2. a. Key question: Estimate the proportion of all females aged 18– 29 who play video games. Another reasonable question is this: Are females aged 18–29 who play video games in the minority (with a proportion less than 0.5 or 50%)? b. The population proportion can be estimated with a confidence interval. Determination of whether the population proportion is less than 0.5 can be addressed with a hypothesis test of the claim that p 6 0.5.
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