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Q. 11

Expert-verifiedFound in: Page 800

Book edition
4th

Author(s)
David Moore,Daren Starnes,Dan Yates

Pages
809 pages

ISBN
9781319113339

A survey ﬁrm wants to ask a random sample of adults in Ohio if they support an increase in the state sales tax from $5\%$% to $6\%$, with the additional revenue going to education. Let $\hat{p}$ denote the proportion in the sample who say that they support the increase. Suppose that $40\%$ of all adults in Ohio support the increase. How large a sample would be needed to guarantee that the standard deviation of $\hat{p}$ is no more than $0.01$?

(a)$1500$

(b) $2400$

(c) $2401$

(d) $2500$

(e) $9220$

The sample that would be needed to guarantee that the standard deviation of $\hat{p}$ is no more than$0.01$ is b) $2400$.

We are given that the $\hat{p}$denote the proportion in the sample who say that they support the increase.

We need to find that sample would be needed to guarantee that the standard deviation of $\hat{p}$ is no more than $0.01$.

First of all , we will use standard deviation for the estimation because it is proportion to population,

$SD$$\left(\hat{p}\right)=\sqrt{\frac{p\times \left(1-p\right)}{n}}$, here $\hat{p}$ denote the proportion in the sample who say that they support the increase, we have to find out $n$.

Now standard deviation should not be more than $0.01$

$\sqrt{\frac{0.4\times 0.6}{n}}\le 0.01\Rightarrow \frac{0.4\times 0.6}{n}={0.01}^{2}$

We will multiply $n$ on other side, by simplifying we will find value of $n$;

$n\ge \frac{0.4\times 0.6}{{0.01}^{2}}\Rightarrow n\ge 2400$

If the sample size is at least $2400$ then condition would be satisfied which is standard deviation would be $0.01$ or less . So answer is $2400$.

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