Suppose I have a random variable $X$ and I conduct a hypothesis test with respect to two hypotheses, $H_0$ and $H_1$ (the null and alternative hypotheses respectively).
I find the critical region (the set of values that will lead to the null hypothesis being rejected), how do I represent this set?
Do I my critical region as: $X > \cdots$ or $x > \cdots$, (for example)?
As far as I am aware, it should be the last one, since this represents the values that the random variable $X$ can take. However, I have seen a lot of material use $X$ when describing the critical region and others use both interchangeably, so I am unsure. Any insight into this is appreciated.