15.1 Group A ranks are 4, 5, 6, 8, 9. Group B ranks are 1, 2, 3, 7, 10.

15.3 H0: No difference in distribution of the number of rooms between the 25 top-ranked hotels and the hotels ranked 26 to 50. Ha: There is a systematic difference in distribution of the number of rooms between the 25 top-ranked and hotels ranked 26 to 50. W=4+5+6+8+9=32.

15.5 μW=27.5, σW=4.7871, W=32, z=0.94, P-value=0.1736.

15.7

1. For exergamers, no TV: 2.14%, some TV: 56.94%, more than 2 hours TV: 40.93%. For non-exergamers, no TV: 5.22%, some TV: 67.03%, more than 2 hours: 27.75%.
2. χ2=20.068, df=2, P-value<0.0005.

15.9 W+=22.

15.11 μW+=27.5, σW+=9.811, W=22, z=−0.61, P-value=0.5418.

15.1 Early class ranks are 3, 4, 7, 8, 10, 12. Late class ranks are 1, 2, 5, 6, 9, 11.

15.3 W=44.

15.5 H0: No difference in distribution of study times between early and late classes. Ha: There is a systematic difference in distribution of study times between early and late classes.

15.7 μW=39, σW=6.245.

15.9 z=0.88, P-value=0.3788. There is not enough evidence to show a systematic difference in the study times between early and late classes.

15.11 Men and women are not significantly different. W=1421, P-value=0.6890 (two-sided). The t test assumes Normality, with small samples this may not be true; t=−0.11, P-value=0.9155.

15.13

1. The differences are 3.0, 4.9, 0.1, 3.0, 1.1, 1.1, 5.0.

• (b–c) There are no negative differences, so the absolute value is the same.

15.15 W+=2.

15.17 H0: No difference in distribution of gas mileage between computer recorded and driver recorded. Ha: There is a systematic difference in distribution of gas mileage between computer recorded and driver recorded.

15.19 μW+=14, σW+=5.916.

15.21 z=2.45, P-value=0.0142. There is enough evidence to show a systematic difference in gas mileage recorded between the computer and the driver.

15.23 W+=119, P-value=0.001.

15.25 If we compute the Haiti content minus factory content (so that a negative difference means that the amount of vitamin C decreased), we find that the mean change is −55, and the median is −53. All five differences are negative; the Wilcoxon statistic is W+=0, for which the P-value<0.03. The differences are systematically negative, so vitamin C content is lower in Haiti.

15.27 We want to compare the attractiveness scores for k=5 independent samples (the 102, 302, 502, 702, and 902 friend groups of subjects). Under the null hypothesis for ANOVA, each population is N(μ, σ). An F test is used to compare the group means. The Kruskal-Wallis test only assumes a continuous distribution in each population, and it uses a chi-square distribution for the test statistic.

15.29 The Kruskal-Wallis hypotheses are H0: all distributions are the same and Ha: some distributions are higher than others. A nonparametric procedure is appropriate because the distribution of the response variable is likely not normal due to the small range of values for the response.

15.31 Minitab gives the median of each group and the average rank for each group. Using the “adjusted for ties” values, H=17.05, P-value=0.002.

15.33

1. For testing H0: The distribution of age at death is the same for all three groups versus Ha: At least one group is systematically higher or lower. From software, H=11.11 with df=2, for which P=0.004.
2. ANOVA yields F=6.56(df=2, 120) and P-value=0.002. The conclusion is the same with either test.

15.35

1. Summary statistics are shown below. We note the outlier in the female distribution at 360 minutes.

   Gender	x¯	s	M
   F	164.8	56.5	170
   M	116.8	74.4	120

2. We test H0: All distributions are the same and Ha: Some distributions are higher than others. From software W=1105.5, P-value=0.0046. This test rejects the null hypothesis, and we conclude that the distributions are not the same.
3. We test H0: H0:   μF=μM versus Ha:   μF≠μM. t=2.82, P-value=0.007(df=54). The conclusions of the two tests are the same.
4. With sample sizes n=30, t tests would be robust to the departures from Normality seen in the boxplots (skew and outlier). Either test would be appropriate here.

15.37

1. Bihai red, bihai yellow, and red-yellow.
2. W1=504, W2=744, W3=127. All P-values are 0.0000.
3. All three are easily significant at the overall 0.05 level.

15.39 For meat, W=15 and P-value=0.4705, and for legumes, W=10.5 and P-value=0.0433.

15.41 Multiple comparisons are appropriate as a follow-up to a significant result from a Kruskal-Wallis test. This means we have three comparisons from each of these exercises, for a total of 6. In order to be significant at the overall 0.05 level, an individual P-value must be less than 0.05/6=0.0083. None of the differences are significant at this level; with such small samples, these tests have low power. (For samples of size 4, W must be between 10 and 26, so five of the six P-values are as small as they can be.)