Which one of these is an assumption we need to use Z in a test of means? a. At least one population standard deviation is known b. Both population standard deviation's are known c. Both sample standard deviation's are known What is the critical value for a one-tailed hypothesis test in which a null hypothesis is tested at the 5% level of significance based on two samples, both sample sizes are 13? Assume the population standard deviations are equal a. 1.711 b. 1.708 c. 2.060 d. 2.064

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!

Explore the assumptions needed for using Z-tests in hypothesis testing, including the significance of population standard deviations. Learn how to determine the critical value for one-tailed tests with sample sizes of 13 at a 5% significance level. This content is suitable for high school students in grades 11-12.

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