In this lecture, we will introduce one method for comparing two population variances. In this case, we have two samples, independently drawn from two populations. Based on the sample variances, we want to investigate if the population variances are equal or not. Like said before, we cannot assess this problem directly. Instead, we approach this problem in an indirect way using the p-value. That is, if wa assume that there is no difference between the two variances from the two populations, what would be the probability of observing the data we have or more extreme?
In order to answer that question, we need to figure out a test statistic and its distribution. Let’s start with the test statistic. Based on the content from previous lectures, the intuitive test statistic would simply be the difference of the two variances. We will demonstrate during the lecture, it is not that useful. Instead, the ratio of the two sample variances is a more useful test statistic, and it has a distribution whose PDF can be derived. This is the $\boldsymbol{\mathcal{F}}$-distribution.