Let's go through the steps to find the mean and the standard deviation of the sampling distribution of the sample proportion.
### Part 1 of 6 (a)
Find the mean [tex]\(\mu_{\hat{p}}\)[/tex]:
Given data:
- Population proportion ([tex]\(p\)[/tex]) = 0.64
- Sample size ([tex]\(n\)[/tex]) = 225
The mean of the sampling distribution of the sample proportion ([tex]\(\mu_{\hat{p}}\)[/tex]) is given by the population proportion [tex]\(p\)[/tex].
[tex]\[
\mu_{\hat{p}} = p = 0.64
\][/tex]
### Part 2 of 6 (b)
Find the standard deviation [tex]\(\sigma_{\hat{p}}\)[/tex]:
The standard deviation of the sampling distribution of the sample proportion ([tex]\(\sigma_{\hat{p}}\)[/tex]) can be calculated using the formula:
[tex]\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}
\][/tex]
Substitute the given values into the formula:
- [tex]\(p = 0.64\)[/tex]
- [tex]\(n = 225\)[/tex]
[tex]\[
\sigma_{\hat{p}} = \sqrt{\frac{0.64(1 - 0.64)}{225}}
\][/tex]
Calculate the expression inside the square root:
[tex]\[
\sigma_{\hat{p}} = \sqrt{\frac{0.64 \times 0.36}{225}}
\][/tex]
Further simplify it:
[tex]\[
\sigma_{\hat{p}} = \sqrt{\frac{0.2304}{225}} = \sqrt{0.001024}
\][/tex]
Finally, take the square root to find the standard deviation:
[tex]\[
\sigma_{\hat{p}} \approx 0.032
\][/tex]
So, the standard deviation [tex]\(\sigma_{\hat{p}}\)[/tex] is approximately 0.032.
