According to a survey in a country, $17 \%$ of adults do not own a credit card. Suppose a simple random sample of 500 adults is obtained. Complete parts (a) through (d) below.
Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2).
(a) Describe the sampling distribution of $\hat{p}$, the sample proportion of adults who do not own a credit card. Choose the phrase that best describes the shape of the sampling distribution of p below.
A. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$
B. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)< 10$
C. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)< 10$
D. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$
Determine the mean of the sampling distribution of $\hat{p}$.
$\mu_{p}=0.17$ (Round to two decimal places as needed.)
Determine the standard deviation of the sampling distribution of $\hat{p}$.
$\sigma_{p}=\square($ Round to three decimal places as needed $)$
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Final Answer: The standard deviation of the sampling distribution of \(\hat{p}\) is \(\boxed{0.017}\).
Step 1 ：The question is asking for the sampling distribution of the proportion of adults who do not own a credit card. The shape of the distribution can be determined by checking the conditions for normality.
Step 2 ：The mean of the sampling distribution is the same as the population proportion, which is \(0.17\).
Step 3 ：The standard deviation of the sampling distribution can be calculated using the formula for the standard deviation of a proportion, which is \(\sqrt{p(1-p)/n}\), where p is the population proportion and n is the sample size.
Step 4 ：Substitute the given values into the formula: p = 0.17 and n = 500.
Step 5 ：Calculate the standard deviation: \(\sigma_p = \sqrt{0.17(1-0.17)/500} = 0.016798809481626965\).
Step 6 ：Round the standard deviation to three decimal places: \(\sigma_p = 0.017\).
Step 7 ：Final Answer: The standard deviation of the sampling distribution of \(\hat{p}\) is \(\boxed{0.017}\).
