when to reject a hypothesis, size and p-value When explaining the reason for rejecting a null hypothesis, I sometimes see "p-value is small", and sometimes it's "for a test of size $\alpha$, we reject it when (some condition)". I was wondering if "p-value is small" is a special case of a commonly used/standard (idk how to phrase this) size $\alpha$ test?
If this is not the case, could you explain when I am supposed to use which?
Thank you!

We set some value, called $\alpha$, as our maximum tolerance for type I error rate. That is, we accept that our work could reject true null hypotheses $100\alpha\%$ of the time the null hypothesis is true. In the common situation of $\alpha=0.05$, we accept that to be $5\%$. In fact, $\alpha=0.05$ is so common that it typically is implied when no $\alpha$ is specified, and we consider p-values of $0.05$ or smaller to be “small” p-values.
Then we run the test and calculate a p-value. If $p\le\alpha$, we reject the null hypothesis in favor of the alternative hypothesis.
