Throughout this chapter we have omitted the theoretical derivations of the null distributions of the various nonparametric test statistics. However, we can use computer simulation to derive approximate rejection regions for the tests. Consider the problem of finding the approximate sampling distribution of the Wilcoxon rank sum statistic for the case n₁ = n₂ = 10.

a. Write a computer program that will randomly order the n = n₁ + n₂ = 20 ranks and compute the corresponding Wilcoxon rank sum T₁. This can be accomplished using a random number generator.

b. Write a computer program that will repeat the instructions of part a N = 1,000 times.

c. Construct a relative frequency distribution for the N = 1,000 computer-generated values of T₁. This simulated distribution represents an approximation to the sampling distribution of T₁. 

d. Use the simulated sampling distribution to determine the value T_.05, such that P(T₁ ≤ T_.05). This value represents the one-tailed critical value of the Wilcoxon rank sum test for α = .05.