12.2 One-Way ANOVA

The Null and Alternative Hypotheses

The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different. For example, if there are k groups:

$H_0:{\mu }_1={\mu }_2={\mu }_3=.\; .\; .{\mu }_k$

$H_a$: At least two of the group means ${\mu }_1,{\mu }_2,{\mu }_3,.\; .\; .\; ,{\mu }_k$ are not equal. That is, ${\mu }_i\ne {\mu }_j$ for some $i\ne j$.

The graphs, a set of box plots representing the distribution of values with the group means indicated by a horizontal line through the box, help in the understanding of the hypothesis test. In the first graph (red box plots), H₀: μ₁ = μ₂ = μ₃ and the three populations have the same distribution if the null hypothesis is true. The variance of the combined data is approximately the same as the variance of each of the populations.

If the null hypothesis is false, then the variance of the combined data is larger which is caused by the different means as shown in the second graph (green box plots).

Figure 12.3 (a) H₀ is true. All means are the same; the differences are due to random variation. (b) H₀ is not true. All means are not the same; the differences are too large to be due to random variation.