AP Stats 3.6: p-Values

What topic 3.6 covers

Topic 3.6 defines the p-value and shows how to interpret it. Assuming the null hypothesis is true, the test statistic follows a known null distribution. The p-value is the probability, under that null distribution, of a test statistic at least as extreme as the one you observed, measured in the direction of the alternative hypothesis. This idea reappears in every hypothesis test in the course, so the interpretation here transfers directly to means and chi-square.

Finding the p-value by direction

The alternative hypothesis sets which tail or tails you use. With $z$ as the observed standardized test statistic:

• If $H_a$ uses $>$, the p-value is $P(Z \ge z)$, the area at or above the observed value.
• If $H_a$ uses $<$, the p-value is $P(Z \le z)$, the area at or below it.
• If $H_a$ uses $\ne$, the p-value is $P(Z \le -|z|) + P(Z \ge |z|)$, both tails.

If instead the null distribution was simulated, the p-value is the proportion of simulated statistics as extreme or more extreme than the observed one. The logic is identical, only the source of the distribution changes.

Reading a p-value correctly

A complete interpretation states that the p-value is computed by assuming the null hypothesis is true, meaning the true proportion equals the value in $H_0$. A small p-value means the observed statistic would be unusual if the null were true, so it provides evidence for the alternative; the smaller the p-value, the stronger that evidence. A p-value that is not small means the observed result would not be surprising under the null, so it does not support the alternative, and it is never evidence that the null is true.

Common misreadings to avoid

A p-value is not the probability that the null hypothesis is true, and it is not the probability that your result happened by chance alone. It is a conditional probability about the data, computed under the assumption that the null holds. A large p-value means the data are consistent with the null, not that the null is proven, since many other values could also be consistent with the same data. Keeping the conditional wording tight is the fastest way to earn full credit on interpretation questions.