# Multi-step hypothesis testing

I have an estimator $$\hat{R}$$ that I am computing for $$N$$ groups. For each group I am performing a hypothesis test with significance ($$\alpha$$) : $$H_0$$: $$\mathbb{E}[\hat{R}]=r$$, $$H_1$$: $$\mathbb{E}[\hat{R}]\neq r$$ where $$r$$ is just an expected number that I provide. Because I am doing $$N$$ hypothesis test, if $$H_0$$ is true for all groups, I would still expect $$N\alpha$$ type I errors where the null hypothesis in incorrectly rejected. In order to correct for this, one can use something like a Bonferroni correction and use a significance of $$\alpha^*=\alpha/N$$ instead for all test.

QUESTION: Is using the Bonferroni correction equivalent in some form to performing a second-step hypothesis testing where after performing the $$N$$ hypothesis tests mentioned above, one performs another test where $$H_0$$ is now: # of violated hypothesis = $$\alpha N$$ ? If so, is there a name for this approach ?

The idea behind Bonferroni is simply how you already explained: The probability of wrongly rejecting $$H_0$$ in one of your tests increases linearity with the number $$N$$ of your tests, so we just make it $$N$$ times harder to wrongly reject the null.
And you are right, if you want to, you can think of this as wrapping your $$N$$ hypothesis tests into another test (a binomial hypothesis test).