p-value, double tail test, hypotheses testing

Question

Can someone explain why intuitively here the p-value for double tail test is $2\min\{\Pr(X \leq x |H),\Pr(X \geq x |H)\}$ rather then $\Pr(X \leq x |H)+\Pr(X \geq x |H)$? The latter would be 1, but what about the strange formula starting with 2?

Answer 1

The value $x$ may be in the right or left tail, and it is not specified beforehand. The tail probability should always be smaller than $0.5$. So, for example, if $x$ is in the right tail

$$ 2\min\{P(X\leq x \mid H), P(X\geq x \mid H) \}=2P(X\geq x \mid H). $$

Because we want to know the probability that $X$ is as extreme as our $x$ test statistic. Even though our test statistic is in the right tail, we want to allow for a similarly extreme test statistic to be in the left tail also. So we multiply by $2$. The left and right tails (given our test statistic $x$ are of equal area. For the Normal distribution and $x$ in the right tail, we get

$$ \begin{aligned} 2\min\{P(X\leq x \mid H), P(X\geq x \mid H) \}&=2P(X\geq x \mid H)\\ &=P(X\geq x \mid H)+P(X\leq -x \mid H). \end{aligned} $$

Often, it should be clear from the context whether a two or one sided test is warranted.