Hypothesis test

A hypothesis test, also called a significance test, assumes a claim about a population parameter and asks how unusual the observed sample would be if that claim held. The machinery is fixed: state $H_0$ and $H_a$, check the conditions, compute a test statistic, turn it into a p-value using the null distribution, and compare that p-value with a significance level $\alpha$ (alpha) fixed beforehand. The output is one of two verdicts, reject $H_0$ or fail to reject $H_0$, and never a probability that either hypothesis is true.

Test $H_0: \mu = 100$ against $H_a: \mu > 100$ with $\sigma$ (sigma) known to be 15 and $n = 25$, so the standard error is $15/\sqrt{25} = 3$. A sample mean of 106.2 gives $z = \frac{106.2 - 100}{3} = 2.07$ and a p-value of 0.0194. At $\alpha = 0.05$ that is a rejection: samples this far above 100 turn up under 2 percent of the time when 100 is the truth.

The misreading that costs points: "the p-value was 0.16, so we accept the null hypothesis and conclude the mean is 100." A test never accepts $H_0$. A sample mean of 103 in that same setup gives $z = 1.00$ and a p-value of 0.1587, and a 95 percent interval of $103 \pm 1.96(3)$, or 97.12 to 108.88. Every value in that range is as consistent with the sample as 100 is, so failing to reject rules nothing in. The asymmetry is deliberate: evidence can contradict one specific value, and no sample can confirm one.

A test also answers only the question its hypotheses posed and does not audit its own scope. Whether a conclusion reaches a population depends on random sampling, and whether it supports a cause depends on random assignment: a rejection from a convenience sample is still a rejection and still worth very little. Significant is not the same as large, the subject of statistical vs practical significance.

In the Fall 2026 AP Statistics course, one-proportion tests are Unit 3 topics 3.5 through 3.7, and tests for means are Unit 4 topics 4.4 and 4.5.