Question

For a one-way ANOVA test, suppose that, in reality, the null hypothesis is false. Does that mean that no two of the populations have the same mean? If not, what does it mean?

Short answer

No, The fact that the null hypothesis is false does not rule out the possibility of the two populations having the same mean.

Step-by-step solution

Concept Introduction 

ANOVA splits noted accumulated variability within a set of data into two components: systematic factors and random factors.

Explanation

The hypothesis for one-way ANOVA is as follows:

Null hypothesis :$H_0:{\mu }_1={\mu }_2={\mu }_3\dots \dots \dots \dots \dots ..={\mu }_k$

Alternative hypothesis: $H_a$

One of the means is different. ANOVA is being used to compare multiple groups of means in a single comparison. The null hypothesis is false because 'at least one of them is different,' but no two populations have the same mean.