Definition:P-Value

Definition

Let $H_0$ and $H_1$ be the null hypothesis and alternative hypothesis of $\delta$ respectively.

Let $T$ be the test statistic which is being used to determine whether $H_0$ or $H_1$ holds.

Let $C$ be the critical region of $\delta$.

Let $\alpha$ be the significance level of $\delta$.

The $p$-value of $\delta$ is the probability that $T$ takes a value greater than or equal to a calculated value when $H_0$ is true.

To test the null hypothesus that $\mu = 0$ against the alternative hypothesis that $\mu \ne 0$, the statistic $t$ has $9$ degrees of freedom.

Suppose $t = 2.41$.

Then:

$\map \Pr {\size t \ge 2.41} = 0.0392$

which indicates a significance at the exact $0.0392$ level.

A result is significant at a pre-chosen level $\alpha$ if and only if $p \le a$.

If a $p$-value is less than a given significance level $\alpha$, the term exact significance level is often used for $p$-value.

In this moderm era, computer software routinely gives exact $p$-values in output.