## 11.1 The ANOVA F-Test

Now that we understand in what kind of situations ANOVA is used, we are ready to learn how it works.

**Stating the Hypotheses**

The null hypothesis claims that there is no relationship between \(X\) and \(Y\). Since the relationship is examined by comparing \(\mu_1, \mu_2,\ldots,\mu_k\) (the means of \(Y\) in the populations defined by the values of \(X\)), no relationship would mean that all the means are equal. Therefore the null hypothesis of the F-testis: \(H_0: \mu_1 = \mu_2 = \cdots = \mu_k\).

As we mentioned earlier, here we have just one alternative hypothesis, which claims that there is a relationship between \(X\) and \(Y\). In terms of the means \(\mu_1, \mu_2,\ldots,\mu_k\) it simply says the opposite of the alternative, that not all the means are equal, and we simply write: \(H_a:\) not all the \(\mu\)’s are equal.

Recall our “Is academic frustration related to major?” example:

**Review: True or False**

The hypothesis that are being test in our example are:

\(H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4\)

\(H_1: \mu_1 \neq \mu_2 \neq \mu_3 \neq \mu_4\)

The correct hypotheses for our example are:

\(H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4\)

\(H_1: \mu_i \neq \mu_j\) for some \(i,j\)

Note that there are many ways for \(\mu_1, \mu_2,\mu_3,\mu_4\) not to be all equal, and \(\mu_1 \neq \mu_2 \neq \mu_3 \neq \mu_4\) is just one of them. Another way could be \(\mu_1 = \mu_2 = \mu_3 \neq \mu_4\) or \(\mu_1 = \mu_2 \neq \mu_3 \neq \mu_4\). The alternative of the ANOVA F-test simply states that not all of the means are equal and is not specific about the way in which they are different.