The ratio depends only on the between sum of squares

Advanced

Topic: Generalised likelihood-ratio tests · more questions on this topic

1 of 2 2 of 2 ›

Two independent normal samples have a common known variance \sigma^2=4, sizes n_1=n_2=8, and sample means 12 and 15. The null hypothesis is that the two means are equal.

1. Compute \mathrm{SS}_B.

2. Compute -2\log\Lambda, and say whether the level-0.05 test rejects. Use \chi^2_{1;0.95}=3.84.

Worked answer

\mathrm{SS}_B=\frac{8\times 8}{16}(15-12)^2=36, \qquad -2\log\Lambda=\frac{\mathrm{SS}_B}{\sigma^2}=\frac{36}{4}=9.

One restriction is being tested, and 9>3.84, so the level-0.05 test rejects equal means. The difference of 3 is 3/(2\sqrt{1/8+1/8})=3 standard errors of the difference, and 3^2=9.