Suppose that $3 \%$ of over 200,000 books borrowed from a library in a year are downloaded. The librarians plan to take an SRS of 75 books from the population of borrowed books to see what proportion of books sampled are downloaded.

What are the mean and standard deviation of the sampling distribution of the proportion of downloaded books?

Choose 1 answer:
(A) $\mu_{\hat{p}}=(75)(0.03)$
$$
\sigma_{\hat{p}}=\sqrt{75(0.03)(1-0.03)}
$$
(B) $\mu_{\hat{p}}=0.03$
$$
\sigma_{\hat{p}}=\sqrt{75(0.03)(1-0.03)}
$$
(c) $\mu_{\hat{p}}=(75)(0.03)$
$$
\sigma_{\hat{p}}=\sqrt{\frac{0.03(1-0.03)}{75}}
$$
(D) $\mu_{\hat{p}}=0.03$
$$
\sigma_{\hat{p}}=\sqrt{\frac{0.03(1-0.03)}{75}}
$$

Question

Suppose that $3 \%$ of over 200,000 books borrowed from a library in a year are downloaded. The librarians plan to take an SRS of 75 books from the population of borrowed books to see what proportion of books sampled are downloaded.

What are the mean and standard deviation of the sampling distribution of the proportion of downloaded books?

Choose 1 answer:
(A) $\mu_{\hat{p}}=(75)(0.03)$
$$
\sigma_{\hat{p}}=\sqrt{75(0.03)(1-0.03)}
$$
(B) $\mu_{\hat{p}}=0.03$
$$
\sigma_{\hat{p}}=\sqrt{75(0.03)(1-0.03)}
$$
(c) $\mu_{\hat{p}}=(75)(0.03)$
$$
\sigma_{\hat{p}}=\sqrt{\frac{0.03(1-0.03)}{75}}
$$
(D) $\mu_{\hat{p}}=0.03$
$$
\sigma_{\hat{p}}=\sqrt{\frac{0.03(1-0.03)}{75}}
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the mean of the sampling distribution of the sample proportion $\hat{p}$, we use the formula $\mu_{\hat{p}} = p$, where $p$ is the population proportion. ‖ ‖ step 2 ⋮ Given that $3\%$ of the books are downloaded, we have $p = 0.03$. Therefore, the mean of the sampling distribution is $\mu_{\hat{p}} = 0.03$. ‖ ‖ step 3 ⋮ To find the standard deviation of the sampling distribution of the sample proportion $\hat{p}$, we use the formula $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size. ‖ ‖ step 4 ⋮ The sample size is $n = 75$. Plugging in the values, we get $\sigma_{\hat{p}} = \sqrt{\frac{0.03(1-0.03)}{75}}$. ‖ ‖ step 5 ⋮ Simplifying the expression for the standard deviation, we get $\sigma_{\hat{p}} = \sqrt{\frac{0.03(0.97)}{75}}$. ‖ ∻[1] Answer∻ ***D*** ∻Key Concept∻ ⚹Sampling Distribution of Proportion⚹ ∻Explanation∻ ⚹The mean of the sampling distribution of the sample proportion is the population proportion, and the standard deviation is calculated using the formula $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$.⚹◊What is the significance of the Central Limit Theorem in determining the shape of the sampling distribution of $\hat{p}$?⍭ Generate me a similar question◊

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