25. Independent random samples o n from k normal w variances are to be used to test the hu σί against the alternati...
25. Independent random samples o n from k normal w variances are to be used to test the hu σί against the alternati ations with unknown means and . . alternative that these variances are not all equal. (a) Show that under the nul hypothesis i the variances likelihood estimates of the means ,41 an and the va are (ni-1)si /n に1 σ2 anded eel. while σ,-are where n Σ ni, while with out restrictions the maximum likelihood estimates of...
12 marks Let independent random samples of sizes n and n2 be taken respectively from two normal distributions with unknown means 1 and 2 and unknown variances oand o. Denote the two samples by . . ,Jn, and y,... , yn2: Which have means T and T, and sample variances s and s2, respectively (a) 4 marks Show that when of = o2, the likelihood ratio test statistic for testing Ho 12 against H 2 can be written as T2...
Suppose that independent samples of sizes n1, n2, . . . , nk are taken from each of k normally distributed populations with means μ1,μ2, . . . , μk and common variances, all equal to σ 2. Let Yi j denote the j th observation from population i, for j = 1, 2, . . . , ni and i = 1, 2, . . . , k, and let n = n1 + n2 + ··· + nk...