Let X1, X2, . . . , Xn and Y1, Y2, . . . , Yn be independent random samples from two normal distributions N(µ1, σ2 ) and N(µ2, σ2 ), respectively, where σ 2 is the common but unknown variance. (a) Find the likelihood ratio Λ for testing H0 : µ1 = µ2 = 0 against all alterna- tives. (b) Rewrite Λ so that it is a function of a statistic Z which has a well-known distribution. (c) Give the distribution of Z under both null and alternative hypotheses.

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