Null vs Alternative Hypothesis (with Examples)

Every statistical test in a research paper or lab report starts with two competing statements: the null hypothesis and the alternative hypothesis. Here is what each means, how to write them, and how to report the result correctly.

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Updated 21 October 2026 · EssayWriting.com editorial team

Key takeaways

The null hypothesis (H0) says there is no effect; the alternative (H1 or Ha) says there is one.
You reject or fail to reject the null; you never “prove” it.
A p-value below your significance level (often 0.05) leads you to reject the null.

Definitions

	Null hypothesis (H₀)	Alternative hypothesis (H₁ or H_a)
Says	There is no effect, no difference or no relationship	There is an effect, difference or relationship
Role	The default assumption the test tries to find evidence against	What the researcher usually expects or wants to show
Contains	An equals sign (=, or ≤ / ≥ in one-tailed tests)	≠, < or >
Example	H₀: μ₁ = μ₂	H₁: μ₁ ≠ μ₂

Why we test the null, not the alternative

Statistical tests calculate how likely your data would be if the null hypothesis were true. If that probability (the p-value) is very small, the data are hard to explain without an effect, so you reject the null. You never prove the alternative directly; you collect evidence against the null.

Worked examples

Research question	H₀	H₁
Does a new study app change exam scores?	Mean score with app = mean score without app	Mean scores differ (two-tailed)
Does caffeine reduce reaction time?	Mean reaction time with caffeine ≥ without	Mean reaction time with caffeine < without (one-tailed)
Is screen time related to sleep duration?	Correlation ρ = 0	ρ ≠ 0
Is a coin fair?	p(heads) = 0.5	p(heads) ≠ 0.5

One-tailed vs two-tailed tests

A two-tailed test asks whether there is a difference in either direction. A one-tailed test asks about one direction only (greater than, or less than). Decide before you see the data; choosing a one-tailed test after looking at results inflates the chance of a false positive. When in doubt, use two-tailed.

p-values and significance level

Before testing, choose a significance level (α). In many fields α = 0.05 is the convention, but your field or instructor may use a different level. If p ≤ α, reject H₀; if p > α, fail to reject H₀.

A p-value is not the probability that the null hypothesis is true, and a statistically significant result is not necessarily a large or important effect. Report effect sizes and confidence intervals where your course expects them.

Type I and Type II errors

	H₀ actually true	H₀ actually false
Reject H₀	Type I error (false positive), probability α	Correct decision (power)
Fail to reject H₀	Correct decision	Type II error (false negative), probability β

How to write hypotheses in a paper

Start from a precise research question.
Name the variables and the population.
Write H₀ as "no difference / no relationship".
Write H₁ as the specific difference or relationship you are testing, including direction if one-tailed.
State α and the test you will use in your methods section.

"We hypothesised that students who used the study app would have different mean exam scores from those who did not (H₁: μ_app ≠ μ_control), against the null hypothesis of no difference (H₀: μ_app = μ_control), tested with an independent-samples t-test at α = .05."

Reporting the result

Write "we rejected the null hypothesis" or "we failed to reject the null hypothesis". Avoid "we accepted the null" and "we proved the alternative". A results sentence in APA style might read: "Students using the app scored higher (M = 74.2, SD = 8.1) than controls (M = 70.5, SD = 9.0), t(98) = 2.17, p = .032." The numbers here are illustrative. See our APA format guide for citation and paper layout.

Hypothesis vs thesis statement

A hypothesis is a testable prediction about data; a thesis statement is the central claim of an essay. Research papers often have both. See how to write a thesis statement.

Common test examples and their hypotheses

Test	Typical H₀	Typical H₁
Independent-samples t-test	The two group means are equal	The group means differ
Paired t-test	The mean difference is zero	The mean difference is not zero
One-way ANOVA	All group means are equal	At least one group mean differs
Chi-square test of independence	The two categorical variables are independent	The variables are associated
Pearson correlation	ρ = 0	ρ ≠ 0
Linear regression (slope)	β = 0	β ≠ 0

Common mistakes

Writing hypotheses about the sample instead of the population.
Putting the equals sign in the alternative hypothesis.
Choosing one-tailed after seeing which direction the data go.
Saying the result "proves" the alternative hypothesis.
Treating a non-significant result as proof of no effect.
Reporting p-values without the test statistic or effect size your course expects.

Quick answers to common questions

Can there be more than one alternative hypothesis?

A study can test several hypotheses, each with its own null and alternative, but each test pairs one null with one alternative.

Is H1 always what I expect?

Usually it states the effect you are investigating, but it should be decided before seeing the data.

Frequently asked questions

What is the difference between the null and alternative hypothesis?

The null hypothesis states there is no effect or difference; the alternative states there is one. Tests look for evidence against the null.

Can you accept the null hypothesis?

Standard practice is to say you fail to reject the null hypothesis, because a non-significant result does not prove there is no effect.

What symbol is used for the alternative hypothesis?

H1 or Ha.

What does p less than 0.05 mean?

If α is 0.05, a p-value at or below 0.05 means the result is statistically significant and you reject the null hypothesis. It does not give the probability that the null is true.

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