Why does a larger between-groups to within-groups ratio support rejecting the null hypothesis?

Conditions

• There are multiple groups being compared.
• The variance has been separated into between-group and within-group components.

Reasoning, step by step

1. Calculate the ratio of between-groups variance to within-groups variance.
2. Observe that a larger ratio indicates more separation between group means relative to the noise within groups.
3. Conclude that a larger ratio makes it more likely that the group means really are different.
4. Reject the null hypothesis when the ratio is sufficiently large.

Example

Slide text states: The larger the ratio, the more likely it is that the groups have different means (reject $H_0$).

Common misconceptions

• Assuming that a larger ratio automatically proves the alternative hypothesis without considering the associated probability.

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