Question: 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. 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.