23 Module 7: Two-Sample Statistical Inference - Question 12, 8.2 .5 Part 2 of 2 HW Score: $51.05 \%, 12.25$ of Points: 0 of 1 \[ H_{a}: \mu_{1}<\mu_{2}, \alpha=0.025, n_{1}=13, n_{2}=11 \] (a) Find the critical value(s) assuming that the population variances are equal. $-2.074$ (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.) (b) Find the critical value(s) assuming that the population variances are not equal. (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.)
Solution
Step 1 ：The problem is asking for the critical value assuming that the population variances are equal. This is a two-sample t-test problem. The critical value is the t-score that corresponds to the given significance level (alpha) in a t-distribution. The degrees of freedom for this t-distribution is \(n_{1} + n_{2} - 2\), where \(n_{1}\) and \(n_{2}\) are the sample sizes.
Step 2 ：Given that \(\alpha = 0.025\), \(n_{1} = 13\), and \(n_{2} = 11\), we can calculate the degrees of freedom as \(df = n_{1} + n_{2} - 2 = 22\).
Step 3 ：The critical value is the t-score that corresponds to the 2.5% significance level in the left tail of a t-distribution with 22 degrees of freedom. The critical value is approximately -2.074.
Step 4 ：Final Answer: The critical value is approximately \(\boxed{-2.074}\).
