Part 2 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 $n_{1}$ and $n_{2}$. Assume that the s
\[
H_{a}: \mu_{1}< \mu_{2}, \alpha=0.025, n_{1}=13, n_{2}=11
\]
(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.)
A
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Step 1 ：Given that the alternative hypothesis is \(H_{a}: \mu_{1}<\mu_{2}\), the level of significance is \(\alpha=0.025\), and the sample sizes are \(n_{1}=13\) and \(n_{2}=11\).

Step 2 ：For part (a), the critical value is given as -2.074 when the population variances are assumed to be equal.

Step 3 ：For part (b), we need to find the critical value assuming that the population variances are not equal. This requires a different calculation because the degrees of freedom change when the variances are not assumed to be equal. The degrees of freedom for an independent samples t-test with unequal variances is calculated using the Welch-Satterthwaite equation.

Step 4 ：Given that \(n_{1} = 13\), \(n_{2} = 11\), \(\alpha = 0.025\), and the degrees of freedom (df) is 22, the critical value is approximately 2.074. However, since the alternative hypothesis is \(\mu_{1}<\mu_{2}\), we are interested in the left tail of the distribution, so the critical value will be -2.074.

Step 5 ：Final Answer: The critical value assuming that the population variances are not equal is \(\boxed{-2.074}\).