Question

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 -2 log (LR) (n1 + n2) log (1 + nin2 where T is the usual two-sample t statistic, given by as T n2 and s(n1)si+ (n2- 1)s3}/(n1n2 (b) [4 marks Assume that 2) 5.12, s = 8.45; 7.25, s = 6.28. n1 20, n2= 30, Find the p-values of the test described in Part (a), using, respectively (i) The exact null distribution (ii) The approximate null distribution, as stated in Theorem 5.10. (c) [4 marks For both ni and n2 large, what are the approximate null distributions of -2 log (LR) for the following two tests. You do not need to derive expressions for -2 log (LR) (i) Ho 2 against H 42 (without assumption o 02); Ho 2and o 02 against H 20r a1 o2. 11

Answer 1

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