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What Is a Null Hypothesis? Examples & How to Write One | CASRAI

Q: Rejecting the null hypothesis proves the alternative hypothesis.

No — rejecting H₀ means the data are inconsistent with it at the chosen threshold; it does not prove H₁. The alternative hypothesis remains tentatively supported, pending replication.

Q: A non-significant result means the null hypothesis is true.

No — it means there is insufficient evidence to reject it at the chosen α. A true effect may simply be undetected because the study is underpowered (too small a sample).

Q: You can test the null hypothesis without specifying H₁.

Only in Fisher's framework. Neyman–Pearson testing requires specifying H₁ in advance — because α, β, and sample size calculations depend on the size of the effect you want to detect.

A null hypothesis (H₀) is the default statement in statistical hypothesis testing — a claim that there is no effect, no difference, or no relationship between variables. Researchers test whether evidence is strong enough to reject it in favour of an alternative hypothesis.

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Fisher vs Neyman–Pearson: two frameworks, one practice

Statistical hypothesis testing evolved from two distinct traditions that most textbooks conflate. Ronald Fisher’s significance testing (1925) treats the p-value as a continuous measure of evidence against a single null hypothesis — not a binary decision. Jerzy Neyman and Egon Pearson’s hypothesis testing (1933) involves specifying both H₀ and H₁ before data collection, choosing α (the Type I error rate), and making a binary accept/reject decision with controlled error rates. Fisher’s approach does not involve H₁ or Type II errors; Neyman–Pearson’s approach does not involve evidence or p-values as continuous quantities. Modern practice typically requires elements of both: specifying H₀ and H₁, computing a p-value, and reporting power — a hybrid that neither Fisher nor Neyman–Pearson would have fully endorsed.

How to write a null hypothesis

A null hypothesis must be specific, testable, and expressed as a statement of no effect or no difference. It should correspond directly to the research question and the statistical test being used. In an experiment testing a blood-pressure medication, H₀ might be: "There is no statistically significant difference in mean systolic blood pressure between the treatment group and the control group at 12 weeks." The alternative hypothesis H₁ can be non-directional ("there is a difference") or directional ("the treatment group has lower blood pressure" — a one-tailed test). Directional hypotheses require prior justification and are less common because they are more prone to Type I error if the direction is specified opportunistically.

Rejecting H₀ and what it means

When the p-value falls below the pre-specified α threshold (usually 0.05), the researcher rejects H₀ — the result is statistically significant. Crucially, rejection of H₀ does not prove H₁; it means only that the data are inconsistent with H₀ at the chosen threshold. Null hypotheses are never "accepted" — a non-significant result means there is insufficient evidence to reject H₀, not that H₀ is true (the study may simply be underpowered). Karl Popper’s falsificationism — that scientific theories must be falsifiable — underpins this logic: hypotheses are corroborated by surviving attempts to disprove them, not proved.

Null hypotheses across disciplines

The form of null hypothesis varies by discipline and statistical test. In a t-test, H₀ states means are equal; in an ANOVA, H₀ states all group means are equal; in a chi-square test, H₀ states the variables are independent (no association). In regression, H₀ for each coefficient is that it equals zero. In ecology, null models posit random species distributions; in genomics, null hypotheses are tested simultaneously for thousands of genes, requiring multiple-comparison corrections (Bonferroni, false discovery rate). In social science, structural equation models test whether observed covariance matrices match a theorised model (H₀: the model fits).

Key facts

At a glance

Definition: Statement of no effect/difference/relationship — the default to be tested
Symbol: H₀; alternative hypothesis is H₁
Two frameworks: Fisher (evidence, p-value) vs Neyman–Pearson (decision, error rates)
Rejection: p < α (e.g. 0.05) allows rejection of H₀, not proof of H₁
Never: Null hypotheses are never "accepted" — only "failed to reject"
Popper link: Falsificationism underpins hypothetico-deductive reasoning
Varies by test: t-test, ANOVA, chi-square, regression, structural equation models

Common misconceptions

What people often get wrong

Often heard: Rejecting the null hypothesis proves the alternative hypothesis.

Actually: No — rejecting H₀ means the data are inconsistent with it at the chosen threshold; it does not prove H₁. The alternative hypothesis remains tentatively supported, pending replication.

Often heard: A non-significant result means the null hypothesis is true.

Actually: No — it means there is insufficient evidence to reject it at the chosen α. A true effect may simply be undetected because the study is underpowered (too small a sample).

Often heard: You can test the null hypothesis without specifying H₁.

Actually: Only in Fisher's framework. Neyman–Pearson testing requires specifying H₁ in advance — because α, β, and sample size calculations depend on the size of the effect you want to detect.

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