# Question: Independent random samples of sizes n1 n2

Independent random samples of sizes n1, n2, . . . , and nk from k normal populations with unknown means and variances are to be used to test the null hypothesis σ21 = σ22 = · · · = σ2k against the alternative that these variances are not all equal.

(a) Show that under the null hypothesis the maximum likelihood estimates of the means µ1 and the variances σ2i are

Where

While without restrictions the maximum likelihood estimates of the means µi and the variances σ2i are

This follows directly from the results obtained in Section 10.8.

(b) Using the results of part (a), show that the likelihood ratio statistic can be written as

(a) Show that under the null hypothesis the maximum likelihood estimates of the means µ1 and the variances σ2i are

Where

While without restrictions the maximum likelihood estimates of the means µi and the variances σ2i are

This follows directly from the results obtained in Section 10.8.

(b) Using the results of part (a), show that the likelihood ratio statistic can be written as

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