Question

There were ten patients who had depression, and each patient tried a medication. Here we compare the control (no medication) and drug. Does the drug offer an improvement in depression? Table 1 shows the depression sores (0 = No depression, 30 = Strong depression), differences, and signed ranks. Test a hypothesis with Wilcoxon signed-rank test at significance level .05.

Table 1: The Psychology Experiments at a Research Institution

Participant	1	2	3	4	5	6	7	8	9	10
Drug	5	6	30	8	9	7	6	17	3	10
Control	16	15	20	15	16	13	14	19	18	18
Difference	-11	-9	+10	-7	-7	-6	-8	-2	-15	-8
Signed rank	-9	-7	+8	-3.5	-3.5	-2	-5.5	-1	-10	-5.5

A. The value of the test statistic for the hypothesis test is _________.

B. The critical value for the hypothesis test is _________.

C. State decision rule for the hypothesis test.

D. State a conclusion for the hypothesis test.

Answer 1

Pair	Sample 1	Sample 2	Difference	Abs. Difference	Sign
1	5	16	-11	11	-1
2	6	15	-9	9	-1
3	30	20	10	10	+1
4	8	15	-7	7	-1
5	9	16	-7	7	-1
6	7	13	-6	6	-1
7	6	14	-8	8	-1
8	17	19	-2	2	-1
9	3	18	-15	15	-1
10	10	18	-8	8	-1

Now, the following table is obtained by removing the ties and organizing the absolute differences in ascending order:

Pair	Sample 1	Sample 2	Difference	Abs. Difference	Sign
8	17	19	-2	2	-1
6	7	13	-6	6	-1
4	8	15	-7	7	-1
5	9	16	-7	7	-1
7	6	14	-8	8	-1
10	10	18	-8	8	-1
2	6	15	-9	9	-1
3	30	20	10	10	+1
1	5	16	-11	11	-1
9	3	18	-15	15	-1

Now that the absolute differences are in ascending order, we assign ranks to them, taking care of assigning the average rank to values with rank ties (same absolute value difference)

Pair	Sample 1	Sample 2	Abs. Difference	Rank	Sign
8	17	19	2	1	-1
6	7	13	6	2	-1
4	8	15	7	3.5	-1
5	9	16	7	3.5	-1
7	6	14	8	5.5	-1
10	10	18	8	5.5	-1
2	6	15	9	7	-1
3	30	20	10	8	+1
1	5	16	11	9	-1
9	3	18	15	10	-1