To find the mean of the sampling distribution of \(\hat{p}\) (the sample proportion), we can rely on basic statistical principles. Here’s a detailed, step-by-step solution:
1. Understanding the Sampling Distribution of \(\hat{p}\):
The sampling distribution of the sample proportion \(\hat{p}\) describes the distribution of sample proportions over many repeated samples of the same size from the same population.
2. Mean of the Sampling Distribution of \(\hat{p}\):
One of the properties of the sampling distribution of the sample proportion \(\hat{p}\) is that its mean equals the true population proportion \(p\). Mathematically this is expressed as:
[tex]\[
\mu_{\hat{p}} = p
\][/tex]
where \(\mu_{\hat{p}}\) is the mean of the sampling distribution and \(p\) is the true population proportion.
3. Given Data:
We are given that the proportion of all high school students who watch national news is \(p = 0.47\).
4. Application:
Using the property that the mean of the sampling distribution of \(\hat{p}\) is equal to \(p\):
[tex]\[
\mu_{\hat{p}} = 0.47
\][/tex]
5. Conclusion:
The mean of the sampling distribution of \(\hat{p}\) is \(\mu_{\hat{p}} = 0.47\).
Hence, the correct choice among the provided options is:
[tex]\[
\mu_{\hat{p}} = p = 0.47
\][/tex]
None of the other options are correct as they misunderstand the basic statistical principle that the mean of the sampling distribution of the sample proportion is simply the population proportion [tex]\(p\)[/tex], without any further calculations or adjustments based on sample size.
