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To solve this problem, we are dealing with a sampling distribution for proportions. Let's break it down:

### 1. Mean of the Sampling Distribution of the Sample Proportion (\(\mu_{\hat{p}}\)):
The mean of the sampling distribution of the sample proportion is equal to the true population proportion \(p\). Given in the problem:

\[
\mu_{\hat{p}} = p = 0.42
\]

So, the exact answer is:
\[
\mu_{\hat{p}} = 0.42
\]

### 2. Standard Deviation of the Sampling Distribution of the Sample Proportion (\(\sigma_{\hat{p}}\)):
The formula for the standard deviation of the sample proportion is:

\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}
\]

Where:
- \(p = 0.42\) (the population proportion),
- \(n = 121\) (the sample size).

Now, plugging in the values:

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.42(1 - 0.42)}{121}}
\]

Let's calculate this.The standard deviation of the sampling distribution of the sample proportion is approximately:

\[
\sigma_{\hat{p}} \approx 0.04
\]

(rounded to two decimal places).

### Summary:
- Mean of the distribution of sample proportion: \(\mu_{\hat{p}} = 0.42\)
- Standard deviation of the distribution of sample proportion: \(\sigma_{\hat{p}} \approx 0.04\)

Would you like further details on this topic, or have any additional questions?

Here are five related questions:
1. How does the sample size affect the standard deviation of the sampling distribution of proportions?
2. What conditions need to be met to use the normal approximation for the sampling distribution of proportions?
3. How would the mean and standard deviation change if the sample size was increased?
4. What would the standard deviation be if the population proportion was 0.5?
5. Can the sample proportion distribution be skewed under certain conditions?

**Tip**: As the sample size increases, the standard deviation of the sampling distribution decreases, making the sample mean more precise.

Suppose the true proportion of voters in the county who support a school levy is 0.42. Consider the sampling distribution for the proportion of supporters with sample size n = 121. What is the mean of this distribution of sample proportion? What is the standard deviation of this distribution of sample proportion?

Learn how to calculate the mean and standard deviation of the sampling distribution of a proportion. With a sample size of 121 and a proportion of 0.42, this problem demonstrates key concepts in statistics including the use of the Central Limit Theorem. Suitable for students in Grades 10-12.

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