Question

what proportion of u.s. residents receive a jury summons each year? a polling organization plans to survey a random sample of 500 u.s. residents to find out. let \\(\hat{p}\\) be the proportion of residents in the sample who received a jury summons in the previous 12 months. according to the national center for state courts, 15% of u.s. residents receive a jury summons each year. suppose that this claim is true.\
\
(a) calculate the mean and standard deviation of the sampling distribution of \\(\hat{p}\\).\
\
\\(\mu_{\hat{p}} = \\) \
(do not round.)\
\\(\sigma_{\hat{p}} = \\) \
(round to 3 decimal places.)\
\
interpret the standard deviation.\
\
in srss of size \\(n = 500\\), the sample proportion of people who received a jury summons in the previous 12 months typically varies from the true proportion by about 0.234.

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 $p$. Here, the population proportion $p = 0.15$ (since 15% of U.S. residents receive a jury summons each year). So, $\mu_{\hat{p}} = p = 0.15$.

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. We know that $p = 0.15$, $1 - p = 1 - 0.15 = 0.85$, and $n = 500$. Plugging these values into the formula:

$$ \sigma_{\hat{p}} = \sqrt{\frac{0.15 \times 0.85}{500}} $$

First, calculate the numerator: $0.15 \times 0.85 = 0.1275$. Then divide by $n$: $\frac{0.1275}{500} = 0.000255$. Now take the square root: $\sqrt{0.000255} \approx 0.01597 \approx 0.016$ (rounded to 3 decimal places).

Answer:

For the mean: $\mu_{\hat{p}} = \boldsymbol{0.15}$

For the standard deviation: $\sigma_{\hat{p}} \approx \boldsymbol{0.016}$