- Write a null and alternative hypothesis for a one-way ANOVA hypothesis test
- Discuss the error sum of squares and group sum of squares
The Null and Alternative Hypotheses
null hypothesis
The null hypothesis for a one-way ANOVA states that all the group/population means are the same. This can be written as:
[latex]H_0: \mu_1 = \mu_2 = ... = \mu_k[/latex]
where [latex]k[/latex] is the number of independent groups or samples.
The alternative hypothesis for a one-way ANOVA is a bit different than the alternative hypothesis we used when comparing only two group means (i.e., two-sample [latex]t[/latex]-test).
When there were only two group means to consider, the null hypothesis that the two means were the same was [latex]H_{0}:\mu_{1}=\mu_{2}[/latex]. If you wanted to show that the two means were different or not equal, the alternative hypothesis would be [latex]H_{A}:\mu_{1}\neq\mu_{2}[/latex]. If we rejected the null hypothesis, we would be able to conclude that the two means were statistically different.
When we reject the null hypothesis for a one-way ANOVA, we cannot simply state that all of the means are not equal. That is, when we reject the null hypothesis, [latex]H_{0}:\mu_{1}=\mu_{2}=\ldots=\mu_{k}[/latex], we are not able to differentiate whether one of the means is different from the others, whether two of the means are different from the others, whether three of the means are different from the others, etc.
alternative hypothesis
To provide flexibility and to account for the multiple outcomes associated with rejecting the null hypothesis, the alternative hypothesis for a one-way ANOVA should be written as:
[latex]H_{A}:[/latex] At least two of the group means are different.