前面文中提到的 $\alpha$ 型一誤差, 是在某一顯著水準下才得到的, 如果以 p-value 可能更能清楚的解釋檢定的問題. 上一個例子中, 在固定的顯著水準 $\alpha=.025$ 之下, 我們得到的 z=-2.52落在棄卻域 $R:Z\leq -1.96$ 中, 所以我們有很強的理由拒絕 H0 的假設. 但有一個問題是, 我們如何決定 $\alpha=.025$? 現在我們考慮 $P[Z\leq -2.52]=.0059$, 也就是說如果觀察到 -2.52 時, 棄卻 H0 犯錯的機會為 .0059, 是一個很小的值. 這就是稱為 significance probability or P-value.

P-value is the probability, calculated under H0, that the test statistic takes a value equal to or more extreme than the value actually observed. The P-value serves as a measure of the strength of evidence against H0. A small P-value means that the null hypothesis is strongly rejected or the result is highly statistically significant.

The Steps for Testing Hypotheses

Formulate the null hypothesis H0 and the alternative hypothesis H1.
Test criterion: State the test statistic and the form of the rejection region.
With a specified $\alpha$, determine the rejection region.
Calculate the test statistic from the data.
Draw a conclusion: State whether or not H0 is rejected at the specified $\alpha$ and interpret the conclusion in the context of the problem. Also, it is a good statistical practice to calculate the P-value and strengthen the conclusion.