This question is related to the Central Limit Theorem (CLT). Let's break it down:

Given:

• Population mean ($\mu$) = 129.7
• Population standard deviation ($\sigma$) = 91.3
• Sample size ($n$) = 147

Questions:

1. What is the mean of the distribution of sample means?

   • The mean of the sample means ($\mu_{\bar{x}}$) is the same as the population mean ($\mu$).
   • So, $\mu_{\bar{x}} = 129.7$.

2. What is the standard deviation of the distribution of sample means (also known as the standard error)?

   • The standard deviation of the sample means (standard error, $\sigma_{\bar{x}}$) is calculated by: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ Substituting the values: $\sigma_{\bar{x}} = \frac{91.3}{\sqrt{147}} = \frac{91.3}{12.124} \approx 7.53$ So, the standard deviation of the sample means is approximately 7.53.

Final Answers:

• $\mu_{\bar{x}} = 129.7$
• $\sigma_{\bar{x}} \approx 7.53$

Would you like further clarification or details on these calculations?

Here are 5 follow-up questions to consider:

1. How does increasing the sample size affect the standard deviation of sample means?
2. What is the importance of the Central Limit Theorem in statistics?
3. Can the Central Limit Theorem be applied to non-normal populations? Why?
4. How would the standard error change if the sample size were halved?
5. What is the significance of a smaller standard error in terms of sample accuracy?

Tip: The Central Limit Theorem states that the distribution of sample means approaches normality as the sample size increases, regardless of the population's distribution.