Statistical Inference Using the F-Distribution

Previously, we looked at confidence intervals and hypothesis tests involving two population means and two population proportions, which allowed us to compare the means or proportions between two populations. But what about confidence intervals and hypothesis tests for two population variances? Sometimes, we need to compare the variability between two populations. For example, a manufacturing process might use two different methods to produce a certain item, and we want to know which method has the smaller variability. In our examination of statistical inference for the two population mean and two population proportion, we looked at the difference between the two population parameters. However, in order to compare the variance between two populations, we have to look at the ratio of the variances in order to use the [latex]F[/latex]-distribution.

Another situation that uses the [latex]F[/latex]-distribution is hypothesis testing on the equality of three or more population means. Many statistical applications in psychology, social science, business administration, and the natural sciences involve several groups. For example, an environmentalist is interested in knowing if the mean amount of pollution varies in several bodies of water. A sociologist is interested in knowing if the amount of income a person earns varies according to his or her upbringing. A consumer looking for a new car might compare the mean gas mileage of several models. Each of these scenarios requires a hypothesis test on three or more population means. Hypothesis tests that compare means between more than two groups employ a method using the [latex]F[/latex]-distribution called analysis of variance (abbreviated ANOVA). The simplest form of ANOVA, called single factor or one-way ANOVA, is the type used to test the equality of three or more population means.

“11.1 Introduction to Statistical Inferences Using the F-Distribution” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.