The rejection region, also known as the critical region, is a set of values for the test statistic that leads to the rejection of the null hypothesis. Its calculation depends on the significance level (alpha), the alternative hypothesis (one-tailed or two-tailed), and the distribution of the test statistic under the null hypothesis. For example, in a right-tailed t-test with a significance level of 0.05 and 20 degrees of freedom, the rejection region would consist of all t-values greater than the critical t-value, which can be found in a t-distribution table (approximately 1.725). Consequently, if the calculated test statistic exceeds this value, the null hypothesis is rejected.
Establishing the rejection region is fundamental in hypothesis testing because it dictates the criteria for deciding whether the evidence from a sample is strong enough to refute the null hypothesis. This process ensures decisions are made with a pre-defined level of confidence, controlling the probability of a Type I error (incorrectly rejecting a true null hypothesis). Historically, this concept emerged from the work of statisticians like Jerzy Neyman and Egon Pearson in the early 20th century, providing a rigorous framework for statistical inference.
