Question

When we carry out a statistical test with significance level α = 5%, the probability of rejecting the null hypothesis when it is true is 5%. Suppose that we independently select 5 random samples of size 100, and for each sample carry out the same statistical test with significance level 5%. We know that the null hypothesis is true. What is the probability that we reject the null hypothesis at most once out of the 5 tests?

Answer 1

According to the question, we know that

P(reject the null hypothesis when it is true) = (5/100) = 0.05

And it is given that the null hypothesis is actually true.

Let define a random variable      which denotes the number of cases where we reject the null hypothesis.

Hence the mass points of      are 0,1,2,3,4,5.

So here  

Hence the pmf of      is  

The probability that we reject the null hypothesis at most once out of the 5 tests is