A survey firm wants to ask a random sample of adults in Ohio if they support an increase in the state sales tax from $5.75\%$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. If the survey firm wants the standard deviation of the sampling distribution of $\hat{p}$to equal $0.01,$how large a sample size is needed?

a.$1500$

b. $2400$

c.$2401$

d.$2500$

e.$9220$

Given in the question that, a survey firm wants to ask a random sample of adults in Ohio if they support an increase in the state sales tax from $5.75\%$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. We need to find the sample size if the survey firm wants the standard deviation of the sampling distribution of $\hat{p}$ to equal$0.01$

A polling firm wants to ask a random sample of Ohio citizens if they support raising the sales tax, with the extra revenue going to education.

It is said as follows:

$p=0.40$

${\sigma }_{\hat{p}}=0.01$

The sampling distribution's standard deviation is calculated as follows:

${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

We'll now evaluate the equation to determine $\mathrm{n}$as follows:

${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

$\Rightarrow n=({)}^{\frac{\sqrt{(1-p)}}{{\sigma }_{\hat{p}}}}$

Now, enter the following values into the formula:

$n=({)}^{\frac{\sqrt{(1-p)}}{{\sigma }_p^p}}$

$=({)}^{\frac{\sqrt{0.40(1-0.00)}}{0.01}}$

$=2400$

As a result, a sample size of$2400$is necessary. As a result, option (b) is the proper choice.