In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.


Left-tailed test, α=0.10, n=38

Verified step by step guidance

1. Step 1: Understand the problem. This is a left-tailed t-test, meaning the rejection region is located in the left tail of the t-distribution. The level of significance (α) is given as 0.10, and the sample size (n) is 38.
2. Step 2: Determine the degrees of freedom (df). For a t-test, degrees of freedom are calculated as df = n - 1. Use the formula: $\mathrm{df}=n-1$.
3. Step 3: Look up the critical value in the t-distribution table. Use the degrees of freedom (df = 38 - 1 = 37) and the level of significance (α = 0.10) for a left-tailed test. The critical value corresponds to the t-score where the cumulative probability equals α.
4. Step 4: Define the rejection region. For a left-tailed test, the rejection region is all t-values less than the critical value. Express this as: $t<\mathrm{critical} \mathrm{value}$.
5. Step 5: Verify the setup. Ensure the degrees of freedom and α are correctly applied, and confirm the critical value and rejection region are consistent with the left-tailed test.

Key Concepts

Critical Value

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. It is determined based on the significance level (alpha) and the type of test being conducted. For a left-tailed test, the critical value corresponds to the point where the cumulative probability equals alpha, indicating the threshold for rejecting the null hypothesis.

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. In a left-tailed test, this region is located to the left of the critical value. 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 determine if there is a significant difference between the means of two groups, or between a sample mean and a known value. It is particularly useful when the sample size is small (typically n < 30) and the population standard deviation is unknown. The type of t-test (one-sample, independent, or paired) depends on the data structure and research question.