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.


Right-tailed test, α=0.02, n=63

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, df = 63 - 1.
2. Identify the level of significance (α) for the test. Here, α = 0.02, which corresponds to the probability of rejecting the null hypothesis when it is true.
3. Since this is a right-tailed test, the critical value corresponds to the t-score where the area to the right under the t-distribution curve equals α. Use a t-distribution table or statistical software to find the t-score for df = 62 and α = 0.02.
4. Define the rejection region. For a right-tailed test, the rejection region consists of all t-scores greater than the critical value found in the previous step.
5. Summarize the results: The critical value and rejection region are determined based on the t-distribution table or software output. The rejection region is t > critical value.

Key Concepts

Critical Value

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (α) and the distribution of the test statistic. For a right-tailed t-test, the critical value is the point beyond which the null hypothesis is rejected, indicating that the observed data is statistically significant.

Rejection Region

The rejection region is the set of values for the test statistic that leads to the rejection of the null hypothesis. In a right-tailed t-test, this region is located to the right of the critical value on the t-distribution curve. If the calculated test statistic falls within this region, it suggests that the sample provides sufficient evidence to reject the null hypothesis at the specified significance level.

T-Test

A t-test is a statistical test used to compare the means of two groups or to compare a sample mean to a known value when the population standard deviation is unknown. It is particularly useful for small sample sizes (typically n < 30) and is based on the t-distribution. The type of t-test (one-sample, independent two-sample, or paired sample) depends on the data structure and research question.