Let us examine how the sampling distribution of the Wilcoxon test statistic is obtained. Consider the sampling distributions of the positive ranks from a sample size of 4. The ranks to be considered are, therefore, 1, 2, 3, and 4. Under the null hypothesis, the differences to be ranked are distributed symmetrically about zero. Thus, each difference is just as likely to be positively as negatively ranked.
a. For a sample size of four, there are 24 = 16 possible sets of signs associated with the four ranks. List the 16 possible sets of ranks that could be positive—that is, (none), (1), (2)…(1, 2, 3, 4). Each of these sets of positive ranks (under the null hypothesis) has the same probability of occurring.
b. Calculate the sum of the ranks of each set specified in part a.
c. Using parts a and b, produce the sampling distribution for the Wilcoxon test statistic when n = 4.