describe the sampling distribution of $ \hat { p } $. assume the size of the population is 25,000$ n = 1000, p = 0.587 $describe the shape of the sampling distribution of $ \hat { p } $. choose the correct answer below$ \bigcirc $a. the shape of the sampling distribution of $ \hat { p } $ is approximately normal because $ n \leq 0.05 n $ and $ n p ( 1 - p ) \geq 10 $$ \bigcirc $b. the shape of the sampling distribution of $ \hat { p } $ is not normal because $ n \leq 0.05 n $ and $ n p ( 1 - p )

Question

describe the sampling distribution of $ \hat { p } $. assume the size of the population is 25,000$ n = 1000, p = 0.587 $describe the shape of the sampling distribution of $ \hat { p } $. choose the correct answer below$ \bigcirc $a. the shape of the sampling distribution of $ \hat { p } $ is approximately normal because $ n \leq 0.05 n $ and $ n p ( 1 - p ) \geq 10 $$ \bigcirc $b. the shape of the sampling distribution of $ \hat { p } $ is not normal because $ n \leq 0.05 n $ and $ n p ( 1 - p ) < 10 $.$ \bigcirc $c. the shape of the sampling distribution of $ \hat { p } $ is not normal because $ n \leq 0.05 n $ and $ n p ( 1 - p ) \geq 10 $.$ \bigcirc $d. the shape of the sampling distribution of $ \hat { p } $ is approximately normal because $ n \leq 0.05 n $ and $ n p ( 1 - p ) < 10 $.determine the mean of the sampling distribution of $ \hat { p } $.$ mu _ { hat { p } } = $ (round to three decimal places as needed.)determine the standard deviation of the sampling distribution of $ \hat { p } $.$ sigma _ { hat { p } } = $ (round to three decimal places as needed.)

Accepted Answer

# Explanation:
## Step1: Check $n\leq0.05N$
Given $n = 1000$, $N=25000$. Calculate $0.05N=0.05\times25000 = 1250$. Since $n = 1000\leq1250$.
## Step2: Check $np(1 - p)\geq10$
Calculate $np(1 - p)=1000\times0.587\times(1 - 0.587)=1000\times0.587\times0.413 = 242.431\geq10$
## Step3: Find the mean of the sampling distribution of $\hat{p}$
By the formula $\mu_{\hat{p}}=p$, so $\mu_{\hat{p}}=0.587$
## Step4: Find the standard deviation of the sampling distribution of $\hat{p}$
Use the formula $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}$. Substitute $p = 0.587$, $n = 1000$
$$
\begin{align*}
\sigma_{\hat{p}}&=\sqrt{\frac{0.587\times(1 - 0.587)}{1000}}\
&=\sqrt{\frac{0.587\times0.413}{1000}}\
&=\sqrt{\frac{0.242431}{1000}}\
&=\sqrt{0.000242431}\
&\approx0.016
\end{align*}
$$

# Answer:
A. The shape of the sampling distribution of $\hat{p}$ is approximately normal because $n\leq0.05N$ and $np(1 - p)\geq10$
$\mu_{\hat{p}} = 0.587$
$\sigma_{\hat{p}}\approx0.016$

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QUESTION IMAGE

Question

Explanation:

Step1: Check \(n\leq0.05N\)

Step2: Check \(np(1 - p)\geq10\)

Step3: Find the mean of the sampling distribution of \(\hat{p}\)

Step4: Find the standard deviation of the sampling distribution of \(\hat{p}\)

Answer: