To solve this problem, we need to determine which graph best represents the sampling distribution of the sample means. We are given the following information:

• The population has a mean ($\mu = 18.5$) and a standard deviation ($\sigma = 11.2$).
• A sample size of 64 is drawn from the population.
• The graphs labeled (a), (b), and (c) represent different sampling distributions.

Key concepts:

1. Mean of the sampling distribution ($\mu_{\bar{x}}$): The mean of the sampling distribution ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$).

   $\mu_{\bar{x}} = \mu = 18.5$

2. Standard deviation of the sampling distribution (standard error) ($\sigma_{\bar{x}}$): The standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size. This is known as the standard error.

   $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{11.2}{\sqrt{64}} = \frac{11.2}{8} = 1.4$

Now, let's analyze the graphs:

• Graph (a): Shows a mean of $\mu_{\bar{x}} = 11.2$ and a standard deviation greater than the population standard deviation, so this does not match.
• Graph (b): Shows a mean of $\mu_{\bar{x}} = 11.2$, and it is not consistent with the population mean either.
• Graph (c): Shows a mean of $\mu_{\bar{x}} = 18.5$ (matching the population mean) and a standard deviation $\sigma_{\bar{x}} = 1.4$, which is the correct standard error based on our calculation.

Conclusion:

• Graph (c) most closely resembles the sampling distribution of the sample means.
• The correct values are: $\mu_{\bar{x}} = 18.5, \quad \sigma_{\bar{x}} = 1.4$

Would you like further explanation on any part of this solution? Here are five related questions to extend your understanding:

1. What is the Central Limit Theorem and how does it apply to this problem?
2. How does increasing the sample size affect the standard error?
3. What would happen to the sampling distribution if the sample size were decreased to 25?
4. How is the shape of the sampling distribution related to the population distribution?
5. Why is it important to distinguish between population standard deviation and standard error?

Tip: The larger the sample size, the smaller the standard error, leading to a narrower sampling distribution.