Let's break down the problem you provided:

Part (a): Describe the $\bar{x}$ distribution and compute the mean and standard deviation of the distribution.

• The Central Limit Theorem (CLT) states that when the sample size $n$ is 30 or larger, the sampling distribution of the sample mean $\bar{x}$ is approximately normal. In this case, the sample size $n = 49$, which is greater than 30, so the distribution of $\bar{x}$ will indeed be approximately normal.

• Mean of $\bar{x}$: The mean of the sampling distribution of $\bar{x}$, denoted $\mu_{\bar{x}}$, is equal to the population mean $\mu$. From the given problem: $\mu_{\bar{x}} = \mu = 85$

• Standard deviation of $\bar{x}$: The standard deviation of the sampling distribution of $\bar{x}$, denoted $\sigma_{\bar{x}}$, is equal to the population standard deviation $\sigma$ divided by the square root of the sample size $n$. So, $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ To compute this, we need the value of $\sigma$ (population standard deviation), which is not directly provided in the image. You’ll need to look up the value of $\sigma$ from any preceding parts of the question or other given data.

Would you like to provide more details, or should I continue with the next parts?

Here are 5 relative questions for further understanding:

1. What is the population standard deviation $\sigma$ in this problem?
2. How does the Central Limit Theorem affect the sampling distribution when the sample size is smaller than 30?
3. Why does the sample size being larger than 30 make the distribution approximately normal?
4. What are the consequences if the sample size is too small for applying the Central Limit Theorem?
5. How would you calculate the probability of obtaining a sample mean less than 73, as required in later parts of the problem?

Tip: Always ensure you have the population standard deviation or sample standard deviation when working on problems involving the CLT, as it is necessary for calculating the standard error.