Null Hypothesis

The null hypothesis, denoted $H_0$, is the hypothesis of no effect, no difference, or no association that a statistical test is constructed to evaluate evidence against. It is typically formulated as a precise, testable statement (e.g. $H_0:\mu ={\mu }_0$ or $H_0:\rho =0$) that allows the sampling distribution of a test statistic to be derived under its assumption. A hypothesis test never directly proves $H_0$ true, failing to reject it means the data didn't provide sufficient evidence against it, not that it has been confirmed, this asymmetry between the null and alternative hypotheses is foundational to the entire Neyman-Pearson hypothesis-testing framework.

• The null hypothesis is chosen to be precise enough that a sampling distribution for the test statistic can be derived under it, this is why H₀ is almost always a single specific value or a simple boundary (e.g. μ = 0) rather than a vague statement, even when the substantive scientific interest lies in the alternative.
• 'Failing to reject the null hypothesis' and 'accepting the null hypothesis' are not the same statement, and conflating them is one of the most common and consequential misinterpretations of hypothesis testing, a non-significant result may simply reflect insufficient statistical power to detect a real effect, not evidence that no effect exists.
• The convention of placing the 'no effect' claim in the null (rather than the hypothesis of interest) reflects a deliberate asymmetry built into the Neyman-Pearson framework: the burden of proof falls on the alternative hypothesis, and Type I error, incorrectly rejecting a true null, is controlled directly by the chosen significance level.