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A statistical hypothesis test evaluates how compatible your data are with a specified null hypothesis under a stated model and decision rule. It can provide evidence against that null; failing to reject it does not prove the null hypothesis true.
This tutorial shows how to formulate hypotheses, choose a method that matches your outcome and design, check method-specific assumptions, and report results without confusing statistical evidence with practical importance.
What a statistical hypothesis test establishes
Start with a null hypothesis (H0) and an alternative hypothesis (HA). A test statistic summarizes how far the observed data are from what H0 predicts. You then apply a rejection rule using either a critical value or a p-value. NIST describes a p-value as the probability, assuming H0 is true, of obtaining a test statistic at least as extreme as the observed one (NIST, “What are statistical tests?”; “Critical values and p values”).
A small p-value is evidence against H0 under the chosen model and procedure. It is not the probability that H0 is true, nor does it by itself show that an effect is large or useful. A non-significant result means the data did not meet the prespecified evidence threshold; it does not establish equality or no effect.
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Set the question and decision rule before looking at the result
- Define the estimand. State whether the target is a mean, variance, proportion or count distribution, and identify the population or process.
- Write H0 and HA. For example, H0: μ = μ0; HA: μ ≠ μ0.
- Choose the direction. A lower-tailed, upper-tailed, or two-sided alternative must follow the substantive question. Do not select a one-sided test after seeing which direction your estimate took (NIST, “Chi-Square Test for the Variance”).
- Prespecify α. Select the significance level before interpreting the data. α is the long-run Type I error rate targeted by the decision rule, not the chance that your particular conclusion is wrong.
- Check the design and assumptions. Confirm independence or pairing, distributional requirements, variance conditions, and sample-size or expected-count requirements for the selected test.
- Compute and interpret. Report the statistic, degrees of freedom when applicable, p-value, and an estimate with an interval—not only “significant” or “not significant.”
Choose a test from the outcome and study design
The label “hypothesis test” is too broad to determine a method. Match the parameter and data structure to a procedure. NIST lists t tests, ANOVA, chi-squared tests, and F tests among classical quantitative techniques (NIST, “Techniques”).
| Research question | Typical design | Representative method and scope | Key checks |
|---|---|---|---|
| Is one population mean equal to a target μ0? | One sample of quantitative observations | One-sample t test; the statistic is T = (Ȳ − μ0)/(s/√N), with N − 1 degrees of freedom under its usual conditions (NIST, “Confidence Limits for the Mean”). | Sampling design, approximate normality of the mean, and absence of influential departures; use the conditions appropriate to the actual data. |
| Do group means differ? | Independent groups, paired observations, or several groups | t tests are commonly used for two-mean comparisons; ANOVA is a classical approach for several means. The exact version depends on pairing, variance structure, and design. | Independence or correct pairing, distributional shape, and variance assumptions required by the chosen procedure. |
| Is a population variance equal to a specified value? | One quantitative sample | Chi-square test for a variance, with lower-, upper-, or two-sided alternatives chosen from the question (NIST, “Chi-Square Test for the Variance”). | The distributional conditions for the variance statistic, especially the relevant normality assumption. |
| Do observed category counts follow a specified distribution? | Counts placed into defined bins or categories | Chi-square goodness-of-fit test (NIST, “Chi-Square Goodness-of-Fit Test”). | Bins must be defined before analysis when possible; expected counts must be sufficient for the chi-square approximation. |
| Is a variance ratio or related model component different? | Design-specific quantitative comparison | F tests are another classical family named by NIST; the appropriate form depends on the model and design. | Use a test-specific reference for the exact statistic, assumptions, and degrees of freedom. |
Understand p-values, critical values, and alternatives
For a prespecified α, the critical-value approach rejects H0 when the test statistic falls in the rejection region. The p-value approach rejects when p ≤ α. These are equivalent representations of the same decision rule when the model and tail definition match.
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A two-sided alternative asks whether the parameter differs in either direction. A one-sided alternative asks specifically whether it is smaller or larger. The direction should be justified by the scientific or operational question, not by the observed estimate. For a one-sample mean, the t statistic above is compared with a t distribution having N − 1 degrees of freedom; the corresponding confidence interval provides the range of mean values compatible with the data and assumptions (NIST, “Confidence Limits for the Mean”).
Check assumptions against the selected test
Assumptions are method-specific. In its process-comparison discussion, NIST describes tests that assume observations arise from one distribution, are approximately normal, and are not correlated over time (NIST, “What assumptions are typically made?”). Those conditions should not be treated as a universal checklist for every test.
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- Distributional shape: Use plots such as a histogram and normal probability plot when normality is relevant. Small departures may be tolerable when data remain roughly bell-shaped and tails are not heavy, but severe skew or outliers can change the result.
- Time dependence: A time-lag plot can reveal correlation. Treating serially correlated observations as independent can make uncertainty look smaller than it is.
- Pairing and independence: Repeated measurements on the same unit require a paired or repeated-measures approach; independent-group methods are not interchangeable.
- Variance conditions: Some comparisons require equal variances or a model that accounts for unequal variances. Check the chosen procedure rather than assuming every t test has the same requirement.
- Counts and bins: For chi-square goodness-of-fit, results depend on how observations are binned, and expected counts must support the approximation (NIST, “Chi-Square Goodness-of-Fit Test”).
Report a result without overstating it
A useful report lets readers reconstruct the decision and judge practical importance. Include:
- the research question, population, sampling or assignment design, and unit of analysis;
- H0, HA, tail direction, and α chosen before analysis;
- the test name, test statistic, degrees of freedom when applicable, and p-value;
- the estimated effect in meaningful units and a confidence interval;
- the assumptions checked, diagnostics used, and any deviations or limitations;
- a conclusion tied to the evidence and context, avoiding “proved,” “chance that H0 is true,” or claims of practical importance based on p alone.
NIST treats hypothesis tests and confidence intervals as complementary tools for comparisons, so provide both whenever the design and method make an interval appropriate (NIST, “Introduction”).
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Common interpretation errors
- “p = 0.03 means there is a 3% chance the null is true.” Incorrect: the p-value is calculated conditional on H0 and measures how extreme the data are under that assumption.
- “Not significant means no effect.” A wide interval may simply indicate limited precision. Report its bounds and the smallest effect that would matter.
- “A tiny p-value means an important effect.” Large samples can make very small differences statistically detectable. Examine the estimate, interval, and practical threshold.
- “The test is valid because the software produced a number.” Software cannot repair dependent observations, inappropriate bins, a mismatched tail, or a violated design assumption.
- “Run many tests and highlight the smallest p-value.” Multiple unplanned comparisons increase false-positive risk. Define the family of comparisons and an appropriate adjustment or analysis plan.
A compact decision checklist
- What parameter or distributional feature is being tested?
- Is there one sample, paired data, independent groups, or several groups?
- Is the alternative lower-tailed, upper-tailed, or two-sided for a substantive reason?
- Which assumptions does the specific method require, and what diagnostics address them?
- Are sample size, expected counts, and dependence structure adequate?
- What estimate and confidence interval accompany the test decision?
- Does the observed difference matter in the real setting, regardless of whether p crosses α?
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