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 \(\boxed{-2.074}\).