A null hypothesis, conventionally denoted (H_0), is a claim about a population or data-generating process evaluated through statistical hypothesis testing. In statistics, it supplies a reference against which observations are assessed. It commonly specifies no difference, no association, or a particular parameter value, but it need not describe an absence of effects. A test determines whether the data are sufficiently incompatible with the null hypothesis, under specified assumptions, to reject it. (itl.nist.gov)
Formulation and scope
The null hypothesis is paired with an alternative hypothesis, usually written (H_1) or (H_a). For a population mean (\mu), a two-sided comparison may use
[ H_0:\mu=\mu_0,\qquad H_1:\mu\ne\mu_0. ]
A directional comparison can instead use (H_0:\mu\le\mu_0) against (H_1:\mu>\mu_0). Thus, the null is not inherently an equality: its form depends on the question and the alternatives the test is intended to detect. Hypotheses concern population characteristics, rather than whether the observed sample mean happens to equal a benchmark. (itl.nist.gov)
More generally, within a statistical model ({P_\theta:\theta\in\Theta}), the null asserts that (\theta) belongs to a specified subset (\Theta_0). A simple hypothesis completely specifies the data’s probability distribution; a composite hypothesis leaves multiple possibilities. For example, a normal distribution with mean zero and variance one is fully specified, whereas normality with unknown mean and variance is composite. Even a fixed-mean hypothesis is composite when other distributional parameters remain unspecified. (stat210a.berkeley.edu)
Testing procedure
A test uses a test statistic to measure a relevant departure from the null. Its sampling distribution under (H_0), together with a chosen significance level (\alpha), defines a rejection region. The probability of entering that region when the null is true is controlled at or below (\alpha). For a composite null, this requirement applies to every distribution covered by the hypothesis:
[ \sup_{\theta\in\Theta_0} P_\theta(\text{reject }H_0)\le\alpha. ]
The supremum is the test’s size; it may be smaller than its nominal level. (itl.nist.gov)
A p-value expresses how extreme the observed statistic is relative to its null distribution. For a statistic whose larger values indicate greater incompatibility, a simple-null example is
[ p=P_{H_0}(T\ge T_{\mathrm{obs}}). ]
Two-sided tests include departures in both directions according to the test’s definition of extremeness. Comparing a valid p-value with a prespecified significance level provides a rejection rule. The p-value is not the probability that (H_0) is true, nor the probability that the results arose through chance alone. (itl.nist.gov)
Example: testing a population mean
Suppose independent observations come from a normal population with unknown standard deviation. To test (H_0:\mu=\mu_0), the one-sample t statistic is
[ T=\frac{\bar X-\mu_0}{S/\sqrt n}, ]
where (\bar X) is the sample mean, (S) is the sample standard deviation, and (n) is the sample size. The denominator estimates the standard error of the mean. Under the null and these assumptions, (T) follows Student’s t-distribution with (n-1) degrees of freedom. A two-sided test rejects for sufficiently large positive or negative values. (itl.nist.gov)
For an illustrative sample with (n=25), (\bar X=102), (S=5), and (\mu_0=100), the statistic is (T=2). At the 5% two-sided level, the critical magnitude for 24 degrees of freedom is approximately 2.064, so the null is not rejected. This outcome concerns the test’s evidence threshold; it does not establish that the population mean is exactly 100. (itl.nist.gov)
Errors, power, and interpretation
The distinction between Type I and Type II errors explains the asymmetry of testing. A Type I error rejects a true null. A Type II error fails to reject a false null. Statistical power is the probability of rejection under a specified alternative, equal to one minus the corresponding Type II error probability. Power depends on the alternative, sample size, variability, and testing procedure; it is not a single property of the null hypothesis alone. (stat.berkeley.edu)
Consequently, failure to reject is not proof of the null hypothesis. Data may be compatible with the null while also being compatible with alternatives that the study has little power to distinguish. Conversely, rejection does not deductively prove an alternative: false rejection remains possible, and incorrect modeling assumptions can undermine the calculation. The conclusion is conditional on both the hypothesis and the assumptions used to obtain the reference distribution. (stat.berkeley.edu)
Estimation and multiple testing
For matching procedures, a two-sided test at level (\alpha) rejects a parameter value precisely when it falls outside the corresponding (100(1-\alpha)%) confidence interval. This correspondence connects hypothesis testing with estimation: the interval displays a range of parameter values compatible with the data under that procedure, rather than only a decision about one value. (itl.nist.gov)
Statistical significance does not measure effect size or practical importance. A small p-value can accompany a small effect, while a large p-value does not establish the absence of an important effect. The American Statistical Association’s 2016 statement emphasizes that scientific conclusions cannot be reduced to whether a p-value crosses a threshold. (doi.org)
When many null hypotheses are examined, multiple testing introduces additional opportunities for false rejection. Procedures such as the Bonferroni correction address error control across a family of comparisons. For (m) tests, testing each at level (\alpha/m) bounds the probability of at least one Type I error by (\alpha), without requiring independence among the tests. (itl.nist.gov)