---
title: "Reject the Null Hypothesis — AP Stats Definition"
description: "Rejecting the null hypothesis means your p-value ≤ α, so the data give convincing evidence for the alternative. The core decision in every AP Stats inference FRQ."
canonical: "https://fiveable.me/ap-stats/key-terms/reject-the-null-hypothesis"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 8"
---

# Reject the Null Hypothesis — AP Stats Definition

## Definition

In AP Statistics, you reject the null hypothesis when the p-value is less than or equal to the significance level α, meaning the observed data would be unlikely if H₀ were true, so you have convincing statistical evidence for the alternative hypothesis.

## What It Is

Rejecting the [null hypothesis](/ap-stats/key-terms/null-hypothesis "fv-autolink") is the formal decision you make at the end of a significance test when your p-value is small enough. The logic works like a proof by contradiction. You assume the null hypothesis (H₀) is true, then ask how likely it is to get a [sample](/ap-stats/unit-3/intro-planning-study/study-guide/YR5NI5ejwMAQ2dglm67s "fv-autolink") result as extreme as yours, or more extreme. That probability is the p-value. If the p-value is less than or equal to your significance level α (usually 0.05), the data are too surprising to chalk up to random chance, so you reject H₀ and conclude there is convincing evidence for the alternative hypothesis (Hₐ).

The decision rule is the same no matter which test you're running. For a two-proportion z-test, the CED says if p-value ≤ α, reject H₀: p₁ = p₂. For a chi-square [goodness of fit](/ap-stats/unit-8 "fv-autolink") test, you compare the p-value from your χ² statistic (with degrees of freedom = categories − 1) to α the exact same way. What changes between tests is the statistic and the distribution, not the decision logic. One thing to lock in early is that rejecting H₀ never *proves* the alternative. It just means the evidence against H₀ is strong enough at your chosen α level.

## Why It Matters

This decision step is the payoff of every inference procedure in Units 6 through 9. It's named explicitly in learning objectives [AP Stats](/ap-stats "fv-autolink") 6.11.C (justify a claim using a two-proportion [z-test](/ap-stats/key-terms/z-test "fv-autolink")) and AP Stats 8.3.D (justify a claim using a chi-square goodness of fit test). Both say the same thing in CED language. A formal decision explicitly compares the p-value to α, and the result of the test serves as the statistical reasoning behind your answer to the research question. On the exam, the conclusion is usually where the points live. You can compute a perfect test statistic and still lose credit if your decision sentence doesn't compare p to α, name the decision, and state the conclusion in context.

## Connections

### [P-Value (Units 6-9)](/ap-stats/key-terms/p-value)

The [p-value](/ap-stats/key-terms/p-value "fv-autolink") is the evidence; rejecting the null is the verdict. Per AP Stats 8.3.C and 6.11.B, the p-value is computed assuming H₀ is true, so a tiny p-value means your data clash with that assumption, which is exactly why you reject it.

### [Chi-Square Statistic (Unit 8)](/ap-stats/key-terms/chi-square-statistic)

In a goodness of fit test, a big χ² = Σ(Observed − Expected)²/Expected means [observed counts](/ap-stats/unit-8/carrying-out-chi-square-test-for-homogeneity-or-independence/study-guide/AmRuD3egBrZMCLGaBpMw "fv-autolink") sit far from what H₀ predicts. Big χ² leads to small p-value leads to rejecting H₀. Same decision rule as a z-test, new distribution.

### [Difference in Two Population Proportions (Unit 6)](/ap-stats/key-terms/difference-in-two-population-proportions)

In Topic 6.11 the null says p₁ = p₂, so rejecting it means you have convincing evidence the two population proportions actually differ. That's why you pool the proportions when computing the [z-statistic](/ap-stats/key-terms/z-statistic "fv-autolink"), because H₀ assumes the groups share one common proportion.

### [Confidence Interval (Unit 6)](/ap-stats/key-terms/confidence-interval)

Intervals and tests are two views of the same logic. If a 95% confidence interval for p₁ − p₂ misses 0, a two-sided test at α = 0.05 would reject H₀: p₁ − p₂ = 0. Exam questions love asking you to connect these.

## On the AP Exam

Multiple choice questions typically hand you a test statistic or p-value and ask which conclusion is correct. For example, a chi-square goodness of fit test with 4 categories and χ² = 9.72 at α = 0.05, or χ² = 9.82 with df = 3 and p-value = 0.02. The wrong answer choices are predictable. They "accept" the null, they reject when p > α, or they state the conclusion about the sample instead of the population. On FRQs, the full significance test is a classic question (the 2017 chi-square FRQ on schizophrenia diagnosis ages and the 2019 two-proportion FRQ on tumbleweed both asked for "convincing statistical evidence"). Your conclusion needs three pieces to earn credit. Compare the p-value to α explicitly, state the decision (reject or fail to reject H₀), and answer the research question in context, using language like "convincing evidence that" rather than "proves."

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

These are the only two legal decisions, and the line between them is the significance level. If p-value ≤ α, reject H₀. If p-value > α, fail to reject H₀. The trap is treating "fail to reject" as "accept H₀" or "prove H₀ is true." It isn't. Failing to reject only means your data didn't provide convincing evidence against H₀. Maybe the null is true, or maybe your sample was too small to detect a real effect. Writing "accept the null hypothesis" on an FRQ conclusion will cost you. Stick to the two official phrases.

## Key Takeaways

- Reject the null hypothesis when the p-value is less than or equal to the significance level α; fail to reject when the p-value is greater than α.
- Rejecting H₀ means you have convincing statistical evidence for the alternative hypothesis, not that you have proven it true.
- The decision rule is identical across every AP inference test, from two-proportion z-tests (Topic 6.11) to chi-square goodness of fit tests (Topic 8.3); only the statistic and distribution change.
- A full-credit FRQ conclusion has three parts: an explicit comparison of the p-value to α, the decision about H₀, and an answer to the research question in context.
- Never write "accept the null hypothesis." Failing to reject H₀ just means the evidence against it wasn't strong enough, not that H₀ is true.
- Your conclusion is about the population that was sampled, not about the sample itself; the sample is the evidence, the population is the claim.

## FAQs

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

It means the p-value from your significance test was less than or equal to α, so the observed data would be unlikely if H₀ were true. You conclude there is convincing evidence for the alternative hypothesis about the population.

### Does rejecting the null hypothesis prove the alternative hypothesis is true?

No. Rejecting H₀ means the evidence supports Hₐ at your chosen significance level, but there's still a chance you rejected a true null (a Type I error, which happens with probability α). On the exam, say "convincing evidence," never "proves."

### What's the difference between reject and fail to reject the null hypothesis?

You reject H₀ when p-value ≤ α and fail to reject when p-value > α. Rejecting means convincing evidence for the alternative; failing to reject means the evidence wasn't strong enough, which is not the same as evidence that H₀ is true.

### Is it ever correct to say "accept the null hypothesis"?

No. A significance test can only show evidence against H₀, never evidence for it. "Fail to reject" is the only acceptable wording when p > α, and graders specifically watch for this.

### Do I always reject the null hypothesis if the p-value is less than 0.05?

Only if α = 0.05, which is the default but not automatic. The rule is p-value ≤ α for whatever α the problem states. A 2024-style MCQ might set α = 0.01, where a p-value of 0.02 means you fail to reject, not reject.

## Related Study Guides

- [8.3 Carrying Out a Chi Square Goodness of Fit Test](/ap-stats/unit-8/carrying-out-chi-square-goodness-fit-test/study-guide/XmvvVf9spR7e6xT6TPEG)
- [6.11 Carrying Out a Test for the Difference of Two Population Proportions](/ap-stats/unit-6/carrying-out-test-for-difference-two-population-proportions/study-guide/mkdLDKdZAuQ6pIsV3aOL)

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