Q.6 - The Practice of Statistics for AP

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 $0.01 in a 99\%$ confidence interval for the population proportion who would say that they attended a religious service? Show your work.

- Construct and interpret a confidence interval for a population proportion.

- Explain how practical issues like nonresponse, under coverage, and response bias can affect the interpretation of a confidence interval.

Step-by-Step Solution

Answer

The sample size is $16577$$16577$

Step-1 Given Information

Given in the question that,

Margin of error$\left(E\right)=1\%=0.01$

Confidence level$=99\%$we have to find  How large a sample would be required to obtain a margin of error and confidence interval for the population proportion who would say that they attended a religious service.

Step-2 Explanation

The formula to compute the sample size is:

$n=\left(\overbrace{p}\right) \left(1-\hat{p}\right){\left(\frac{z}{E}\right)}^2$

Calculation:

From the standard normal table, the z-score at $99\%$ confidence level is $2.576$.

Consider, sample proportion to be $0.05$ .

The sample size is calculated as:

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

   $=0.50\left(1-0.50\right){\left(\frac{2.576}{0.01}\right)}^2$$=0.50 \left(1-0.50\right) {\left(\frac{2.576}{0.01}\right)}^2$

   $=16577$

Thus, the sample size is $16577$.