F-Test for Equality of Two Variances

F-Test for Equality of Two Variances

Purpose: To test if two population variances are equal.

Assumptions:

• If two populations are normal $\rightarrow$ F-test
• If two populations are not normal $\rightarrow$ Levene’s Test

Hypotheses:

$H_o:\sigma_1^2=\sigma_2^2$ “The two populations variances are equal.”

$H_a:\sigma_1^2\ne\sigma_2^2$ “The two populations variances are not equal.”

Test Statistic:

Important: The larger sample variance $s_L^2$ should be put in the numerator, and the smaller sample variance $s_s^2$ is put in the denominator. This forces the F-test to always be a right-sided test, which makes things a lot easier.

If the variances are equal $s_L^2=s_S^2$, then $F=1$.

The F-test has two degrees of freedom:

• $numerator\ \ df=n_L-1$
• $denominator\ df=n_s-1$

P-value method:

• If you are able to solve for the p-value using software, ensure that the p-value is doubled for two-tailed tests.
• In order to reject $H_o$, the p-value needs to be smaller than the significance level $\alpha$.

Critical value method:

• Without software, use to F-table to find the critical value.
• For two-tailed tests, divide $\alpha$ by 2 to find the critical value $F_{\frac{\alpha}{2},df_1,df_2}$.
• In order to reject $H_o$, the test statistic needs to be bigger than the critical value.

Example

$H_o:\sigma_1^2=\sigma_2^2$ “The two populations variances are equal.”

$H_a:\sigma_1^2\ne\sigma_2^2$ “The two populations variances are not equal.”

$\alpha=0.10$

• $numerator\ \ df=9$
• $deno\min ator\ df=20$

Find the critical value.