When deciding the Hypothesis, do I, in some way, define the rejection region?

Question

This has always troubled me a bit. When I choose my hypothesis, do I define in some way the rejection region [RR], or, do I do that by choosing the test statistic I want to use?

By fixing the significance level, I'm in a way determining the area/volume of the RR.

In two different contexts(different null hypothesis), I've seen the same statistic being used with two different RR. In some books, the authors give the sense that once we decide the hypothesis, we've chosen the RR. Others, state explicitly that at least in some hypothesis, the RR is not completely defined, and we need other criteria... I would like to structure this as best as possible.

Any help would be appreciated

Answer 1

Theoretically when having $H_0$ and wanting to prove $H_1$ you should define ahead your validation of certainty as in $\alpha =const$. When You have both $\alpha$ and the decision rule, then you have a unique Rejection Region. But with real life experience, you usually choose $\alpha$ as the infimum of all validation of certainty that prove your $H_1$ claim.

Answer 2

The rejection region depends on the null hypothesis, but also on the level of the test (i.e. how big a probability of Type I error you tolerate) and on what test statistic is used.