# 12. Hypothesis Tests Concerning Differences between Variances

## j. identify the appropriate test statistic and interpret the results for a hypothesis test concerning 1) the variance of a normally distributed population, and 2) the equality of the variances of two normally distributed populations based on two independent random samples;

What is the math notation for the test statistic to test the equality of variances between two normally distributed populations?

$$F = \frac{S_1^2}{S_2^2}$$

- `F`

= Fisher (the name of the person who formulated this distribution)

- `S`

= Sample variances from the two populations

Which statistic is the F-test most like: z-test, t-test, chi-square?

The F-test is asymmetrical and bounded at zero, like chi-square.

What does the F-test check about the test statistic `(S_1)^2 / (S_2)2`

?

If the test statistic ratio is close to 1 you will get a non-rejection of the null hypothesis (`H_0`

), otherwise you will get a rejection.

Unlike the chi-square test which is only determined by one parameter, the F-test is determined by two parameters. What are they?

The degrees of freedom in the numerator and the degrees of freedom in the denominator.

In an F-test, do both populations need to be normally distributed and independent?

Yes

##### Source:

- CFA