Question: Let \(X_{1}, \ldots, X_{n}\) be a random sample from a \(N\left(\theta_{X}, \sigma_{X}^{2}\right)\) population and \(Y_{1}, \ldots, Y_{n}\) be a random sample from a \(N\left(\theta_{Y}, \sigma_{Y}^{2}\right)\) population, with \(\sigma_{X}^{2}\) and \(\sigma_{Y}^{2}\) known. Consider testing \(H_{0}: \theta_{X}=\theta_{Y}\) vs. \(H_{1}: \theta_{X} \neq \theta_{Y}\)

1. Show that the random variables \(W_{t}=X_{r} Y_{t}\) are iid \(N\left(\theta_{W}, \sigma_{W}^{2}\right)\)
2. Show that the above test can be derived as an LRT with rejection region \(|\bar{W}|>Z_{\alpha / 2} \sqrt{\sigma_{W}^{2} / n}\)

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