To compare the variations in the weight of four breeds of dogs, researchers took independent…

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To compare the variations in weight of four breeds of dogs, researchers took independent random samples of sizes n1 = 8, n2 = 10, n3 = 6, and n4 = 8, and got
= 16,
= 25,
= 12, and
= 24. Assuming that the populations sampled are normal, use the formula of part

(b) of Exercise 25 to calculate -2 · ln ? and test the null hypothesis = = = at the 0.05 level of significance. Explain why the number of degrees of freedom for this approximate chi-square test is 3.

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

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