---
title: "Null and Alternative Hypotheses — AP Stats Guide"
description: "Null (H₀) and alternative (Hₐ) hypotheses are the competing claims every significance test starts with. Learn how to write them and how p-values depend on Hₐ."
canonical: "https://fiveable.me/ap-stats/key-terms/null-and-alternative-hypotheses"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 3"
---

# Null and Alternative Hypotheses — AP Stats Guide

## Definition

In AP Statistics, the null hypothesis (H₀) is the claim of no change or no effect about a population parameter, while the alternative hypothesis (Hₐ) is the claim of a change or effect; every significance test measures evidence against H₀ in favor of Hₐ.

## What It Is

Every [significance test](/ap-stats/key-terms/significance-test "fv-autolink") starts with two competing claims about a **population parameter** (like p or μ), never about a sample statistic. The **null hypothesis (H₀)** is the boring [claim](/ap-stats/unit-3/justifying-claim-based-on-confidence-interval-for-population-proportion/study-guide/YeTpyj6nyq03j0AJO3Bm "fv-autolink"), the status quo, the assumption that nothing has changed. It always contains an equals sign, like H₀: p = 0.5. The **alternative hypothesis (Hₐ)** is the claim the researcher suspects or hopes is true, written with <, >, or ≠ depending on whether you're looking for a decrease, an increase, or just any difference.

Here's the part that makes everything else in [Unit 3](/ap-stats/unit-3 "fv-autolink") click. The whole test runs *assuming H₀ is true*. You build the null distribution (a randomization distribution or a theoretical z distribution) under that assumption, then ask how surprising your sample would be in that world. If your result would be really rare assuming H₀ is true, that's evidence against H₀ and in favor of Hₐ. The hypotheses aren't decoration. They define what "extreme" means and what your p-value is actually measuring.

## Why It Matters

This term lives in **the relevant current topics (Interpreting [p-Values](/ap-stats/unit-3/interpreting-p-values/study-guide/b0FEXf5MDjyQtz4Skz70 "fv-autolink"))** in **Unit 3: Inference for Categorical Data: Proportions**, supporting learning objectives **the relevant learning objective** and **the relevant learning objective**. The [null hypothesis](/ap-stats/key-terms/null-hypothesis "fv-autolink") supplies the value p₀ that goes directly into the z-statistic formula, z = (p̂ - p₀) / √(p₀(1-p₀)/n), and the alternative hypothesis tells you which tail(s) of the null distribution to use for the p-value. Per the CED, the p-value is computed as the proportion at or above the test statistic if Hₐ uses >, at or below if Hₐ uses <, and in both tails if Hₐ uses ≠. So you literally cannot calculate or interpret a p-value correctly without nailing the hypotheses first. The same H₀/Hₐ framework then repeats for means, slopes, and chi-square tests through the rest of the course.

## Connections

### [p-Value (Unit 3)](/ap-stats/key-terms/p-value)

The [p-value](/ap-stats/key-terms/p-value "fv-autolink") is defined entirely in terms of the hypotheses. It's the probability of getting a result as extreme as yours, assuming H₀ is true, with "extreme" pointing in the direction Hₐ specifies. A one-sided Hₐ uses one tail; a ≠ alternative doubles up with both tails.

### [Null Distribution (Unit 3)](/ap-stats/key-terms/null-distribution)

The [null distribution](/ap-stats/key-terms/null-distribution "fv-autolink") is what the test statistic looks like in a world where H₀ is true. Whether it's a randomization distribution or a theoretical z curve, H₀ is the assumption that builds it.

### Type I and Type II Errors (Unit 3)

Both errors are defined relative to H₀. A [Type I error](/ap-stats/key-terms/type-i-error "fv-autolink") is rejecting H₀ when it's actually true; a Type II error is failing to reject H₀ when Hₐ is actually true. If you can't state the hypotheses, you can't describe the errors.

### Tests for Means and Chi-Square Tests (Units 7-8)

The H₀/Hₐ setup you learn for proportions is the same skeleton used for t-tests about means in Unit 4 and chi-square tests in Unit 3. Only the parameter and the distribution change; the logic of assuming H₀ and weighing evidence against it never does.

## On the AP Exam

Stating hypotheses is usually the very first scored step of a significance-test FRQ, and it shows up almost every year. The 2021 FRQ (coupon offer to increase repeat purchases), the 2023 FRQ (omega-3 supplement vs. placebo), and the 2024 FRQ all required setting up or reasoning from null and alternative hypotheses in context. To earn credit you need three things. First, write hypotheses about the population parameter (p or μ), not the sample statistic (p̂ or x̄). Second, define the parameter in context. Third, match the direction of Hₐ to the research question ("increase" means >, "differs" means ≠). On multiple choice, expect stems asking which pair of hypotheses fits a scenario, or which p-value calculation matches a given Hₐ. Conclusions are also graded against the hypotheses, so you reject or fail to reject H₀, and you never "accept" H₀.

## Null and Alternative Hypotheses vs Null Hypothesis vs. Alternative Hypothesis (which is which)

The null hypothesis is the skeptic's claim of no effect or no change, and it always gets the equals sign (H₀: p = p₀). The alternative is the researcher's claim of an effect, written with <, >, or ≠. A classic trap is putting the claim you want to support into H₀. The thing you're trying to find evidence FOR goes in Hₐ; the test only ever gathers evidence against H₀.

## Key Takeaways

- The null hypothesis (H₀) states no effect or no change and always contains an equals sign, like H₀: p = 0.5.
- The alternative hypothesis (Hₐ) is the claim the study is trying to find evidence for, written with <, >, or ≠.
- Hypotheses are always about population parameters like p or μ, never about sample statistics like p̂ or x̄.
- The direction of Hₐ determines how the p-value is calculated: one tail for < or >, both tails for ≠.
- The null value p₀ from H₀ plugs directly into the z-statistic formula, z = (p̂ - p₀) / √(p₀(1-p₀)/n).
- Your conclusion is always 'reject H₀' or 'fail to reject H₀'; you never accept or prove the null hypothesis.

## FAQs

### What are null and alternative hypotheses in AP Stats?

They are the two competing claims about a population parameter that start every significance test. H₀ says there's no effect or change (with an equals sign), and Hₐ says there is one (with <, >, or ≠). The test measures how much evidence the sample provides against H₀.

### Does a small p-value prove the alternative hypothesis is true?

No. A small p-value gives convincing evidence against H₀ in favor of Hₐ, but it doesn't prove anything. It just says your sample result would be unlikely if H₀ were true. There's always a chance you've made a Type I error.

### Can you accept the null hypothesis if the p-value is large?

No, and this costs points every year. A large p-value means you fail to reject H₀, which only says you lack convincing evidence for Hₐ. It never proves H₀ is true.

### How do I know if the alternative hypothesis should be <, >, or ≠?

Read the research question. Words like 'increased' or 'more than' mean >, 'decreased' or 'less than' mean <, and 'changed' or 'differs from' mean ≠. The 2021 FRQ about increasing repeat purchases, for example, calls for a one-sided > alternative.

### Do hypotheses use p̂ or p?

Always p (or μ for means). Hypotheses are claims about the unknown population parameter. Writing H₀: p̂ = 0.5 is a common error because you already know p̂ from your sample, so there's nothing to test about it.

## Related Study Guides

- [3.6 p-Values](/ap-stats/unit-3/interpreting-p-values/study-guide/b0FEXf5MDjyQtz4Skz70)

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