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 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
![](https://test.transtutors.com/qimg/fad14538-cc44-4522-9e77-cde993c50f96.png)
(b) Using the results of part (a), show that the likelihood ratio statistic can be written as
![](https://test.transtutors.com/qimg/50e952e9-1687-4144-97e7-369075e1476c.png)