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We are given a normally distributed population with a mean (\(\mu\)) of 183.4 and a standard deviation (\(\sigma\)) of 88.3, and a random sample size (\(n\)) of 201. The problem asks for:

### a. The mean of the distribution of sample means (\(\mu_{\bar{x}}\))
The mean of the distribution of sample means (\(\mu_{\bar{x}}\)) is the same as the population mean. So:

\[
\mu_{\bar{x}} = \mu = 183.4
\]

### b. The standard deviation of the distribution of sample means (\(\sigma_{\bar{x}}\))
The standard deviation of the distribution of sample means (also known as the **standard error of the mean**) is given by the formula:

\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]

Substituting the given values (\(\sigma = 88.3\) and \(n = 201\)):

\[
\sigma_{\bar{x}} = \frac{88.3}{\sqrt{201}} \approx \frac{88.3}{14.18} \approx 6.23
\]

### Final answers:
- \(\mu_{\bar{x}} = 183.4\)
- \(\sigma_{\bar{x}} \approx 6.23\) (rounded to two decimal places)

Would you like any more details or have further questions? 

Here are 5 related questions you can explore:
1. How does the standard error change with different sample sizes?
2. What happens to \(\sigma_{\bar{x}}\) if the population standard deviation increases?
3. Why does the mean of the sample means equal the population mean?
4. How is the Central Limit Theorem relevant to this problem?
5. How would the calculation change if \(n\) were smaller?

**Tip:** The standard error decreases as the sample size increases because larger samples tend to provide more accurate estimates of the population mean.

A normal distributed population has parameters μ = 183.4 and σ = 88.3. If a random sample of size n = 201 is selected, a. What is the mean of the distribution of sample means? b. What is the standard deviation of the distribution of sample means? Round to two decimal places.

This problem involves calculating the mean and standard deviation of sample means for a normally distributed population with a mean of 183.4 and a standard deviation of 88.3, given a sample size of 201. It demonstrates the use of key statistical concepts including the Central Limit Theorem and the standard error formula.

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