The test statistic for a one-way between-subjects ANOVA is used to determine whether the variance associated with having different groups is significantly larger than the variance attributed to chance or error.

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This is correct. The test statistic in a one-way between-subjects ANOVA, typically F, is used to test whether the variance between groups is significantly larger than the variance within groups, or due to chance/error.

The Test Statistic for a One-Way Between-Subjects ANOVA Is Used