Assuming no ties, obtain the exact null distribution of

Chapter 15, Problem 36E

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Problem 36E

Problem

Assuming no ties, obtain the exact null distribution of the Kruskal–Wallis H statistic for the case k = 3, n1 = n2 = n3 = 2. [Because the sample sizes are all equal, if ranks 1 and 2 are assigned to treatment 1, ranks 3 and 4 are assigned to treatment 2, and ranks 5 and 6 are assigned to treatment 3, the value of H is exactly the same as if ranks 3 and 4 are assigned to treatment 1, ranks 5 and 6 are assigned to treatment 2, and ranks 1 and 2 are assigned to treatment 3. That is, for any particular set of ranks, we may interchange the roles of the k populations and obtain the same values of the H statistic. Thus, the number of cases that we must consider can be reduced by a factor of 1/ k!. Consequently, H must be evaluated only for (6!/[2! · 2! · 2!])/3! = 15 distinct arrangements of ranks.]