Rejection Region

Definition

The rejection region, also known as the critical region, is a key concept in hypothesis testing. It refers to the set of values for the test statistic that lead to the rejection of the null hypothesis, indicating that the observed data is unlikely to have occurred by chance if the null hypothesis is true.

5 Must Know Facts For Your Next Test

1. The rejection region is determined by the significance level (α), which represents the maximum probability of rejecting the null hypothesis when it is true (Type I error).
2. The size of the rejection region is inversely related to the significance level, as a smaller significance level results in a larger rejection region.
3. The location of the rejection region depends on the alternative hypothesis, which can be one-tailed (left-tailed, right-tailed) or two-tailed.
4. In a one-tailed test, the rejection region is located on one side of the distribution, while in a two-tailed test, the rejection region is split between the two tails of the distribution.
5. The rejection region is an essential component in determining whether to reject or fail to reject the null hypothesis based on the observed test statistic.

Review Questions

• Explain the purpose of the rejection region in the context of hypothesis testing.
  • The rejection region in hypothesis testing serves to define the set of values for the test statistic that would lead to the rejection of the null hypothesis. It represents the range of values that are considered statistically significant, meaning the observed data is unlikely to have occurred by chance if the null hypothesis is true. The size and location of the rejection region are determined by the significance level and the alternative hypothesis, respectively, and are crucial in deciding whether to reject or fail to reject the null hypothesis based on the calculated test statistic.
• Describe the relationship between the significance level and the size of the rejection region.
  • The significance level (α) and the size of the rejection region are inversely related. A smaller significance level, which represents a lower probability of rejecting the null hypothesis when it is true (Type I error), results in a larger rejection region. Conversely, a larger significance level leads to a smaller rejection region. This relationship is important because it allows researchers to balance the trade-off between the risk of making a Type I error and the power of the statistical test to detect a significant effect if it exists.
• Analyze the impact of the alternative hypothesis on the location of the rejection region.
  • The location of the rejection region is directly influenced by the alternative hypothesis. In a one-tailed test, where the alternative hypothesis specifies the direction of the effect (e.g., the population mean is greater than the hypothesized value), the rejection region is located on one side of the distribution. In a two-tailed test, where the alternative hypothesis does not specify the direction of the effect (e.g., the population mean is different from the hypothesized value), the rejection region is split between the two tails of the distribution. Understanding the relationship between the alternative hypothesis and the location of the rejection region is crucial in correctly interpreting the results of a hypothesis test and making appropriate inferences.

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.