### 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.