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Suppose that independent samples of sizes ni,n2...., nu are taken from each of k normally distributed populations with means H1, M2, ..., Me and common variances, all equal to o?. Let Y, denote the jth observation from population i, for j = 1,2,...,n; and i = 1, 2, ...,k, and let n = n +12+...+nk. a Recall that 1 SSE = (n-1)? where ? (yy-Y..)? n; -1 Argue that SSE/o2 has a y distribution with (n-1)+(12-1)+...+(n-1)=n-kdf. b Argue that under the null hypothesis, H:41 = 42 = ... = My all the Yij's are inde- pendent, normally distributed random variables with the same mean and variance. Use Theorem 7.3 to argue further that, under the null hypothesis, Total SS = EKY , 752 is such that (Total SS)/0? has a x? 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 Ho: 41 = H2 = ... = Hk, SST/o has a xa distribution with k - 1df. d Use the results of parts (a)(c) to argue that, under the hypothesis Ho: 41 = 42 = ... = Hks F = MST/MSE has an F distribution with k - 1 and n-k numerator and denominator degrees of freedom, respectively. Suppose that independent samples of sizes ni,n2...., nu are taken from each of k normally distributed populations with means H1, M2, ..., Me and common variances, all equal to o?. Let Y, denote the jth observation from population i, for j = 1,2,...,n; and i = 1, 2, ...,k, and let n = n +12+...+nk. a Recall that 1 SSE = (n-1)? where ? (yy-Y..)? n; -1 Argue that SSE/o2 has a y distribution with (n-1)+(12-1)+...+(n-1)=n-kdf. b Argue that under the null hypothesis, H:41 = 42 = ... = My all the Yij's are inde- pendent, normally distributed random variables with the same mean and variance. Use Theorem 7.3 to argue further that, under the null hypothesis, Total SS = EKY , 752 is such that (Total SS)/0? has a x? 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 Ho: 41 = H2 = ... = Hk, SST/o has a xa distribution with k - 1df. d Use the results of parts (a)(c) to argue that, under the hypothesis Ho: 41 = 42 = ... = Hks F = MST/MSE has an F distribution with k - 1 and n-k numerator and denominator degrees of freedom, respectively