In statistics, a hypothesis is an assertion about a population. A null hypothesis, usually denoted by H0, is a particular assertion that is to be either rejected or not rejected. The alternative hypothesis H1 specifies some alternative to H0. To decide whether H0 is to be rejected or not rejected, a significance test tests whether a sample taken from the population could have occurred by chance, given that H0 is true.
From the sample, the test statistic is calculated. The test partitions the set of possible values into the critical (or rejection) region and its complement. It is not uncommon for this complement to be referred to as the ‘acceptance region’, though this language is imprecise. These depend upon the choice of the significance level α, which is the probability that the test statistic lies in the critical region if H0 is true. Often α is chosen to be 5%. If the test statistic falls in the critical region, the null hypothesis H0 is rejected; otherwise, the conclusion is that there is no evidence for rejecting H0 and H0 is not rejected.
A Type I error, or false positive, occurs if H0 is rejected when it is in fact true. The probability of a Type I error is α, so this depends on the choice of significance level. A Type II error, or false negative, occurs if H0 is not rejected when it is in fact false. If the probability of a Type II error is β, the power of the test is defined as 1−β. This depends upon the choice of alternative hypothesis. The null hypothesis H0 normally specifies that a certain parameter has a certain value. If the alternative hypothesis H1 says that the parameter is not equal to this value, then the test is said to be two‐tailed (or two‐sided). If H1 says that the parameter is greater than this value (or less than this value), then the test is one‐tailed (or one‐sided). See also p‐value.