Let \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{n}\) be
independent random samples from two normal distributions \(N\left(\mu_{1},
\sigma^{2}\right)\) and \(N\left(\mu_{2}, \sigma^{2}\right)\), respectively,
where \(\sigma^{2}\) is the common but unknown variance.
(a) Find the likelihood ratio \(\Lambda\) for testing \(H_{0}: \mu_{1}=\mu_{2}=0\)
against all alternatives.
(b) Rewrite \(\Lambda\) 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.