15.3 Comparing clinical therapies A clinical psychologist wants to choose between two therapies for treating severe mental depression. She selects six patients who are similar in their depressive symptoms and overall quality of health.

She randomly selects three patients to receive Therapy 1.

The other three receive Therapy 2. After one month of treatment, the improvement in each patient is measured by the change in a score for measuring severity of mental depression—the higher the change score, the better.

The improvement scores are Therapy 1: 25, 40, 45 Therapy 2: 10, 20, 30

a. Show all possible ways the ranks from 1 to 6 could be distributed between the two treatments. (Hint: There are 20 allocations.)

b. For each possible allocation of ranks, find the mean rank for each treatment and the difference between the mean ranks.

c. Consider the null hypothesis of identical response distributions for the two treatments. Presuming H0 is true, construct the sampling distribution of the difference between the sample mean ranks for the two treatments.

d. For the actual data shown above, find and interpret the P-value for the alternative hypothesis that the two treatments have different effects. (You can check your answers by entering the data in the Permutation Test for Means web app, selecting Wilcoxon rank sum as the test statistic, and selecting the option for generating all possible permutations.)