10.6:
Bonferroni Test
The Bonferroni test is a type of multiple comparison test that reduces Type 1 error by dividing the significance alpha value by the number of pairwise comparisons in a dataset.
Consider comparing the students' test scores from three samples with unequal means.
Begin by stating the null hypotheses for each sample pair as follows.
Calculate the modified t-statistic and P-values for all pairs. Compare the P-values with an adjusted alpha, calculated as alpha value divided by the number of pairs, which is three, here.
The P-values of pairs 1 and 2, and 1 and 3 are less than the adjusted alpha value. We infer that these pairs have significantly different means and reject the null hypotheses for both.
The P-value of the pair 2 and 3 is greater than the adjusted alpha. We infer that the means of this pair aren't significantly different and fail to reject the null hypothesis.
We can conclude that sample 1 has a significantly different mean among the three samples in the dataset.
10.6:
Bonferroni Test
The Bonferroni test is a statistical test named after Carlo Emilio Bonferroni, an Italian mathematician best known for Bonferroni inequalities. This statistical test is a type of multiple comparison test to determine which means are different than the rest. Bonferroni test can minimize the Type 1 error by reducing the significance level alpha, which otherwise increases with sample pairs.
The means of different samples are first paired in all possible combinations.
The null hypothesis of the Bonferroni test assumes that means in each pair are the same. The t-statistic and P-value are separately calculated for each sample pair. If the P-value for a particular sample pair is less than the adjusted P-value, then that sample pair is considered to have significantly different sample means. This is done for all the sample pairs, and finally, the sample pair with the significantly different mean is identified.