: Two-Sample Statistical Inference - Question 11, 8.2.3 Part 1 of 2 Use the t-distribution table to find the critical value(s) for the indicated alternative hypotheses, level of significance $\alpha$, and sample sizes $\mathrm{n}$ \[ H_{a}: \mu_{1} \neq \mu_{2}, \alpha=0.01, n_{1}=12, n_{2}=10 \] (a) Find the critical value(s) assuming that the population variances are equal. $-2.845$ (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.)

Step 1 ：The problem is asking for the critical value(s) for a two-tailed t-test with a significance level of 0.01 and sample sizes of 12 and 10, assuming equal variances.

Step 2 ：The degrees of freedom for a t-test with equal variances is calculated as the sum of the two sample sizes minus 2. In this case, the degrees of freedom would be \(12 + 10 - 2 = 20\).

Step 3 ：We can then look up the critical value for a two-tailed t-test with 20 degrees of freedom and a significance level of 0.01 in a t-distribution table.

Step 4 ：The critical value for a two-tailed t-test with 20 degrees of freedom and a significance level of 0.01 is approximately 2.845.

Step 5 ：However, since this is a two-tailed test, the critical values are both positive and negative, so the critical values are -2.845 and 2.845.

Step 6 ：Final Answer: The critical values are \(\boxed{-2.845}\) and \(\boxed{2.845}\).