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

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 Bio. 6.36 The effect of Benzedrine on the heart rate of dogs in beats per minute was examined in an experiment on 14 dogs chosen for the study. Each dog was to serve as its own control, with half of the dogs assigned to receive Benzedrine during the first study period and the other half assigned to receive a placebo saline solution. All dogs were examined to determine the heart rates after 2 hours on the medication. After 2 weeks in which no medication was given, the regimens for the dogs were switched for the second study period. The dogs previously on Benzedrine were given the placebo and the others received Benzedrine. Again heart rates were measured after 2 hours. The following sample data are not arranged in the order in which they were taken but have been summarized by regimen. Use these data to test the research hypothesis that the distribution of heart rates for the dogs when receiving Benzedrine is shifted to the right of that for the same animals when on the placebo. Use a one-tailed Wilcoxon signed-rank test with a ⫽ .05. Dog Placebo Benzedrine Dog Placebo Benzedrine 1 250 258 8 296 305 2 271 285 9 301 319 3 243 245 10 298 308 4 252 250 11 310 320 5 266 268 12 286 293 6 272 278 13 306 305 7 293 280 14 309 313 6.6 Choosing Sample Sizes for Inferences about M 1 ⴚ M 2 Med. 6.37 A study is being planned to evaluate the possible side effects of an anti-inflammatory drug. It is suspected that the drug may lead to an elevation in the blood pressure of users of the drug. A pre- liminary study of two groups of patients, one receiving the drug and the other receiving a placebo, provides the following information on the systolic blood pressure in mm Hg of the two groups: Group Mean Standard Deviation Placebo 129.9 18.5 Users of drug 135.5 18.7 Assume that both groups have systolic blood pressures that have a normal distribution with stan- dard deviations relatively close to the values obtained in the pilot study. Suppose the study plan provides for the same number of patients in the placebo as in the treatment group. Determine the sample size necessary for an a ⫽ .05 t test to have power of .80 to detect an increase of 5 mm Hg in the blood pressure of the treatment group relative to the placebo group. Med. 6.38 Refer to Exercise 6.37. Suppose that the agency sponsoring the study specifies that the group receiving the drug should have twice as many patients as the placebo group. Determine the sample sizes necessary for an a ⫽ .05 t test to have power of .80 to detect an increase of 5 mm Hg in the blood pressure of the treatment group relative to the placebo group. Med. 6.39 Refer to Exercise 6.37. The researchers also need to obtain precise estimates of the mean difference in systolic blood pressure for people who use the anti-inflammatory drug versus those who do not.

a. Suppose the sample sizes are the same for both groups. What sample size is needed

to obtain a 95 confidence interval for the mean difference in systolic blood pressure between the users and nonusers having a width of at most 5 mm Hg.

b. Suppose the user group will have twice as many patients as the placebo group. What

sample size is needed to obtain a 95 confidence interval for the mean difference in sys- tolic blood pressure between the users and nonusers having a width of at most 5 mm Hg. Env. 6.40 An environmental impact study was performed in a small state to determine the effective- ness of scrubbers on the amount of pollution coming from the cooling towers of a chemical plant.