Question

Math

Posted 5 months ago

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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) $\mu_{\hat{p}}=0.03$
$\sigma_{\hat{p}}=\sqrt{\frac{0.03(1-0.03)}{75}}$
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Answer from Sia

Posted 5 months ago

Solution by Steps

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 $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$.

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