How small a variance?

Topic: Hypothesis tests · more questions on this topic
‹ Power of a serve test Paired or two-sample? ›

In the tuning-fork example, 7 forks are tested, H_0:\sigma^2=1.1 and H_1:\sigma^2<1.1, at level 0.05. Using \chi^2_{6;0.05}=1.635, find the largest sample variance that leads to rejection. What is the power of this test if the true variance is 0.3?

Worked answer (click to show)

The test rejects when 6s^2/1.1\le1.635, that is when s^2\le1.635\times1.1/6=0.300.

If the true variance is 0.3, then 6s^2/0.3\sim\chi^2_6, and the power is

\mathbb{P}(s^2\le0.300)=\mathbb{P}\Bigl(\frac{6s^2}{0.3}\le\frac{6\times0.300}{0.3}\Bigr)=\mathbb{P}(\chi^2_6\le5.995)\approx0.58.

Even a reduction of the variance to less than a third of its old value is detected only a little more than half the time with 7 forks.