Welch t-test is done before to apply the Bonferroni approach.
It is used to investigate the significance of the difference between the means of 2 populations.
Welch's T-test is a two-sample location test which is used to test the hypothesis that 2 populations have equal means. It is an adaptation of Student's t-test and is more reliable when two samples have unequal variances and/or unequal sample sizes (our case).
These tests are often referred to as "unpaired" or "independent samples" t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping.
$$ t = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{\frac{s^2_1}{N_1} + \frac{s^2_2}{N_2}}} $$
Unlike the Student's T-test the denominator is not base on the pooled variance estimate.
Siano $H_1, ... H_n$una famiglia di ipotesi e $p_1, ... p_n$ i loro corrispettivi valori $p$. Siano $m$ il numero totale di ipotesi nulle e $m_0$ il numero di ipotesi nulle vere. Il tasso di errore familiare (FWER) è la probabilità di rifiutare almeno una $H_i$ vera, cioè di commettere almeno un errore di tipo I (falso positivo).
La correlazione di Bonferroni respinge l'ipotesi nulla per ciascun
$$ p_i \le \frac{\alpha}{m} $$
controllando in tal modo il $FWER \le \alpha$.