# 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

## Answer to relevant Questions

Show that for k = 2 the likelihood ratio statistic of Exercise 12.25 can be expressed in terms of the ratio of the two sample variances and that the likelihood ratio test can, therefore, be based on the F distribution. The average drying time of a manufacturer’s paint is 20 minutes. Investigating the effectiveness of a modification in the chemical composition of her paint, the manufacturer wants to test the null hypothesis µ = 20 ...With reference to Example 12.1, what would have been the probabilities of type I and type II errors if the acceptance region had been x > 16 and the corresponding rejection region had been x ≤ 16? Example 12.1 Suppose that ...Let X1 and X2 constitute a random sample from a normal population with σ2 = 1. If the null hypothesis µ = µ0 is to be rejected in favor of the alternative hypothesis µ = µ1 > µ0 when > µ0 + 1, what is the size of ...Show that the rule on page 372 for calculating the expected cell frequencies applies also when we test the null hypothesis that we are sampling r populations with identical multinomial distributions.Post your question