In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.


Two-tailed test, α=0.02, n=12

Verified step by step guidance

1. Determine the degrees of freedom (df) for the t-test. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, n = 12, so df = 12 - 1 = 11.
2. Identify the level of significance (α) for the two-tailed test. Since α = 0.02, divide it equally between the two tails. This means each tail will have an area of α/2 = 0.02/2 = 0.01.
3. Use a t-distribution table or statistical software to find the critical t-value(s) corresponding to df = 11 and a cumulative probability of 1 - α/2 = 1 - 0.01 = 0.99 for the upper tail. For the lower tail, the critical t-value will be the negative of the upper tail value.
4. Define the rejection region(s) based on the critical t-values. For a two-tailed test, the rejection regions are t < -t_critical and t > t_critical, where t_critical is the positive critical value obtained in the previous step.
5. Summarize the results: The critical t-values and rejection regions are determined, and these will guide the decision to reject or fail to reject the null hypothesis based on the test statistic.

Key Concepts

Critical Value

A critical value is a point on the scale of the test statistic that separates the region where the null hypothesis is rejected from the region where it is not rejected. In hypothesis testing, critical values are determined based on the significance level (α) and the type of test being conducted, such as a t-test. For a two-tailed test, critical values are found at both tails of the distribution.

Rejection Region

The rejection region is the set of all values of the test statistic that would lead to the rejection of the null hypothesis. In a two-tailed test, this region is divided into two parts, corresponding to the critical values on either side of the distribution. The size of the rejection region is determined by the significance level (α), which indicates the probability of making a Type I error.

Two-Tailed Test

A two-tailed test is a statistical test that evaluates whether a sample mean is significantly different from a population mean in either direction (higher or lower). This type of test is used when the alternative hypothesis does not specify a direction of the effect. In the context of the given question, a two-tailed test with α=0.02 means that the rejection regions will be located in both tails of the t-distribution, each containing 1% of the total area.