Short Answer

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) $2400$ .

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Step by Step Solution

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TABLE OF CONTENTS

Step 1: Given Information

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 $0.01$ .

Step 2: Simplify

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

$S D$ $(\hat{p}) = \sqrt{\frac{p \times (1 - p)}{n}}$ , 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 $0.01$

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

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

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

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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