Acceptance region is a statistical concept that plays a crucial role in hypothesis testing. It refers to the range of values within which we can accept the null hypothesis as true. In other words, if the test statistic falls within this region, we consider the results to be consistent with the null hypothesis. On the other hand, if the test statistic falls outside this region, we reject the null hypothesis. The definition of the acceptance region may vary depending on the type of test we are conducting and the level of significance we choose. 

Methods To Determine Acceptance Region

Typically, the acceptance region is determined by two factors: the level of significance and the degrees of freedom. The level of significance, denoted by α, is the probability of rejecting the null hypothesis when it is actually true. The degrees of freedom, on the other hand, depend on the sample size and the number of parameters being estimated. 

Using Critical Values

One common method for determining the acceptance region is to use the critical values of the test statistic. These values are calculated from a probability distribution, such as the t-distribution or the F-distribution, and depend on the level of significance and the degrees of freedom.

Using P-Values

Another method is to use p-values, which give the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. It is important to note that the acceptance region is not necessarily a fixed interval. Rather, it is a region that will vary depending on the sample size, the number of parameters being estimated, and the level of significance. Additionally, the acceptance region is only one part of the hypothesis testing process. In order to draw meaningful conclusions, we must also consider the power of the test, the effect size, and other factors that may impact the validity of our results. 

Example

A term associated with statistical significance tests, that gives the set of values of a test statistic for which the null hypothesis is not rejected. Suppose, for example, a 𝑧 test is being used to test the null hypothesis that the mean blood pressure of men and women is equal against the alternative hypothesis that the two means are not equal. If the chosen significance level of the test is 0.05 then the acceptance region consists of values of the test statistic 𝑧 between 1.96 and 1.96. That is, the concept of acceptable region is majorly utilised in hypothesis testing under statistical analyses. When the hypotheses are tested, the test procedures segregate the possible sample outcomes into two subsets, whether the observed value of the test statistic is smaller or not than the pre-determined threshold value. The subset that favours the null hypothesis is recognised as the acceptable region, while the other subset is known as the rejection or critical region. That implies that if the outcome of the sample corresponds with the acceptable region, then the null hypothesis is accepted, or else rejected. Thus, the acceptance region will always correspond to the null hypothesis, denoted by H0. It is further noted that the acceptance region relates to the probability of 1 –  , such that  is recognised as the significance level of the test procedure. One can also define the acceptance region as the condition when the probability of making a Type 1 error equates to the level of significance for the test procedure. The acceptance region is always a complement to the rejection region, implying that if the sample outcome does not fall under the rejection region, it will definitely fall under the acceptance region, and vice-versa. The acceptance region is defined for the null hypothesis, and also depends on whether the hypothesis tested is one-tailed or two-tailed.