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The problem you've shared involves calculating the mean and standard deviation of the distribution of sample proportions for a random sample of size \( n = 623 \) from a population with proportion parameter \( p = 0.81 \).

### Part (a) - Mean of the Distribution of Sample Proportions

The mean of the distribution of sample proportions, denoted as \( \mu_{\hat{p}} \), is simply equal to the population proportion \( p \). Hence:
\[
\mu_{\hat{p}} = p = 0.81
\]

So, the mean \( \mu_{\hat{p}} \) is:
\[
\mu_{\hat{p}} = 0.81
\]

### Part (b) - Standard Deviation of the Distribution of Sample Proportions

The standard deviation of the distribution of sample proportions, denoted as \( \sigma_{\hat{p}} \), is calculated using the formula:
\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}
\]
Substitute \( p = 0.81 \) and \( n = 623 \):
\[
\sigma_{\hat{p}} = \sqrt{\frac{0.81(1 - 0.81)}{623}} = \sqrt{\frac{0.81 \times 0.19}{623}} = \sqrt{\frac{0.1539}{623}} \approx \sqrt{0.000247} \approx 0.0157
\]

So, the standard deviation \( \sigma_{\hat{p}} \) is approximately:
\[
\sigma_{\hat{p}} \approx 0.02
\]

### Final Answers:
- (a) The mean \( \mu_{\hat{p}} \) is \( 0.81 \).
- (b) The standard deviation \( \sigma_{\hat{p}} \) is approximately \( 0.02 \).

Would you like more details or explanations about this?

Here are five related questions for practice:
1. What is the probability that the sample proportion will be within one standard deviation of the population proportion?
2. How would the sample size affect the standard deviation of the sample proportions?
3. If \( p \) were different, how would the mean and standard deviation change?
4. Can we use this normal approximation for smaller sample sizes? Why or why not?
5. What is the interpretation of the standard deviation in the context of sample proportions?

**Tip:** Remember that as the sample size increases, the standard deviation of the sample proportion decreases, leading to more precise estimates of the population proportion.

A random sample of size n = 623 from a population whose parameter is p = 0.81.

(a) What is the mean of the distribution of sample proportions? Round the answer accurate to 2 decimal places.
(b) What is the standard deviation of the distribution of sample proportions? Round the answer accurate to 2 decimal places.

Learn how to find the mean and standard deviation of sample proportions from a population parameter p = 0.81 with a sample size of 623. This solution demonstrates the use of statistical formulas for calculating the mean (μp̂) and standard deviation (σp̂), providing a clear and concise explanation suitable for college-level students.

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