# 10.2 - A Statistical Test for One-Way ANOVA

Hypotheses

Before we go into the details of the test, we need to determine the null and alternative hypotheses. Recall that for a test for two independent means, the null hypothesis was $$\mu_1=\mu_2$$. In one-way ANOVA, we want to compare $$t$$ population means, where $$t>2$$. Therefore, the null hypothesis for analysis of variance for $$t$$ population means is:

$$H_0\colon \mu_1=\mu_2=...\mu_t$$

The alternative, however, cannot be set up similarly to the two-sample case. If we wanted to see if two population means are different, the alternative would be $$\mu_1\ne\mu_2$$. With more than two groups, the research question is “Are some of the means different?." If we set up the alternative to be $$\mu_1\ne\mu_2\ne…\ne\mu_t$$, then we would have a test to see if ALL the means are different. This is not what we want. We need to be careful how we set up the alternative. The mathematical version of the alternative is...

$$H_a\colon \mu_i\ne\mu_j\text{ for some }i \text{ and }j \text{ where }i\ne j$$

This means that at least one of the pairs is not equal. The more common presentation of the alternative is:

$$H_a\colon \text{ at least one mean is different}$$ or $$H_a\colon \text{ not all the means are equal}$$

Test Statistic

Recall that when we compare the means of two populations for independent samples, we use a 2-sample t-test with pooled variance when the population variances can be assumed equal.

Test Statistic for One-Way ANOVA

For more than two populations, the test statistic, $$F$$, is the ratio of between group sample variance and the within-group-sample variance. That is,

$$F=\dfrac{\text{between group variance}}{\text{within group variance}}$$

Under the null hypothesis (and with certain assumptions), both quantities estimate the variance of the random error, and thus the ratio should be close to 1. If the ratio is large, then we have evidence against the null, and hence, we would reject the null hypothesis.

In the next section, we present the assumptions for this test. In the following section, we present how to find the between group variance, the within group variance, and the F-statistic in the ANOVA table.