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. What sample size is required to obtain a $90\%$confidence interval with a margin of error of at most $0.04$?

Short answer

Sample size is$423$.

Step-by-step solution

Given Information

Confidence Level $=90\%$

$E=0.04$

Corresponding to $90\%$confidence level role="math" localid="1654454884514" $Z_{\frac{\alpha }{2}}=1.645$

Calculation of Sample Size

The formula used is:

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

Regarding $\hat{p}$, no detail is given assume it as $0.50$

Hence, $n=\hat{p}(1-\hat{p}){\left(\frac{z_{\frac{a}{2}}}{E}\right)}^2$

$=0.50(1-0.50){\left(\frac{1.645}{0.04}\right)}^2$

$=422.816\approx 423$

Sample Size is$423$