# Solved (Free): How many people should be surveyed if we want to estimate the proportion of students who prefer using the test center

#### ByDr. Raju Chaudhari

Mar 30, 2021

How many people should be surveyed if we want to estimate the proportion of students who prefer using the test center? Assume we want 90% confidence we are within 5% of the true population proportion and that a prior study showed 53% preferred it.

#### Solution

The formula to estimate the sample size required to estimate the proportion is

 $$n =p*(1-p)\bigg(\frac{z}{E}\bigg)^2$$
where $p$ is the proportion of success, $z$ is the $Z_{\alpha/2}$ and $E$ is the margin of error.

Given that margin of error $E =0.05$. The confidence coefficient is $1-\alpha=0.9$. The sample proportion is $p =0.53$.

The critical value of $Z$ is $Z_{\alpha/2} =Z_{0.05}= 1.64$.

The minimum sample size required to estimate the proportion is

 \begin{aligned} n&= p(1-p)\bigg(\frac{z}{E}\bigg)^2\\ &= 0.53(1-0.53)\bigg(\frac{1.64}{0.05}\bigg)^2\\ &=267.992\\ &\approx 268. \end{aligned}

Thus, the sample of size $n=268$ will ensure that the $90$% confidence interval for the proportion will have a margin of error $0.05$.