To find the mean of the sampling distribution of [tex]\(\hat{p}\)[/tex], it is important to understand the properties of the sampling distribution of a sample proportion.
Given:
- [tex]\( p \)[/tex] (the population proportion) = 0.71
- [tex]\( n \)[/tex] (the sample size) = 55
The mean of the sampling distribution of [tex]\(\hat{p}\)[/tex] is actually just the population proportion [tex]\( p \)[/tex]. This is because [tex]\(\hat{p}\)[/tex] is an unbiased estimator of [tex]\( p \)[/tex]. The symbol [tex]\(\hat{p}\)[/tex] typically denotes the sample proportion, and because we are dealing with a proportion, the mean of the sampling distribution (expected value) is just the population proportion [tex]\( p \)[/tex].
Therefore, the correct interpretation for the mean of the sampling distribution of [tex]\(\hat{p}\)[/tex] is:
[tex]\[
\mu_{\hat{p}} = p = 0.71
\][/tex]
So, the correct answer is:
[tex]\[
\mu_{\hat{p}} = p = 0.71
\][/tex]
