Rejection region

Definition

The rejection region is the set of values in hypothesis testing that leads to the rejection of the null hypothesis. It is determined based on the significance level, which defines the probability of making a Type I error, and usually falls in the tails of the sampling distribution. The size and placement of the rejection region depend on whether the test is one-tailed or two-tailed, influencing how we interpret sample data in relation to population parameters.

5 Must Know Facts For Your Next Test

1. The rejection region is determined by the significance level (commonly set at 0.05 or 0.01), which reflects how much risk of a Type I error is acceptable.
2. In a one-tailed test, the rejection region is located entirely in one tail of the distribution, while in a two-tailed test, it is split between both tails.
3. The critical value(s) defining the boundaries of the rejection region are calculated based on the sampling distribution and corresponding significance level.
4. If a test statistic falls within the rejection region, it provides strong evidence against the null hypothesis, leading researchers to favor the alternative hypothesis.
5. Understanding where the rejection region lies helps in making informed decisions about hypotheses based on sample data and its relation to population parameters.

Review Questions

• How does the significance level affect the placement and size of the rejection region in hypothesis testing?
  • The significance level directly impacts both the size and placement of the rejection region. A lower significance level, like 0.01, creates a smaller rejection region compared to a higher level, such as 0.05, making it harder to reject the null hypothesis. This means fewer values will lead to rejection under a stricter significance level, increasing confidence that any observed effects are genuine.
• In what scenarios would you use a one-tailed test versus a two-tailed test when establishing your rejection region?
  • A one-tailed test is used when you have a specific direction for your alternative hypothesis (e.g., testing if a mean is greater than a certain value), thus placing all of the rejection region in one tail. In contrast, a two-tailed test is appropriate when you are testing for any difference without a specific direction (e.g., testing if a mean differs from a certain value), dividing the rejection region across both tails of the distribution.
• Evaluate how changes in sample size impact the determination of the rejection region and hypothesis testing outcomes.
  • Increasing sample size typically leads to more accurate estimates of population parameters and narrows the sampling distribution's standard error. This can make it easier to fall into the rejection region when testing hypotheses because smaller sample sizes may not provide enough evidence against the null hypothesis due to wider confidence intervals. As sample size increases, even small differences may become statistically significant, which can shift conclusions about rejecting or failing to reject the null hypothesis.

Related terms

Null Hypothesis:

A statement that assumes no effect or no difference in the population, serving as the basis for testing.

Significance Level:

The probability threshold (commonly denoted as alpha, \(\alpha\)) set before conducting a test, used to determine the rejection region.

Type I Error:

The error made when rejecting a true null hypothesis, often denoted by the significance level.