---
title: "Fail to Reject the Null Hypothesis — AP Stats Definition"
description: "Failing to reject H₀ means the p-value is bigger than α, so there's not enough evidence for Hₐ. It never proves H₀ true. Core to AP Stats Unit 6 conclusions."
canonical: "https://fiveable.me/ap-stats/key-terms/fail-to-reject-the-null-hypothesis"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 3"
---

# Fail to Reject the Null Hypothesis — AP Stats Definition

## Definition

In AP Statistics, failing to reject the null hypothesis is the decision you make when the p-value is greater than the significance level α. It means there is insufficient statistical evidence to support the alternative hypothesis, not that the null hypothesis is true.

## What It Is

Every [significance test](/ap-stats/key-terms/significance-test "fv-autolink") ends with one of two decisions. If the [p-value](/ap-stats/key-terms/p-value "fv-autolink") ≤ α, you reject the null hypothesis. If the p-value > α, you **fail to reject** the null hypothesis. That's it. Those are the only two options, and the comparison to α is what makes the decision "formal" in the CED's language.

Here's the part that trips people up. Failing to reject H₀ does NOT mean H₀ is true. It means your [sample](/ap-stats/unit-1/random-sampling-data-collection/study-guide/nQz8XwRMmIKKBS59qrew "fv-autolink") didn't produce evidence strong enough to convince you it's false. Think of a courtroom verdict. A jury says "not guilty," never "innocent." A not-guilty verdict means the prosecution didn't make its case, not that the defendant definitely didn't do it. Same logic here. A p-value of 0.13 with α = 0.05 says "the evidence isn't there," so you write that there is insufficient evidence to conclude the alternative hypothesis is true, always stated in context.

## Why It Matters

This term lives in **Topic 6.6 (Concluding a Test for a Population Proportion)** in Unit 6, and it directly supports learning objective **[AP Stats](/ap-stats "fv-autolink") 6.6.A**, which asks you to justify a claim about a [population](/ap-stats/key-terms/population "fv-autolink") based on a significance test. The essential knowledge spells out the exact rule. Compare the p-value to α. If p > α, fail to reject H₀, which means there is insufficient statistical evidence to support the alternative hypothesis.

The wording matters enormously on the exam. Graders draw a hard line between "fail to reject H₀" (correct) and "accept H₀" or "H₀ is true" (wrong, and it can sink an otherwise perfect FRQ conclusion). And this isn't just a Unit 6 skill. The same reject / fail-to-reject decision structure carries through every inference procedure you'll learn afterward, including tests for means, [chi-square tests](/ap-stats/key-terms/chi-square-tests "fv-autolink"), and tests for regression slopes.

## Connections

### [P-value (Unit 6)](/ap-stats/key-terms/p-value)

The p-value is the trigger for this whole decision. You [fail to reject](/ap-stats/key-terms/fail-to-reject "fv-autolink") H₀ exactly when the p-value lands above α. The p-value measures how surprising your sample is if H₀ were true, so a big p-value means your data isn't surprising at all, which is why it can't count as evidence against the null.

### [Significance Level (Unit 6)](/ap-stats/key-terms/significance-level)

α is the cutoff you pick before running the test, and it's defined as the [probability](/ap-stats/unit-2/intro-probability/study-guide/gfnBWfyMANOxF3vWLrbA "fv-autolink") of rejecting H₀ when H₀ is actually true. Whether you fail to reject can depend entirely on this choice. A p-value of 0.078 fails to reject at α = 0.05 but rejects at α = 0.10, same data, different verdict.

### [Alternative Hypothesis (Unit 6)](/ap-stats/key-terms/alternative-hypothesis)

Your written conclusion is always a statement about Hₐ, not H₀. Failing to reject means insufficient evidence FOR the alternative. AP conclusions never say anything is proven about the null itself, which is exactly why "accept H₀" is banned phrasing.

### [Hypothesis Test (Units 6-9)](/ap-stats/key-terms/hypothesis-test)

Reject vs. fail to reject is the universal ending of every hypothesis test in the course. Once you nail the logic with one-proportion z-tests in Unit 6, you reuse the identical decision rule for two proportions, means, chi-square, and slopes. Only the test statistic changes, never the conclusion structure.

## On the AP Exam

Multiple-choice questions love handing you a p-value and an α and asking which conclusion is "most appropriate." A classic setup gives a p-value like 0.078 or 0.13 against α = 0.05 and lists tempting wrong answers like "accept the null hypothesis" or "there is convincing evidence the proportion has not changed." The right answer always says there is insufficient evidence to support the alternative, stated in context.

On FRQs, the four-step significance test asks you to (1) compare p-value to α, (2) state the decision, and (3) write a conclusion in context about the alternative hypothesis. To earn full credit when p > α, write something like "Because the p-value of 0.13 is greater than α = 0.05, we fail to reject H₀. There is not convincing statistical evidence that the proportion of adults who exercise regularly has increased from 25%." All three pieces (comparison, decision, contextual conclusion) need to be there.

## Fail to Reject the Null Hypothesis vs Accepting the Null Hypothesis

These sound interchangeable but only one exists in AP Statistics. "Accepting H₀" would claim the null is true, and a significance test can never establish that. The test only measures evidence against H₀. Failing to reject just means the evidence against H₀ wasn't strong enough, the way a "not guilty" verdict isn't a declaration of innocence. Writing "accept H₀" or "H₀ is true" on an FRQ conclusion costs you credit, so train yourself to say "fail to reject" every single time.

## Key Takeaways

- Fail to reject H₀ whenever the p-value is greater than the significance level α; reject H₀ whenever the p-value is less than or equal to α.
- Failing to reject the null hypothesis 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 only two valid decisions are reject and fail to reject.
- Your written conclusion must be a statement about the alternative hypothesis, in the context of the problem, like "there is not convincing evidence that the proportion of voters supporting the proposal is less than 40%."
- The same decision can flip depending on α, so a p-value of 0.078 fails to reject at α = 0.05 but rejects at α = 0.10.
- This decision rule from Topic 6.6 repeats in every later inference unit, including tests for means, chi-square tests, and regression slopes.

## FAQs

### What does fail to reject the null hypothesis mean in AP Stats?

It's the decision you make when the p-value is greater than α. It means your sample didn't provide enough statistical evidence to support the alternative hypothesis, so you can't conclude the effect or difference in Hₐ exists.

### Does failing to reject the null hypothesis mean the null is true?

No. A significance test can only measure evidence against H₀, never for it. Failing to reject means the evidence wasn't strong enough, like a "not guilty" verdict that doesn't prove innocence.

### Why can't I say "accept the null hypothesis" on the AP exam?

Because "accept" implies you've shown H₀ is true, and the test logic doesn't allow that conclusion. Graders specifically look for "fail to reject H₀" plus a statement that there is insufficient evidence for Hₐ, and "accept H₀" can cost you the conclusion point on an FRQ.

### How do I know whether to reject or fail to reject the null hypothesis?

Compare the p-value to α. If the p-value ≤ α, reject H₀. If the p-value > α, fail to reject H₀. For example, a p-value of 0.078 with α = 0.05 means you fail to reject, because 0.078 > 0.05.

### What's the difference between failing to reject H₀ and having a small p-value?

They're opposites. A small p-value (at or below α) means your data would be very unlikely if H₀ were true, so you reject H₀. A large p-value means your data is consistent with H₀, so you fail to reject it.

## Related Study Guides

- [3.7 Carrying Out a Test for a Population Proportion](/ap-stats/unit-3/concluding-test-for-population-proportion/study-guide/THZeUpkm11DAwnNb6p4g)

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