Use the given data to find the minimum sample size required to estimate the population proportion.
-Margin of error: 0.04; confidence level: 99%; from a prior study, $\hat { \mathrm { p } }$ is estimated by 0.12.
A) 526
B) 254
C) 18
D) 438

Question

Quizplus · Accepted Answer

The formula for minimum sample size is given by 
$$n=\left(\dfrac{z_{\alpha/2}}{E}\right)^2\hat{p}(1-\hat{p})$$
where $E$ is the margin of error, $\hat{p}$ is the estimated population proportion and $z_{\alpha/2}$ is the critical value of the standard normal distribution at the given confidence level. 
Here, $E=0.04$, $\hat{p}=0.12$, and the confidence level is 99%, which means $\alpha=0.01$. The critical value $z_{\alpha/2}$ can be obtained from standard normal distribution tables, or by using a calculator or software. For a 99% confidence level, $z_{\alpha/2}=2.576$. 
Substituting the values, we get 
$$n=\left(\dfrac{2.576}{0.04}\right)^2(0.12)(1-0.12)\approx 438$$
Therefore, the minimum sample size required to estimate the population proportion is 438.

Note: Choice A can be obtained by using the same formula but with a different estimated population proportion of $\hat{p}=0.5$ (when no prior estimate is available). However, since the question mentions a prior study and a specific estimate of $\hat{p}=0.12$, choice D is the more appropriate answer.

Use the Given Data to Find the Minimum Sample Size p^\hat { \mathrm { p } }p^​

Use the Given Data to Find the Minimum Sample Size $\hat { \mathrm { p } }$