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When the null hypothesis is true, the computed $$F$$ ratios in an analysis of variance follow a specific probability curve known as the $$F$$ distribution. This distribution is strictly unimodal and positively skewed, with the majority of values clustering around $$1$$. The precise shape of the $$F$$ distribution is dynamic and depends entirely on two parameters: the between-groups degrees of freedom ($$df_B$$) and the within-groups degrees of freedom ($$df_W$$). By knowing the exact shape of this distribution under the null hypothesis, researchers can accurately determine the $$p$$-value associated with any computed $$F$$ statistic.

Distribution of the F Ratio

To determine if there are significant differences among group means in an ANOVA, researchers evaluate the computed $$F$$ statistic by finding its corresponding $$p$$-value or comparing it against a predetermined critical value from an $$F$$ table. If the computed $$F$$ ratio is greater than the critical value, the associated $$p$$-value is less than the alpha level (typically $$.05$$). In this outcome, the researcher rejects the null hypothesis and concludes that the population means differ. Conversely, if the $$F$$ ratio is less than the critical value, resulting in a $$p$$-value greater than $$.05$$, the researcher retains the null hypothesis, concluding there is insufficient statistical evidence to claim any differences exist.

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