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 (n1 + n2) ! / n1 ! n2 !. Calculate this value for n1 = 2 and n2 = 3. Now calculate the probability of any one of the possible Mann–Whitney U -values.
c. List all the possible Mann–Whitney U -values you obtained in part a. Then using part b, calculate the probability that each of these U -values occurs, thereby producing the sampling distribution for the Mann–Whitney U statistic when n1 = 2 and n2 = 3.