Question: "If the overall ANOVA F-test is significant, will the Tukey's HSD test be certain to find a statistically significant difference between at least one pair of means?" This question has puzzled John, the statistician, a lot these days. Recently, John worked on a balanced one-way ANOVA experiment in which \(t=3\) (three groups) and \(n=5\) (each group has five observations). The group means are: \(\bar{Y}_{1}=10, \bar{Y}_{2}=12\) and \(\bar{Y}_{3}=20 .\) When John ran the F-test in ANOVA, the p-value is significant at \(\alpha=0.05\). However, when Tukey's HSD procedure is applied, none of the pairwise comparisons were significant (familywise error rate is set at \(\alpha=0.05\) ). Find the range of MSE in this study.

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