Critical Region

Definition

The critical region, also known as the rejection region, is a specific range of values for a test statistic that leads to the rejection of the null hypothesis in a statistical hypothesis test. It represents the area of the sampling distribution where the test statistic is considered extreme enough to provide evidence against the null hypothesis.

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 making a Type I error (rejecting the null hypothesis when it is true).
2. The size of the critical region is inversely related to the significance level, meaning a smaller significance level results in a larger critical region.
3. The critical region is located in the tail(s) of the sampling distribution, depending on the alternative hypothesis (one-tailed or two-tailed test).
4. If the test statistic falls within the critical region, the null hypothesis is rejected, and the alternative hypothesis is supported by the evidence.
5. The critical region is an essential component in making a decision about the null hypothesis and drawing conclusions from the hypothesis test.

Review Questions

• Explain the relationship between the significance level and the size of the critical region in a hypothesis test.
  • The significance level (α) and the size of the critical region are inversely related. A smaller significance level, which represents a lower acceptable probability of making a Type I error, results in a larger critical region. Conversely, a larger significance level leads to a smaller critical region. This relationship is important because it allows researchers to balance the risk of making a Type I error (rejecting the null hypothesis when it is true) with the power of the test to detect a significant effect if one exists.
• Describe how the location of the critical region is determined based on the alternative hypothesis in a hypothesis test.
  • The location of the critical region is determined by the alternative hypothesis being tested. In a one-tailed test, the critical region is located in either the upper or lower tail of the sampling distribution, depending on the direction of the alternative hypothesis. For example, in a one-tailed test with the alternative hypothesis $H_a: \theta > \theta_0$, the critical region would be in the upper tail of the sampling distribution. In a two-tailed test, the critical region is split between the upper and lower tails of the sampling distribution, representing the area where the test statistic is considered extreme enough to provide evidence against the null hypothesis.
• Explain the role of the critical region in the decision-making process of a hypothesis test and the implications of the test statistic falling within or outside the critical region.
  • The critical region is a crucial component in the decision-making process of a hypothesis test. If the test statistic falls within the critical region, the null hypothesis is rejected, and the alternative hypothesis is supported by the evidence. This means that the observed data provides sufficient evidence to conclude that the parameter of interest is significantly different from the value specified in the null hypothesis. Conversely, if the test statistic falls outside the critical region, the null hypothesis is not rejected, indicating that the data does not provide enough evidence to conclude that the parameter of interest is significantly different from the value specified in the null hypothesis. The decision made based on the location of the test statistic relative to the critical region has important implications for the conclusions drawn from the hypothesis test.

Related terms

Null Hypothesis: The null hypothesis is a statement that there is no significant difference or relationship between the variables being tested.

Alternative Hypothesis:

The alternative hypothesis is a statement that there is a significant difference or relationship between the variables being tested.

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.