Critical Region

Definition

The critical region, also known as the rejection region, is a range of values for a test statistic that leads to the rejection of the null hypothesis in a statistical hypothesis test. It represents the set of outcomes that are considered statistically significant and unlikely to have occurred by chance under the assumption that the null hypothesis is true.

5 Must Know Facts For Your Next Test

1. The critical region is determined by the significance level (α) chosen for the hypothesis test, which represents the maximum acceptable probability of rejecting the null hypothesis when it is true (Type I error).
2. The size of the critical region is inversely related to the significance level; a smaller significance level results in a larger critical region, making it more difficult to reject the null hypothesis.
3. The critical region is located in the tail(s) of the sampling distribution of the test statistic, depending on the alternative hypothesis (one-tailed or two-tailed test).
4. The critical value is the boundary value that separates the critical region from the non-rejection region, and it is used to determine whether the observed test statistic falls within the critical region.
5. Defining the critical region is a crucial step in the hypothesis testing process, as it determines the decision rule for rejecting or failing to reject the null hypothesis based on the observed data.

Review Questions

• Explain the relationship between the critical region and the null hypothesis in a hypothesis test.
  • The critical region is the set of values for the test statistic that lead to the rejection of the null hypothesis. If the observed test statistic falls within the critical region, the null hypothesis is rejected, indicating that the observed data is unlikely to have occurred by chance under the assumption that the null hypothesis is true. The size and location of the critical region are determined by the significance level and the alternative hypothesis, which together define the decision rule for the hypothesis test.
• Describe how the significance level (α) affects the size of the critical region and the probability of making a Type I error.
  • The significance level (α) represents the maximum acceptable probability of rejecting the null hypothesis when it is true (Type I error). A smaller significance level results in a larger critical region, making it more difficult to reject the null hypothesis. Conversely, a larger significance level leads to a smaller critical region, increasing the likelihood of rejecting the null hypothesis even when it is true. The choice of significance level involves a trade-off between the risk of making a Type I error and the power of the test to detect a significant effect if it exists.
• Analyze how the alternative hypothesis (one-tailed or two-tailed) affects the location and size of the critical region in a hypothesis test.
  • The alternative hypothesis, whether one-tailed or two-tailed, determines the location of the critical region within the sampling distribution of the test statistic. In a one-tailed test, the critical region is located in only one tail of the distribution, corresponding to the direction of the alternative hypothesis (either greater than or less than the null hypothesis value). In a two-tailed test, the critical region is split between the two tails of the distribution, reflecting the possibility of the observed statistic being significantly different from the null hypothesis value in either direction. The size of the critical region is also affected by the choice of a one-tailed or two-tailed test, with a two-tailed test generally requiring a larger critical region to maintain the same significance level.

Related terms

Null Hypothesis:

The null hypothesis is a statement that there is no significant difference or relationship between the variables being studied. It is the hypothesis that the researcher aims to test.

Alternative Hypothesis:

The alternative hypothesis is a statement that there is a significant difference or relationship between the variables being studied. It is the hypothesis that the researcher believes to be true if the null hypothesis is rejected.

Test Statistic:

The test statistic is a numerical value calculated from the sample data that is used to determine whether to reject or fail to reject the null hypothesis.