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Show that for k = 2 the likelihood ratio statistic of Exercise 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.
Exercise 25
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 = =···= 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 are
(b) Using the results of part (a), show that the likelihood ratio statistic can be written as
Let P be a regular transition matrix and let w be the unique non-zero fixed vector of P. Show that no entry of w is 0.
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lowering the price does not decrease this behavior and we should not decide to lower prices based on this research." Comment on the executive's conclusion. What type of inference is being made? Will the decision not to lower prices be a good one.
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