Let s 2

i

, i = 1, …, k be k independent mean squares calculated from independent normal random variables. Suppose E (s 2

i ) = σ

2 i

, var (s 2

i ) = 2σ

2 i /mi where mi is the number of degrees of freedom for s 2

i

. Derive the likelihood ratio statistic, W, for testing the hypothesis H0 : σ

2 i = σ

2 against the alternative that leaves the variances unspecified. Using the results of Section 7.5.1, show that under H0, E (W) = k − 1 +

1 3 ∑m

−1 i −

1 3

m

−1

•

where m•

is the total degrees of freedom. Hence derive Bartlett’s test for homogeneity of variances (Bartlett, 1937).