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=0.05, n=20

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 = 20, so df = 20 - 1.
2. Identify the level of significance (α) for the test. For a two-tailed test, the significance level is split equally between the two tails of the t-distribution. Here, α = 0.05, so each tail will have α/2 = 0.025.
3. Use a t-distribution table or statistical software to find the critical t-value(s) corresponding to df = 19 and a cumulative probability of 1 - α/2 = 0.975 for the upper tail and α/2 = 0.025 for the lower tail.
4. The rejection region(s) for the two-tailed test are defined as the areas in the tails of the t-distribution where the test statistic falls beyond the critical t-values. Specifically, the rejection regions are t < -t_critical and t > t_critical.
5. Summarize the critical values and rejection regions. For example, if the critical t-values are ±t_critical, the rejection regions are t < -t_critical and t > t_critical. These regions indicate where the null hypothesis would be rejected.

Key Concepts

Critical Value

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. In a two-tailed test, there are two critical values that correspond to the significance level (α). For example, with α = 0.05, the critical values define the boundaries of the rejection regions in both tails of the distribution.

Rejection Region

The rejection region is the area in the tails of the probability distribution where we reject the null hypothesis. For a two-tailed test with α = 0.05, the rejection regions are located in both tails, each containing 2.5% of the total probability. If the test statistic falls within these regions, we conclude that the observed data is statistically significant.

t-Test

A t-test is a statistical test used to determine if there is a significant difference between the means of two groups, especially when the sample size is small (n < 30) and the population standard deviation is unknown. The t-test uses the t-distribution, which accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.