(Bonferroni Correction, Multiple Testing)

For each j = 1, …, J, let be iid Bernoulli random variables with (unknown) success probability p j

∈ (0, 1), and n > 1. Note that the variables are independent for fixed j and varying i, but could dependent for fixed i and varying j (e.g. X ij may be a yes/no answer of the ith individual to the jth question of a customer survey).

We wish to test the hypothesis pair:

(in our example: are customers on average satisfied on all J issues, or do there exist issues where customers are dissatisfied, on average?). Construct a test for the hypothesis pair that respects a given level α ∈ (0, 1).