In AP Stats, "fail to reject" is the decision you make when the p-value is greater than the significance level α, meaning the sample data do not provide convincing statistical evidence for the alternative hypothesis, so the null hypothesis remains plausible (but is never proven true).
"Fail to reject" is one of only two possible decisions in a significance test, and it follows a mechanical rule. Compare your p-value to α. If the p-value is greater than α, you fail to reject the null hypothesis (H₀). If the p-value is less than or equal to α, you reject it. That's the whole decision step, straight from the CED for Topic 6.6.
The meaning is the trickier part. Failing to reject does NOT mean H₀ is true. It means your data were the kind of result that could plausibly happen by random chance if H₀ were true, so you don't have sufficient evidence for the alternative hypothesis (Hₐ). Think of it like a jury verdict. "Not guilty" doesn't mean innocent; it means the prosecution didn't bring enough evidence. Same idea here. Your conclusion sentence on the exam should say something like "we do not have convincing evidence that [Hₐ in context]," never "H₀ is true" or "we accept H₀."
This term lives in Topic 6.6 (Concluding a Test for a Population Proportion) in Unit 6 and directly supports learning objective AP Stats 6.6.A, justifying a claim about a population based on a significance test. The essential knowledge spells out the exact decision rule (p-value ≤ α means reject; p-value > α means fail to reject) and the exact meaning (failing to reject means insufficient evidence for the alternative). Once you learn this language in Unit 6, it never goes away. Every significance test in the rest of the course, including tests for means, chi-square tests, and slope tests, ends with the same two-option decision. Getting the wording right is one of the easiest points to earn on inference FRQs, and getting it wrong ("accept the null") is one of the easiest points to lose.
Null Hypothesis (Unit 6)
Failing to reject keeps H₀ alive as a plausible explanation, but it never confirms it. The null is the assumption you test against, not a claim you can prove with sample data.
P-value (Unit 6)
The p-value is the entire basis for the decision. A large p-value says your sample result is unsurprising if H₀ is true, and unsurprising results don't count as evidence against anything.
Significance Level / Alpha Level (Unit 6)
α is the cutoff you set before testing. The same p-value of 0.03 leads to rejecting at α = 0.05 but failing to reject at α = 0.01, so the decision depends on the threshold, not just the data.
Test Statistic and Z-Score (Unit 6)
A test statistic close to zero, like z = 1.4, means your sample proportion sits comfortably within normal sampling variability. Small |z| produces a large p-value, which produces a fail-to-reject decision. The whole chain links together.
On multiple choice, the classic stem gives you a p-value and an α and asks for the most appropriate conclusion. For example, a manufacturer's defect-rate test with a p-value of 0.103 at α = 0.05 means you fail to reject because 0.103 > 0.05. The wrong answer choices are designed to catch the two big mistakes, saying you "accept H₀" or claiming the data prove the null is true. Watch boundary cases too. If the p-value is 0.0099 and α = 0.01, you reject, because the rule is p-value ≤ α, not strictly less than.
On FRQs, significance test questions require a full conclusion in context. The scoring expectation is a sentence that makes the decision, links it to the p-value and α comparison, and states there is not convincing evidence for the alternative hypothesis in the context of the problem. "Because the p-value of 0.103 is greater than α = 0.05, we fail to reject H₀; we do not have convincing evidence that less than 5% of the products are defective." That sentence pattern earns the point.
These sound interchangeable, but AP graders treat them very differently. "Accept H₀" claims you've shown the null is true, which a significance test can never do. A sample can fail to contradict H₀ without confirming it, the same way a "not guilty" verdict doesn't prove innocence. "Fail to reject" is the only correct phrasing because it says exactly what happened, the evidence wasn't strong enough to support the alternative. Writing "accept the null" on an FRQ conclusion can cost you the point even if your calculations are perfect.
Fail to reject the null hypothesis whenever the p-value is greater than α; reject whenever the p-value is less than or equal to α.
Failing to reject means there is insufficient statistical evidence to support the alternative hypothesis, not that the null hypothesis is true.
Never write "accept the null hypothesis" on the AP exam; the correct phrase is always "fail to reject."
Your conclusion must be stated in terms of the alternative hypothesis and in the context of the problem, like "we do not have convincing evidence that the defect rate is less than 5%."
The boundary case counts as rejection. If the p-value exactly equals α, you reject the null hypothesis.
The same decision rule applies to every significance test in the course, from proportions in Unit 6 to means, chi-square, and slope tests later on.
It's the decision you make when the p-value is greater than the significance level α. It means your sample didn't provide convincing evidence for the alternative hypothesis, so the null hypothesis remains a plausible explanation for the data.
No. Failing to reject only means you lack sufficient evidence against H₀, not that you've confirmed it. A sample can be consistent with many possible parameter values, so the test can't prove the null is the right one.
"Fail to reject" says the evidence wasn't strong enough to support the alternative, while "accept" wrongly claims the null was proven true. AP scoring guidelines expect "fail to reject," and writing "accept H₀" can cost you the conclusion point on an FRQ.
Whenever the p-value is greater than α. For example, a p-value of 0.103 at α = 0.05 means you fail to reject, because a result like yours would happen about 10.3% of the time by chance alone if H₀ were true.
You reject. The CED's decision rule is p-value ≤ α means reject, so a p-value of exactly 0.05 at α = 0.05 leads to rejecting H₀. Only a p-value strictly greater than α produces a fail-to-reject decision.
Connect this key term to the AP exam workflow: review the course, practice questions, and check related study tools.
Review units, study guides, and course resources.
Check this vocabulary in multiple-choice context.
Apply key concepts in written AP responses.
Estimate the exam score you are working toward.
Review the highest-yield facts before practice.
Put the full course together before test day.