To calculate the test statistic for the given hypothesis test, we can use the formula for the z-test statistic when the population standard deviation is known:

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

Where:

• $\bar{x}$ is the sample mean,
• $\mu_0$ is the population mean under the null hypothesis,
• $\sigma$ is the population standard deviation,
• $n$ is the sample size.

Given:

• $\bar{x} = 80$
• $\mu_0 = 75$
• $\sigma = 30$
• $n = 100$

Plugging these values into the formula:

$z = \frac{80 - 75}{30 / \sqrt{100}} = \frac{5}{30 / 10} = \frac{5}{3} \approx 1.67$

Thus, the value of the test statistic is approximately 1.67.

If you need further details or have any questions, let me know! Here are some related questions for your consideration:

1. What is the significance level typically used in hypothesis testing?
2. How do you interpret the z-test statistic in the context of this hypothesis test?
3. What would be the critical value for a one-tailed test at a 0.05 significance level?
4. How would you determine whether to reject or fail to reject the null hypothesis?
5. What are the potential consequences of a Type I and Type II error in this context?

Tip: Always check the assumptions of the test (e.g., normality, independence) before applying the hypothesis test.