Question

Starting a nightclub A college student organization wants to start a nightclub for students under the age of $21$. To assess support for this proposal, they will select an SRS of students and ask each respondent if he or she would patronize this type of establishment. They expect that about $70$% of the student body would respond favorably. What sample size is required to obtain a $90$% confidence interval with an approximate margin of error of $0.04$? Show your work

Short answer

From the given information, the required sample size is$356$.

Step-by-step solution

Given Information

It is given in the question that,

The margin of error$E=0.04$

Sample proportion$p=70\%$

Confidence interval =$90\%$

What sample size is required to obtain a $90$% confidence interval with an approximate margin of error of $0.04$?

Step 2: Explanation

The confidence interval is $90\%$.

Convert $90$% into decimal.

$\frac{90}{100}=0.90$

For confidence interval $0.90$, use table A.

$z_{\alpha /2}=1.645$

Sample proportion $\stackrel{\wedge }{p}is70\%$

Convert $70\%$into decimal.

$\frac{70}{100}=0.70$

Calculate the sample size. Use the formula $n=\frac{{\left[z_{\alpha /2}\right]}^2\hat{p}(1-\hat{p})}{E^2}$

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

$n=\frac{{1.645}^2\times 0.70\times (1-0.70)}{{0.04}^2}$

$=356$

Hence, the required sample size is$356$