# 12Analysis of Variance (ANOVA)

## 12.1 Introduction

Let's suppose that we want to determine if there is a difference in the mean miles per gallon of cars that have three different types of motors. How do we answer this question through hypothesis testing? More generally, how do we conduct hypothesis testing when we want to compare the mean of 5 populations? Or to compare the mean of 10 populations?

In this chapter, we will study comparing means of three or more populations. There are various points to consider that make the process distinct to hypothesis testing of one mean. The extension to subdividing populations more than one way will also be discussed, as well as some other features.

*Hypothesis Testing When Comparing Three or More Populations, Intuition*

To conduct hypothesis testing, when comparing three or more populations, it is useful to think of the procedure when we compare two populations. For example, we had established that if we wish to analyze the following null hypothesis and alternative hypothesis:

We may write equivalently:

Let's go back to the example of the mean ...

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