Question: Test for the equality of means of serval normal populations.

Let \(X_{i j}, i=1,2, \ldots, k ; j=1,2, \ldots, n_{i}\) be \(k\) independent random samples from \(k\) normal populations with means \(\mu_{1}, \mu_{2}, \ldots, \mu_{k}\), respectively and unknown but common variance \(\sigma^{2}\). Find a likelihood ratio test at the significance level of \(\alpha\) for the following test:

\(H_{0}: \mu_{1}=\mu_{2}=\ldots=\mu_{k}\) (unspecified) with \(\sigma_{1}^{2}=\sigma_{2}^{2}=\ldots=\sigma_{k}^{2}=\sigma^{2}\) (unspecified); \(H_{1}: \mu_{i}^{\prime} s\) (unspecified) are not all equal with \(\sigma_{1}^{2}=\sigma_{2}^{2}=\ldots=\sigma_{k}^{2}=\sigma^{2}\) (unspecified). Assume that \(n_{1}=n_{2}=\ldots,=n_{k}=n\). That is the sample size is the same for all \(k\) samples.

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