Employing Bonferronis test in our example

In an ANOVA involving more than two groups, we estimate the underlying variability from more than two samples, and yet we are interested in the extent to which (only) two of the means differ from each other. Therefore, when comparing the means of two samples, the pooled standard deviation from the two-sample case, is replaced by an estimate that captures the variability across all groups in the analysis - the mean square error or the within-samples mean square. Recall from Section 11.4 that this quantity has the same interpretation as the pooled standard deviation, the typical spread of data across all observations. 

When using Bonferroni s method, the null hypothesis associated with a pairwise comparison is rejected if the calculated test statistic, that is. 

Remember that comes from the ANOVA table and it is the mean square error, which has also been referred to as the within-samples variability or, more informally, the background noise. This is analogous to sj in the two-sample case. As we assume equal variances, we use the estimator that uses the most data and therefore gives the most precise estimate. 

The critical value can be determined from a table or software (using a two-sided test of size 

Once the value of MSD has been determined, the absolute value of the difference in means will be compared with the MSD. If the absolute value of the difference in means, (Xj - x,) / is greater than or equal to the MSD the null hypothesis will be rejected. 

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