Question

question 1 of 38
according to data from the 2010 united states census, 11.4% of all housing units in the united states were vacant. maria, a researcher, selects a random sample of 200 housing units in the united states and finds that 15 are vacant. let \\( \hat{p} \\) represent the sample proportion of housing units that were vacant.
what are the mean and standard deviation of the sampling distribution of \\( \hat{p} \\)?
\\( \mu_{\hat{p}} = \\) \\( \quad \\) (give your answer to 3 decimal places)
\\( \sigma_{\hat{p}} = \\) \\( \quad \\) (give your answer to 3 decimal places)

Explanation:

Step1: Find the mean of the sampling distribution of \(\hat{p}\)

The mean of the sampling distribution of the sample proportion \(\hat{p}\) (denoted as \(\mu_{\hat{p}}\)) is equal to the population proportion \(p\). From the problem, the population proportion \(p = 0.114\) (since 11.4% = 0.114). So, \(\mu_{\hat{p}}=p = 0.114\).

Step2: Find the standard deviation of the sampling distribution of \(\hat{p}\)

The formula for the standard deviation (standard error) of the sampling distribution of \(\hat{p}\) is \(\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}\), where \(p\) is the population proportion and \(n\) is the sample size. Here, \(p = 0.114\), \(1-p=1 - 0.114 = 0.886\), and \(n = 200\).

First, calculate \(p(1 - p)\): \(0.114\times0.886=0.114\times0.886 = 0.100004\)

Then, divide by \(n\): \(\frac{0.100004}{200}=0.00050002\)

Finally, take the square root: \(\sigma_{\hat{p}}=\sqrt{0.00050002}\approx0.022\) (rounded to three decimal places)

Answer:

\(\mu_{\hat{p}} = 0.114\)
\(\sigma_{\hat{p}} \approx 0.022\)