Bonferroni method of multiple comparisons is used when there is not interest in all pairwise comparisons but of some comparisons that are not based on the data.\\ \\ It depends on the number of comparisons of interest how wide the confidence interval will be.\\ \\ For c comparisons the Bonferroni confidence intervals become: \[\hat{d} \pm t_{1-\frac{\alpha}{2c},N-I} \times \sqrt{MSE \bigg(\frac{1}{n_i}+\frac{1}{n_j} \bigg)}\]
Examples
If the interest in the rat experiment had been to only compare fat and carbohydrate diet and fat and protein based diet a method of Bonferroni could have been use. This decision would though had to been made before data analysis. The 95\% confidence intervals for those two difference would then become. \[53.6-62.6 \pm 2.53 \times\sqrt{4.19\bigg(\frac{1}{5}+\frac{1}{5}}\bigg)\] \[-12.3 \le \mu_1-\mu_2 \le -5.7\] \[53.6-61.3 \pm 2.53 \times\sqrt{4.19\bigg(\frac{1}{5}+\frac{1}{6}}\bigg)\] \[-10.8 \le \mu_1-\mu_3 \le -4.6\] Note that these confidence intervals are narrower than the Tukey intervals. If one the other hand we would have done all pairwise comparisons using the Bonferroni method the intervals would have been wider than when using the Tukey method, thats why the Tukey method is used for all pairwise comparisons.