Question

In a study of the effectiveness of physical exercise in weight reduction, 12 subjects followed a program of physical exercise for two months. Their weights (in pounds) before and after this program are shown in the table. Use Wilcoxon's signed-ranks test and a significance level of 0.05 to test the claim that the exercise program has no effect on weight.

Before

162

190

188

152

148

127

195

164

175

156

180

136

After

157

194

179

149

135

130

183

168

168

148

170

138

What would be the signed rank for the column with values of 175 and 168?

		8.5
		10
		8
		9

Answer 1

The following table is obtained:

Pair	Sample 1	Sample 2	Difference	Abs. Difference	Sign
1	162	157	5	5	+1
2	190	194	-4	4	-1
3	188	179	9	9	+1
4	152	149	3	3	+1
5	148	135	13	13	+1
6	127	130	-3	3	-1
7	195	183	12	12	+1
8	164	168	-4	4	-1
9	175	168	7	7	+1
10	156	148	8	8	+1
11	180	170	10	10	+1
12	136	138	-2	2	-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
12	136	138	-2	2	-1
4	152	149	3	3	+1
6	127	130	-3	3	-1
2	190	194	-4	4	-1
8	164	168	-4	4	-1
1	162	157	5	5	+1
9	175	168	7	7	+1
10	156	148	8	8	+1
3	188	179	9	9	+1
11	180	170	10	10	+1
7	195	183	12	12	+1
5	148	135	13	13	+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
12	136	138	2	1	-1
4	152	149	3	2.5	+1
6	127	130	3	2.5	-1
2	190	194	4	4.5	-1
8	164	168	4	4.5	-1
1	162	157	5	6	+1
9	175	168	7	7	+1
10	156	148	8	8	+1
3	188	179	9	9	+1
11	180	170	10	10	+1
7	195	183	12	11	+1
5	148	135	13	12	+1

The sum of positive ranks is:

W+ = 2.5+6+7+8+9+10+11+12 = 65.5

and the sum of negative ranks is:

W- = 1+2.5+4.5+4.5 = 12.5

Hence, the test statistic is

.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H0​: Median (Difference) = 0

Ha​: Median (Difference) ≠ 0

(2) Rejection Region

The critical value for the significance level α=0.05 provided, and the type of tail specified is , and the null hypothesis is rejected if

(3) Decision about the null hypothesis

Since in this case T=12.5≤13, there is enough evidence to claim that the population median of differences is different than 0, at the α=0.05 significance level.

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