B of exercise 25 to calculate -2 middot ln lambda and test

Question

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 λ and test the null hypothesis  =  =  =  at the 0.05 level of signifi- cance. 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 n_k 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