Question: Test for the equality of variances of two normal populations.

Let \(X_{i j}, i=1,2 ; j=1,2, \ldots, n_{i}\) be two independent random samples from two normal populations \(N\left(\mu_{i}, \sigma_{i}^{2}\right), i=1,2\), respectively. Find a likelihood ratio test at the significance level of \(\alpha\) for the following test:

\(H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}\) (unspecified) with \(\mu_{1}, \mu_{2}\) (unspecified);

\(H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2}\) (unspecified) with \(\mu_{1}, \mu_{2}\) (unspecified).

Assume that \(n_{1}=n_{2}=n\). That is the sample size is the same for both samples.

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