Non-parametric Hypothesis Test

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Nonparametric Test: Wilcoxon Signed Rank Test
The mean weight change during the control period is equal to that during the lack-of-consistency period.

The mean weight change during the control period is different from that during the lack-of-consistency period.

We are supposed to analyze the data using non-parametric hypothesis test. For this, we shall use Wilcoxon Signed Rank Test for analyzing the given hypothesis.
Let us consider significance level,
Since the alternate hypothesis is this is a two-tailed test.
Number of observations, N = 23.

Weight change, control period: Xi
Weight change, lack-of-consistency period: Yi

(Xi-Yi) Sign of (Xi-Yi) Absolute (Xi-Yi) Rank of Absolute Positive Negative
-12.6 -1 12.6 18 18
-10.3 -1 10.3 15 15
-5.9 -1 5.9 11 11
-5.4 -1 5.4 8.5 8.5
-4.5 -1 4.5 6 6
-2.7 -1 2.7 4 4
-1.8 -1 1.8 2 2
0.3 1 0.3 1 1
2.2 1 2.2 3 3
3.5 1 3.5 5 5
4.8 1 4.8 7 7
5.4 1 5.4 8.5 8.5
5.8 1 5.8 10 10
6.0 1 6.0 12 12
6.7 1 6.7 13 13
9.6 1 9.6 14 14
11.5 1 11.5 16 16
12.2 1 12.2 17 17
13.9 1 13.9 19 19
14.2 1 14.2 20 20
18.0 1 18.0 21 21
18.6 1 18.6 22 22
21.7 1 21.7 23 23
Total 211.5 64.5

The test statistic is given as the sum of all the values in the Signed Rank column.

The smaller of the two sums is 64.5. From the p-value table the two tailed p-value for N=23 and S = 64.5 is given as:

As the p-value (0.02542) is smaller than the significance level (0.05), we reject the null hypothesis. Therefore, our conclusion is that the mean weight change during the control period is different from that during the lack-of-consistency period. In other words, there are significant differences between the weight change during the control period and that during the lack-of-consistency period.

Reference:
http://comp9.psych.cornell.edu/Darlington/wilcoxon/wilcox1.htm
http://comp9.psych.cornell.edu/Darlington/wilcoxon/wilcox51.htm

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