To answer your questions:

1. Assumption for using Z in a test of means: The correct option is a. At least one population standard deviation is known. This is because a Z-test can be used when the population standard deviation is known or when the sample size is large enough (typically n > 30) to invoke the Central Limit Theorem.

2. Critical value for a one-tailed hypothesis test: Given a significance level of 5% and sample sizes of 13 for both samples, you can use the t-distribution since the sample size is small (n < 30). The degrees of freedom for two samples is calculated as $n_1 + n_2 - 2 = 13 + 13 - 2 = 24$. For a one-tailed test at the 5% significance level with 24 degrees of freedom, you look up the t-value:

   • The critical t-value for $\alpha = 0.05$ and $df = 24$ is approximately 1.711.

Thus, the answer is a. 1.711.

If you have any questions or would like further details, please let me know! Here are some related questions you might consider:

1. What is the difference between a Z-test and a t-test?
2. How do you determine the degrees of freedom for different statistical tests?
3. What are the implications of using a Z-test versus a t-test?
4. How do you calculate the sample size required for a specific level of confidence?
5. What are the consequences of violating the assumptions of a Z-test or t-test?

Tip: Always ensure the assumptions for your statistical tests are met to obtain valid results!