808 APPENDIX D 7. R1 = 253.5, R2 = 124.5, mR = 182, sR = 20.607, test statistic: z = 3.47. P-value: 0.0005. Critical values: z = {1.96. Reject the null hypothesis that the populations have the same median. There is sufficient evidence to reject the claim that for those treated with 20 mg of Lipitor and those treated with 80 mg of Lipitor, changes in LDL cholesterol have the same median. It appears that the dosage amount does have an effect on the change in LDL cholesterol. 9. R1 = 7236.5, R2 = 7298.5, mR = 7267.5, sR = 320.868, test statistic: z = -0.10. P-value: 0.923. Critical values: z = {2.575. Fail to reject the null hypothesis that the populations have the same median. There is not sufficient evidence to warrant rejection of the claim that cars in two lines have a median waiting time equal to that of cars in a single line. 11. R1 = 501, R2 = 445, mR = 484, sR = 41.15823, test statistic: z = 0.41. P-value: 0.3409. Critical value: z = 1.645. Fail to reject the null hypothesis that the populations have the same median. There is not sufficient evidence to support the claim that subjects with medium lead levels have a higher median of the full IQ scores than subjects with high lead levels. Based on these data, it does not appear that lead level affects full IQ scores. 13. Using n1 = 12 and n2 = 15, we get U = 131 and z = 2.00. The test statistic from the Mann-Whitney U test is the same test statistic from the Wilcoxon rank-sum test but with opposite sign. Section 13-5 1. R1 = 103.5, R2 = 48.5, R3 = 90, R4 = 83 3. n1 = 7, n2 = 5, n3 = 6, n4 = 7, and N = 25 5. Test statistic: H = 2.029. Critical value: x2 = 7.815. (Tech: P@value = 0.566.) Fail to reject the null hypothesis of equal medians. There is not sufficient evidence to warrant rejection of the claim that small, midsize, large, and SUV vehicles have the same median HIC measurement in car crash tests. 7. Test statistic: H = 16.949. Critical value: x2 = 9.210 (Tech: P@value = 0.0002.) Reject the null hypothesis of equal medians. There is sufficient evidence to warrant rejection of the claim that pages from books by those three authors have the same median Flesch Reading Ease score. At least one of the books has a median different from the others. 9. Test statistic: H = 2.745. Critical value: x2 = 11.071. (Tech: P@value = 0.739.) Fail to reject the null hypothesis of equal medians. There is not sufficient evidence to warrant rejection of the claim that the six different colors of M&M candies have the same median weight. It does not seem feasible that the color would have much of an effect on the weight, so the result is as expected. 11. Test statistic: H = 2.5999. Critical value: x2 = 7.815. (Tech: P-value = 0.458.) Fail to reject the null hypothesis of equal medians. It appears that the four hospitals have birth weights with the same median. 13. The values of t are 2, 2, 2, 2, and 4, so the values of T are 6, 6, 6, 6, and 60 and ΣT = 84. Using ΣT = 84 and N = 19, the corrected value of H is 0.703, which is not substantially different from the uncorrected value of 0.694. In this case, the large numbers of ties do not appear to have a dramatic effect on the test statistic H. Section 13-3 1. a. The only requirements are that the matched pairs be a simple random sample and the population of differences be approximately symmetric. b. There is no requirement of a normal distribution or any other specific distribution. c. The Wilcoxon signed-ranks test is “distribution-free” in the sense that it does not require a normal distribution or any other specific distribution. 3. The sign test uses only the signs of the differences, but the Wilcoxon signed-ranks test uses ranks that are affected by the magnitudes of the differences. 5. Test statistic: T = 22. Critical value: T = 8. Fail to reject the null hypothesis that the population of differences has a median of 0. There is not sufficient evidence to warrant rejection of the claim of no difference between measured weights and reported weights of females. 7. Convert T = 612 to the test statistic z = -1.52. P-value: 0.1285 (Table: 0.1286). Critical values: z = {1.96. There is not sufficient evidence to warrant rejection of the claim of no significant difference between the numbers of words spoken by males and females in couple relationships. Based on the given data, there does not appear to be a significant difference. 9. Convert T = 491.5 to the test statistic z = -5.77. P-value: 0.0000. Critical values: z = {2.575. There is sufficient evidence to warrant rejection of the claim that the median is equal to 98.6°F. 11. Convert T = 19 to the test statistic z = -26.01. P-value: 0.0000. Critical values: z = {2.575. There is sufficient evidence to warrant rejection of the claim that smokers have a median cotinine level equal to 2.84 ng>mL. 13. a. 0 and 1275 b. 637.5 c. 1110 d. n1n + 12 2 - k Section 13-4 1. Yes. The two samples are independent. (The heights from ANSUR I and ANSUR II are not matched in any way.) Each sample has more than 10 values. 3. H0: The sample of heights from ANSUR I and the sample of heights from ANSUR II are from populations with the same median. H1: The sample of heights from ANSUR I and the sample of heights from ANSUR II are from populations with different medians. H1: The ANSUR I sample is from a population with a lower median than the median of the second population. H1: The ANSUR I sample is from a population with a higher median than the median of the second population. 5. R1 = 172, R2 = 206, mR = 168, sR = 20.4939, test statistic: z = 0.20. P-value: 0.845. Critical values: z = {1.96. Fail to reject the null hypothesis that the populations have the same median. There is not sufficient evidence to warrant rejection of the claim that the sample of female heights from ANSUR I and the sample of female heights from ANSUR II are from populations with the same median.
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