Illuminating Statistical Analysis Using Scenarios and Simulations

33Variance Ratios and the F-Distribution

Statistical Scenario—Variance Difference

Random samples of 30 female and 30 male community members were surveyed for their opinions about a new policy using the following scale.

Strongly agree

1

2

3

4

5

6

7

Strongly disagree

You want to know whether the evidence suggests that there is a difference in the variety of opinions among the community's males and females.

We will use the sample variances as the sample statistic of variety.

Which of the survey sample variance ratio values in Table 33.1 convince you that there is a difference between the female and male population variances?

Table 33.1 Scenario cases.

Female sample variance	Males sample variance	Sample variance ratio value
3	1	3
2	1	2
3	2	1.5
2	2	1

Said another way: How likely is it that you would get values for the sample variance ratio value as large as each of those shown in the table when randomly sampling from two populations that have equal population variances?

Null hypothesis: There is no difference in the variance of community opinion between males and females. That is, the population variance ratio equals 1.

To compare two sample variances, we need to (re)invent a sample statistic and the corresponding probability distribution. To make a sample statistic we can simply take the ratio of two sample variances and see if it is much different than 1, since a ratio value close to 1 suggests that the population variances may well be equal. We'll denote ...