Q. 51 - The Practice of Statistics for AP

Question

51. You want to design a study to estimate the proportion of students at your school who agree with the statement, “The student government is an effective organization for expressing the needs of students to the administration.” You will use a $95\%$ confidence interval, and you would like the margin of error to be $0.05$ or less. The minimum sample size required is
(a) $22$. (b) $271$. (c) $385$. (d) $769$. (e) $1795$.

Step-by-Step Solution

Answer

The correct answer is option (c) $385$.

Step 1: Given information

To find the minimum sample size required for $95\%$ confidence interval, and  the margin of error to be $0.05$ or less.

Step 2: Explanation

Let, mirror of error is $E=0.05$.

Determine the sample size as:

$\hat{p}$ unknown: $n=\frac{{\left[z_{\alpha /2}\right]}^{2}0.25}{E^2}$

Using table A, for confidence level $95\%$,
$z_{\alpha /2}=1.96$
$\hat{p}$ is unknown, then the sample size is:
$n=\frac{{\left[z_{\alpha /2}\right]}^{2}0.25}{E^2}$

$=\frac{1.{96}^2\times 0.25}{0.{05}^2}$

$\approx 385$

As a result, option (c) $385$ is correct.