810 APPENDIX D 6. The test statistic of x = 3 is not less than or equal to the critical value of 1 (from Table A-7). There is not sufficient evidence to warrant rejection of the claim that the sample is from a population with a median equal to 15 minutes. 7. Test statistic T = 15.5 is not less than or equal to the critical value of 8. There is not sufficient evidence to warrant rejection of the claim that the sample is from a population with a median equal to 15 minutes. 8. n1 = 14, n2 = 16, G = 17, critical values: 10, 22. Fail to reject randomness. There is not sufficient evidence to warrant rejection of the claim that odd and even digits occur in random order. The lottery appears to be working as it should. 9. R1 = 204.5, R2 = 230.5, mR = 255, sR = 22.58318, test statistic: z = -2.24. Tech: P-value: 0.025. Critical values: z = {1.96. Reject the null hypothesis that the populations have the same median. There is sufficient evidence to warrant rejection of the claim that the recent eruptions and past eruptions have the same median time interval between eruptions. The conclusion does change with a 0.01 significance level. 10. rs = 0.714. Critical values: {0.738. Fail to reject the null hypothesis of rs = 0. There is not sufficient evidence to support the claim that there is a correlation between the student ranks and the magazine ranks. When ranking colleges, students and the magazine do not appear to agree. Chapter 13: Cumulative Review Exercises 1. x = 4.5574 g, median = 4.5185 g, range = 0.3860 g, s = 0.1192 g, s2 = 0.0142 g2 2. The normal quantile plot shows that the points approximate a straight-line pattern, so the weights do appear to be from a population having a normal distribution. 3. H0: m = 4.5333 g. H1: m ≠ 4.5333 g. Test statistic: t = 0.640. P@value = 0.5384 (Table: P@value 7 0.20). Critical values: t = {2.262. Fail to reject H0. There is not sufficient evidence to warrant rejection of the claim that the sample is from a population with mean equal to 4.533 g. The claim of 340 g appears to be valid. 4. The test statistic of x = 4 is not less than or equal to the critical value of 1 (from Table A-7). There is not sufficient evidence to warrant rejection of the claim that the sample of weights is from a population with a median of 4.5333 g. 4. rs = 1. There is sufficient evidence to conclude that there is a correlation between the time of travel and the distance traveled. 5. Rank correlation can be used in a wider variety of circumstances than linear correlation. Rank correlation does not require a normal distribution for any population. Rank correlation can be used to detect some (not all) relationships that are not linear. 6. The sign test can be used to test claims involving matched pairs of sample data, it can be used to test claims involving nominal data with two categories, and it can be used to test claims about the median of a single population. 7. Because there are only two runs, all of the values below the mean occur at the beginning and all of the values above the mean occur at the end, or vice versa. This indicates the presence of an upward (or downward) trend. 8. Because the sign test uses only signs of differences while the Wilcoxon signed-ranks test uses ranks of the differences, the Wilcoxon signed-ranks test uses more information about the data and tends to yield conclusions that better reflect the true nature of the data. 9. The Wilcoxon signed-ranks test is used to test a claim that a population of matched pairs has the property that they have differences with a median equal to zero or to test a claim that a single population of individual values has a median equal to some claimed value, whereas the Wilcoxon rank-sum test is used to test the null hypothesis that two independent samples are from populations having equal medians. 10. One-way analysis of variance can be used instead of the KruskalWallis test. Like many other nonparametric tests, the KruskalWallis test has no requirement that the populations have a normal distribution or any other particular distribution. Chapter 13: Review Exercises 1. The test statistic of x = 0 is less than or equal to the critical value of 0 (from Table A-7). There is sufficient evidence to warrant rejection of the claim that for the population of freshman male college students, there is not a significant difference between the weights in September and the weights in the following April. Based on the given data, there does appear to be a significant difference. The test does not address the specific weight gain of 15 lb. 2. T = 0. The critical value is T = 2. There is sufficient evidence to warrant rejection of the claim that the median of the differences is equal to 0. There does appear to be a difference. The test does not address the specific weight gain of 15 lb. 3. rs = 0.983. Critical values: -0.700 and 0.700. Reject the null hypothesis of rs = 0. There is sufficient evidence to support the claim of a correlation between the September weights and the April weights. The presence of a correlation tells us nothing about the belief that college students gain 15 lb (or 6.8 kg) during their freshman year. 4. Test statistic: H = 0.465. Critical value: x2 = 5.991 (Tech: P@value = 0.792.) Fail to reject the null hypothesis of equal medians. There is not sufficient evidence to warrant rejection of the claim that the lengths of stay at the three hospitals have the same median. 5. Test statistic: z = -1.59. P@value = 0.1113. The P-value is not small, so fail to reject the null hypothesis of p = 0.5. There is not sufficient evidence to warrant rejection of the claim that in each World Series, the American League team has a 0.5 probability of winning.
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