Null and Alternative Hypotheses: Definitions and Examples

A null hypothesis (H0) states there is no effect, difference, or relationship in a population; an alternative hypothesis (Ha) states that one exists. Statistical tests compare sample evidence against H0 and use a p-value to decide whether to reject it.

What is a Null Hypothesis?

The null hypothesis (H0) is the default, “no effect” claim in a statistical test. It states there is no difference, no relationship, or no change between the groups or variables being studied.

Researchers assume H0 is true at the outset. A test then checks whether the sample data are unusual enough, under that assumption, to justify rejecting it in favour of a real effect.

H0 is written as an equality, such as “the population mean equals 65” or “there is no correlation between two variables.” It is always the statement being tested, never the one being proven.

The term was popularised by statistician Ronald Fisher in the early twentieth century. Later work by Jerzy Neyman and Egon Pearson added the alternative hypothesis and formalised the reject/fail-to-reject framework still used today.

What is an Alternative Hypothesis?

The alternative hypothesis (Ha, sometimes written H1) is the claim a researcher actually wants to investigate — that a genuine effect, difference, or relationship exists. It directly contradicts H0.

Ha can be non-directional (“a difference exists”) or directional (“the effect is higher” or “lower”). This choice, made before data collection, decides whether the test is two-tailed or one-tailed.

Why the Null Hypothesis Matters in Research

Science works by trying to disprove claims, not confirm them. Testing against H0 forces a researcher to show that a result is unlikely to be chance before claiming an effect is real.

This burden of proof sits with Ha, not H0. Without a properly stated null hypothesis, there is no fixed baseline against which to judge whether a finding is meaningful or just noise.

How the Null and Alternative Hypotheses Work Together

Hypothesis testing never proves H0 true. It only asks whether the sample data give enough evidence to reject it. Weak evidence means you fail to reject H0 — not that H0 is confirmed.

The full process runs through fixed steps: writing both hypotheses, setting a significance level, running the test, and comparing the resulting p-value against that threshold. Our hypothesis testing guide covers each step in detail.

Each step depends on the one before it. Skip the significance-level step, for example, and there is no fixed threshold for judging the p-value, so the decision becomes arbitrary.

Null Hypothesis vs Alternative Hypothesis at a Glance

The table below compares the two hypotheses directly, since students often confuse which one carries the burden of proof and which one is assumed true by default.

Null Hypothesis Examples

The table below shows how H0 and Ha are written for different types of research question, from comparing two groups to testing a single proportion.

Null Hypotheses Across Common Statistical Tests

The wording of H0 changes slightly depending on the test, but the underlying idea — no effect, no difference, no relationship — stays the same across every method below.

• t-test: H0 states the two group means are equal.
• Chi-square test: H0 states two categorical variables are independent.
• ANOVA: H0 states all group means are equal.
• Correlation: H0 states the correlation coefficient is zero.
• Regression: H0 states a predictor’s coefficient is zero (no effect on the outcome).

How to Write a Null Hypothesis Step by Step

Start with your research question, then identify the population and the exact variable you plan to measure. Every hypothesis should refer to a single, testable statistic.

1. Identify the variable (e.g. exam score, reaction time, heart rate).
2. State the population or groups being compared.
3. Write H0 as an equality: no difference, no effect, no relationship.
4. Write Ha as the contradiction: a difference, effect, or relationship exists.
5. Decide the test direction (one-tailed or two-tailed) before collecting data.

Example: testing whether online and in-person students get different average grades. H0: μ(online) = μ(in-person). Ha: μ(online) ≠ μ(in-person) — a two-tailed test, since no direction is predicted.

One-Tailed vs Two-Tailed Alternative Hypotheses

A two-tailed Ha predicts a difference in either direction (≠). A one-tailed Ha predicts a specific direction (> or <). The choice must be fixed before data collection, based on the research question.

Two-tailed tests split the significance level across both ends of the distribution. One-tailed tests place the whole rejection region on one side, making it easier to reject H0 in that direction.

The diagram shows shaded rejection regions in each tail. If the test statistic falls inside a shaded region, the result is statistically significant and H0 is rejected.

How to Reject the Null Hypothesis

You reject H0 when the p-value is less than or equal to the chosen significance level (α), commonly 0.05. This signals the sample result would be unlikely if H0 were actually true.

Rejecting H0 does not prove Ha is true. It means the evidence favours Ha over H0 at the chosen confidence level — not certainty, and never a guarantee.

Two error types can occur: rejecting a true H0 (Type I error) or failing to reject a false H0 (Type II error). Neither test result eliminates this risk entirely.

Larger samples and well-designed tests give more reliable p-values, but no single study proves a hypothesis beyond doubt. Replication across studies builds the stronger case.

Common Mistakes When Writing Hypotheses

A frequent error is writing Ha first and reverse-engineering H0 to match. Always state H0 as the “no effect” claim, then derive Ha from the actual research question.

Another mistake is treating “fail to reject H0” as proof of no effect. It may simply mean the sample was too small or too noisy to detect a real difference.

Vague hypotheses that skip a measurable variable — “this method is better” — are hard to test. Tie H0 and Ha to a specific statistic, such as a mean, proportion, or correlation.

A fourth mistake is changing the hypothesis or the tail direction after seeing the data. Both must be fixed before the test runs, or the resulting p-value is no longer trustworthy.

Where Hypothesis Testing Fits in Your Research

Hypothesis testing runs through the results chapter of most quantitative dissertations, from a single t-test to multiple regression models. Our dissertation writing service supports the methodology and results chapters around your data.

If you are running these tests in SPSS, our SPSS data analysis help walks through output tables and interpretation. Browse more explainers in the statistical analysis guide hub.

Key Takeaways

• H0 claims no effect; Ha claims an effect exists.
• You never “prove” H0 — you only fail to reject it.
• Rejecting H0 requires p ≤ α, a threshold set before testing.
• Choose one-tailed or two-tailed before collecting data, not after.
• Report an effect size alongside any decision to reject or not.

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