Question

Do you go to church? The Gallup Poll plans to ask a random sample of adults

whether they attended a religious service in the last 7 days. How large a sample would be required to obtain a margin of error of at most 0.01 in a 99% confidence interval for the population proportion who would say that they attended a religious service?

Short answer

The required sample size is$3152$

Step-by-step solution

Given information

Confidence level$=99\%$

Margin of error $\left(E\right)=0.01$

The objective is to find out the sample size to have a margin of error of at most 0.01at 99%a confidence level.

We know,

The formula to compute the sample size is:

$n=\hat{p}(1-\hat{p}){\left(\frac{z_{{\displaystyle \frac{\alpha }{2}}}}{E}\right)}^2$

There is no information about $\hat{p}$provided here.

Hence, assume is to be $0.50.$

According to the standard normal table, the value of $z_{\frac{a}{2}}$corresponding to the $99\%$confidence level is$2.576$

The sample size can be calculated as:

localid="1654402152379" $$\begin{gathered} n=\hat{p}(1-\hat{p}){\left(\frac{z_{{\displaystyle \frac{\alpha }{2}}}}{E}\right)}^2 \\ =0.50(1-0.50){\left(\frac{2.576}{0.01}\right)}^2 \\ =3151.576 \\ \approx 3152 \end{gathered}$$

Therefore, the sample size is$3152$