To solve these questions, we'll use the following concepts from statistics regarding the sampling distribution of the sample mean:

1. Mean of the distribution of sample means ($\mu_{\bar{x}}$): This is the same as the population mean ($\mu$).
2. Standard deviation of the distribution of sample means ($\sigma_{\bar{x}}$): This is also known as the standard error of the mean and is calculated using the formula:

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma$ is the population standard deviation
• $n$ is the sample size

Given:

• Population mean ($\mu$) = 160.6
• Population standard deviation ($\sigma$) = 33.4
• Sample size ($n$) = 197

Let's calculate the required values.The results are:

• The mean of the distribution of sample means, $\mu_{\bar{x}}$, is 160.6.
• The standard deviation of the distribution of sample means, $\sigma_{\bar{x}}$, is 2.38.

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to explore further:

1. How does increasing the sample size affect the standard error of the mean?
2. What would be the standard error if the sample size was doubled?
3. What is the probability that a sample mean falls within one standard deviation of the population mean?
4. How would the standard deviation of the sample means change if the population standard deviation were different?
5. What is the relationship between sample size and the accuracy of the sample mean as an estimate of the population mean?
6. What assumptions are required for the Central Limit Theorem to apply?
7. How would the sampling distribution change if we sampled without replacement?
8. What is the difference between the population standard deviation and the standard deviation of the sample means?

Tip: When calculating the standard deviation of the distribution of sample means (standard error), remember it decreases as the sample size increases, providing a more accurate estimate of the population mean.