Suppose that independent samples of sizes n1, n2,..., nk are taken from each of k

Chapter 13, Problem 13.6

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Suppose that independent samples of sizes n1, n2,..., nk are taken from each of k normally distributed populations with means 1, 2,...,k and common variances, all equal to 2. Let Yi j denote the jth observation from population i, for j = 1, 2,..., ni and i = 1, 2,..., k, and let n = n1 + n2 ++ nk . a Recall that SSE = k i=1 (ni 1)S2 i where S2 i = 1 ni 1 ni j=1 (Yi j Yi) 2 . Argue that SSE/2 has a 2 distribution with (n11)+(n21)++(nk 1) = nk df b Argue that under the null hypothesis, H0 :1 = 2 = = k all the Yi js are independent, normally distributed random variables with the same mean and variance. Use Theorem 7.3 to argue further that, under the null hypothesis, Total SS = k i=1 ni j=1 (Yi j Y ) 2 is such that (Total SS)/2 has a 2 distribution with n 1 df. c In Section 13.3, we argued that SST is a function of only the sample means and that SSE is a function of only the sample variances. Hence, SST and SSE are independent. Recall that Total SS = SST + SSE. Use the results of Exercise 13.5 and parts (a) and (b) to show that, under the hypothesis H0 :1 = 2 == k , SST/2 has a 2 distribution with k 1 df. d Use the results of parts (a)(c) to argue that, under the hypothesis H0 :1 = 2 == k , F = MST/MSE has an F distribution with k 1 and n k numerator and denominator degrees of freedom, respectively