# Intuition of p-value from theorem in Wasserman

I am trying to intuitively understand a theorem on $$p$$-values in Wasserman's book.

Let $$X$$ be a random variable. The theorem says that if $$T(X)$$ is a statistic such that the null hypothesis is rejected for $$T(X)$$ large, i.e. $$T(X) > c$$ for some constant c, then for an observation $$x$$, the p-value is the supremum of $$P(T(X) \ge T(x))$$ among all parameter values in the null hypothesis.

So if the p-value is small, it means that the observed $$T(x)$$ is pretty large, i.e. it lies along the "extreme" end of the range of values of $$T(X)$$. Since the observed $$T(x)$$ is large, according to the rejection rule in the theorem above, we reject the null hypothesis.

Is my understanding correct as to why a small p-value based on the above definition implies rejecting the null hypothesis?

Edit: What I'm really confused about is : what is the importance of the rejection rule being $$T(X)>c$$? Why can't the rejection rule be $$T(X) with this definition?