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Hypothesis testing uses sample data to evaluate a claim about a population. It can provide evidence to reject a specified null hypothesis, but it cannot prove that claim—or its alternative—true. A sound test starts with a clear question, a suitable method and a decision rule chosen before interpreting the results.
What is hypothesis testing?
Hypothesis testing is a statistical procedure for assessing whether sample data are sufficiently inconsistent with a specified claim about a population. The claim is represented by a null hypothesis, written H0. A competing explanation is represented by an alternative hypothesis, written Ha.
The procedure calculates a test statistic and evaluates it under a model that assumes H0 is true. The result is a decision about whether the data justify rejecting H0 under the chosen test and its assumptions—not proof that either hypothesis is true. NIST/SEMATECH explains the role of a hypothesis pair in its overview of statistical tests.
How to conduct a hypothesis test
- State the question and the population parameter. Identify the quantity or comparison you want to understand, such as a population mean, proportion, or difference between groups. Define the population and outcome clearly.
- Write H0 and Ha. The null states the benchmark or no-difference claim being evaluated. The alternative states what evidence would support instead. Use a one-sided alternative only when a direction is justified by the question in advance; otherwise, a two-sided alternative may be appropriate.
- Choose a test that fits the data and design. Consider the outcome, parameter, sampling structure, and assumptions. For example, determine whether observations are independent, paired, or clustered, and whether the test’s reference distribution is appropriate. Tests for means, standard deviations, and proportions are common examples, but no single test fits every design.
- Set the significance level, α, in advance. Alpha is the procedure’s tolerated probability of rejecting H0 when it is true, subject to the test assumptions. NIST lists 0.05, 0.1, and 0.01 as common examples; the choice is context-dependent and should be justified. Do not select the threshold after seeing which result is convenient. See NIST’s discussion of p-values and critical values.
- Calculate the test statistic and p-value, or use a critical value. The statistic summarizes the sample evidence relative to the null model. A p-value is the probability, assuming H0 is true, of obtaining a test statistic at least as extreme as the observed one. It is not the probability that H0 is true, nor does it measure the size or practical importance of an effect.
- Apply the decision rule. With the p-value approach, reject H0 if p is at or below the preselected α; otherwise, fail to reject H0. With the critical-value approach, compare the statistic with the rejection-region cutoff for the test and α. These approaches express the same decision rule when they use the same test and significance level.
- Explain what the result means in context. Describe the estimated effect’s direction and size, its uncertainty, the assumptions and data-quality limits, and whether the effect matters in practice. A threshold crossing alone is not the full conclusion.
How to interpret the decision
Reject H0
Rejecting H0 means the observed data meet the test’s rejection rule under the stated assumptions. It is evidence against the null model, not proof that the alternative is true. Report the test, decision rule, estimate and uncertainty so readers can understand the strength and practical meaning of the evidence.
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Fail to reject H0
Failing to reject means the data did not meet the chosen rule for rejecting H0. It does not establish that H0 is true or that there is no effect. The study may have limited information or power to detect the effect of interest. Use “fail to reject” rather than “accept” the null; NIST discusses this distinction and the limits of non-rejection in its quantitative-techniques guidance.
Errors, power, and practical importance
- Type I error: Rejecting H0 when it is true. Alpha sets the procedure’s bound on this risk, subject to its assumptions.
- Type II error: Failing to reject a false H0. Its probability, β, depends on a particular alternative; it is not a single fixed property of a test in every situation.
- Power: The probability of rejecting H0 when a specified alternative is true, equal to 1 − β. Sample size and the effect being considered influence whether a test is likely to detect it.
Statistical significance and practical significance are different. A result that meets a statistical threshold may still represent a small or unimportant effect; a result that misses the threshold does not by itself show that an effect is absent. Interpret the estimate and uncertainty alongside the test decision.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How tests and confidence intervals differ
A hypothesis test assesses a specified claim using a decision rule. A confidence interval instead gives a range of plausible values for a population quantity under its method and assumptions. They address related sampling uncertainty but provide different outputs: the test focuses on a claim, while the interval helps show the estimated magnitude and precision. NIST discusses both approaches in its overview of quantitative techniques.
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Common mistakes to avoid
- Choosing Ha, the test, or α after looking at the results.
- Using a one-sided test when the direction was not justified before analysis.
- Reading the p-value as the chance that the null hypothesis is true.
- Calling a non-significant result proof of no effect or saying the null was accepted.
- Reporting only whether p crossed α, without the estimated effect, uncertainty, assumptions, or practical context.
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