As an example of how the sampling distribution for the Mann–Whitney Test is derived, consider two samples with sample sizes n1  2 and n2  3. The distribution is obtained under the assumption that the two variables, say x and y, are identically distributed. Under this assumption, each measurement is equally likely to obtain one of the ranks between 1 and n1  n2.

a. List all the possible sets of two ranks that could be obtained from five ranks. Calculate the Mann–Whitney U-value for each of these sets of two ranks.

b. The number of ways in which we may choose n1 ranks from n1  n2 is given by (n n )! n !n !. 1 2 1 2 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.

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