F Tests for Comparing Two Population Variances

All flashcards

What does population variance measure?

It measures how spread out population values are around their population mean; lower variance generally indicates greater consistency.

What assumptions does the classical F test for two variances require?

Both populations should be approximately normally distributed, and the two samples should be independent.

Why is normality especially important for the F test of variances?

The test is highly sensitive to departures from normality, which can bias the test statistic and conclusion.

What alternative hypothesis expresses that population 1 is more consistent than population 2?

Greater consistency means less variability, so the alternative is $H_a: \sigma_1^2<\sigma_2^2$.

How does a general variance-ratio hypothesis include the equal-variance test?

The null specifies $\sigma_1^2/\sigma_2^2=\delta_0$; setting $\delta_0=1$ gives equal variances.

Quiz questions

1. For a test of equal population variances, how is the F statistic formed from sample variances $s_1^2$ and $s_2^2$?Answer: $F_c=s_1^2/s_2^2$.
2. If sample variance $s_1^2$ is in the numerator of an F statistic, what are its numerator degrees of freedom?Answer: $n_1-1$.
3. For a two-tailed F test with significance level $\alpha$, how is the error probability allocated between the tails?Answer: Use $\alpha/2$ in each tail.
4. How can a lower-tail F critical value be obtained from an upper-tail table?Answer: Reverse the degrees of freedom and take the reciprocal of the corresponding upper-tail value.
5. What does failing to reject the null hypothesis in an F test establish?Answer: The evidence is insufficient to reject the null; it does not prove the null is true.