Acceptance region, also known as critical region, is a specific area in a statistical hypothesis test where the null hypothesis is rejected. It is a range of values of the test statistic that leads to the rejection of the null hypothesis. In other words, if the calculated test statistic falls within the acceptance region, the null hypothesis is rejected and the alternative hypothesis is accepted.

For example, consider a study to determine if a new drug is effective in reducing blood pressure. The null hypothesis states that the drug has no effect on blood pressure, while the alternative hypothesis states that the drug does have an effect on blood pressure. In this case, the acceptance region would be a range of values of the test statistic that indicates that the drug is effective in reducing blood pressure.

In order to determine the acceptance region, the researcher must first select a significance level, also known as the alpha level. This is the probability of rejecting the null hypothesis when it is true. A common significance level is 0.05, which means that there is a 5% chance of rejecting the null hypothesis when it is true.

Once the significance level is selected, the researcher must then determine the critical value of the test statistic. This is the value that separates the acceptance region from the rejection region. For example, if the researcher is using a two-tailed test with a significance level of 0.05, the critical value would be the value that is 1.96 standard deviations away from the mean in each direction. Any test statistic that falls outside of this range would be considered significant and the null hypothesis would be rejected.

In the case of the drug study mentioned above, the acceptance region would be the range of values of the test statistic that indicates that the drug is effective in reducing blood pressure. If the calculated test statistic falls within this range, the null hypothesis would be rejected and the alternative hypothesis would be accepted.

Another example of acceptance region can be seen in a study to determine if there is a difference in the average height between men and women. The null hypothesis states that there is no difference in the average height between men and women, while the alternative hypothesis states that there is a difference in the average height between men and women. In this case, the acceptance region would be the range of values of the test statistic that indicates a significant difference in the average height between men and women.

In order to determine the acceptance region, the researcher must first select a significance level, such as 0.05. The researcher must then determine the critical value of the test statistic. For example, if the researcher is using a two-tailed test with a significance level of 0.05, the critical value would be the value that is 1.96 standard deviations away from the mean in each direction. Any test statistic that falls outside of this range would be considered significant and the null hypothesis would be rejected.

In the case of the height study mentioned above, the acceptance region would be the range of values of the test statistic that indicates a significant difference in the average height between men and women. If the calculated test statistic falls within this range, the null hypothesis would be rejected and the alternative hypothesis would be accepted.

In conclusion, acceptance region is a specific area in a statistical hypothesis test where the null hypothesis is rejected. It is a range of values of the test statistic that leads to the rejection of the null hypothesis. The acceptance region is determined by selecting a significance level and determining the critical value of the test statistic. If the calculated test statistic falls within the acceptance region, the null hypothesis is rejected and the alternative hypothesis is accepted.