I've got the following data values:
POW |157 days after release | 379 days after release 1 | 2.46 | 3.73 2 | 4.11 | 5.46 3 | 3.93 | 7.04 4 | 4.51 | 4.73 5 | 4.96 | 4.71 6 | 4.42 | 6.19 7 | 1.02 | 1.42 8 | 4.30 | 8.70 9 | 7.56 | 7.37 10 | 7.07 | 8.46 11 | 8.00 | 7.16
The values above are the test scores on brain function for Prisoners of War. The higher the test scores, the worse their brain is performing. I am trying to see whether the prisoners' brains perform worse 379 days after release than compared to 157 days after release.
Because the samples are performed on the same subjects, and the samples are assumed to be not normal and are less than 25 in size, I am supposed to use a wilcoxon signed rank test. The test is with a significance level of 0.05.
I realize how to rank and assign the differences, and how to get the value of 14 from the Wilcoxon table. I also realize that the negative rank sum is 9 and the positive rank sum is 57. What I do not understand is why, in the solution below, we use the negative rank sum (9) instead of the positive rank sum (57) in order to test against the value of 14 (we got from the Wilcoxon table). As well, why are we checking whether 9 is less than or equal to 14 (e.g., why don't we test whether 57 is greater than or equal to 14?)?
The solution to this question is apparently the following below: