Use the following computer output to evaluate whether there is a difference in the Estimate the size of the difference in the mean final grades of the students in academic

c. Do the conditions for using the t procedures appear to be satisfied for these data? d. Does it appear that using twins in this study to control for variation in final scores was effective as compared to taking a random sample of 30 students in both types of home environments? Justify your answer. 6.5 A Nonparametric Alternative: The Wilcoxon Signed-Rank Test 6.30 Set up the rejection regions of the Wilcoxon signed-rank procedure for testing the following:

a. H : M ⫽ 0 versus H

a : M 0, with n 1 ⫽ 11, n 2 ⫽ 14, and a ⫽ .05

b. H : M ⱕ 0 versus H

a : M ⬎ 0, with n 1 ⫽ n 2 ⫽ 17, and a ⫽ .01

c. H : M ⱖ 0 versus H

a : M ⬍ 0, with n 1 ⫽ 8, n 2 ⫽ 12, and a ⫽ .025

6.31 Consider the data given in Exercise 6.23. a. Conduct a Wilcoxon signed-rank test of H

: M ⱕ 0 versus H a : M ⬎ 0 with d ⫽ y 1 ⫺ y 2 . Use a ⫽ .05.

b. Compare your conclusions here to those given in Exercise 6.23. Does it matter which

test t or Wilcoxon signed-rank test is applied to these data?

6.32 Refer to the data of Exercise 6.31. a. Give the level of significance for your test.

b. Place a 95 confidence interval on the median difference, M.

⫽ .999 .99 .95 .80 .50 .20 .05 .01 .001 –5 5 10 Differences Probability Boxplot of differences with H and 95 t confidence interval for the mean [ ] y H –5 5 10 Differences Normal probability plot of differences Two-Sample T-Test and Confidence Interval Two-sample T for Academic vs Nonacademic N Mean StDev SE Mean Academic 30 75.2 13.3 2.4 Nonacademic 30 71.4 11.4 2.1 95 CI for mu Academic mu Nonacademic: 2.6, 10.2 T-Test mu Academic mu Nonacademic vs not : T 1.19 P 0.24 DF 56 Paired T-Test and Confidence Interval Paired T for Academic Nonacademic N Mean StDev SE Mean A cademic 30 75.23 13.29 2.43 Nonacademic 30 71.43 11.42 2.09 Difference 30 3.800 4.205 0.768 95 CI for mean difference: 2.230, 5.370 T-Test of mean difference 0 vs not 0: T-value 4.95 P-Value 0.000

6.33 Use the level and power values for the paired t test and Wilcoxon signed-rank test given in

Table 6.18 to answer the following questions. a. For small sample sizes, n ⱕ 20, does the actual level of the t test appear to deviate from the nominal level of a ⫽ .05?

b. Which type of deviations from a normal distribution, skewness or heavy-tailedness,

appears to have the greater affect on the t test?

c. For small sample sizes, n ⱕ 20, does the actual level of the Wilcoxon signed-rank test

appear to deviate from the nominal level of a ⫽ .05?

d. Which type of deviations from a normal distribution, skewness or heavy-tailedness,

appears to have the greater effect on the Wilcoxon signed-rank test?

6.34 Use the level and power values for the paired t test and Wilcoxon signed-rank test given in

Table 6.18 to answer the following questions: a. Suppose a level .05 test is to be applied to a paired data set that has differences which are highly skewed to the right. Will the Wilcoxon signed-rank test’s “actual” level or the paired t test’s actual level be closer to .05? Justify your answer.

b. Suppose a boxplot of the differences in the pairs from a paired data set has many out-

liers, an equal number above and below the median. If a level a ⫽ .05 test is applied to the differences, will the Wilcoxon signed-rank test’s “actual” level or the paired t test’s actual level be closer to .05? Justify your answer. Soc. 6.35 A study was conducted to determine whether automobile repair charges are higher for female customers than for male customers. Ten auto repair shops were randomly selected from the telephone book. Two cars of the same age, brand, and engine problem were used in the study. For each repair shop, the two cars were randomly assigned to a man and woman participant and then taken to the shop for an estimate of repair cost. The repair costs in dollars are given here. Repair Shop 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

19 20

Female customers 871 684 795 838 1,033 917 1,047 723 1,179 707 817 846 975 868 1,323 791 1,157 932 1,089 770 Male customers 792 765 511 520 618 447 548 720 899 788 927 657 851 702 918 528 884 702 839 878 a. Which procedure, t or Wilcoxon, is more appropriate in this situation? Why? b. Are repair costs generally higher for female customers than for male customers? Use a ⫽ .05. 500 400 300 200 100 –100 .001 .01 .05 .20 .50 .80 .95 .99 .999 Probability Difference Normal probability plot of differences in cost [ ] y H –100 100

200 300

400 500 Difference Boxplot of differences with H and 95 t confidence interval for the mean