A university dean is interested in determining the proportion of students who receive some sort of financial aid.Rather than examine the records for all students,the dean randomly selects 200 students and finds that 118 of them are receiving financial aid.If the dean wanted to estimate the proportion of all students receiving financial aid to within 3% with 99% reliability,how many students would need to be sampled?
A) n = 1,503
B) n = 1,435
C) n = 1,784
D) n = 1,844

Question

Quizplus · Accepted Answer

To estimate the proportion of all students receiving financial aid to within a certain percentage (margin of error, E) with a certain level of confidence, we use the formula for sample size in estimating proportions: $n = \left(\frac{Z^2 \cdot p \cdot (1-p)}{E^2}\right)$, where $Z$ is the Z-score corresponding to the desired level of confidence (for 99% reliability, $Z \approx 2.576$), $p$ is the estimated proportion from the preliminary sample (118/200 = 0.59), and $E$ is the margin of error expressed as a decimal (3% = 0.03). Plugging these values into the formula gives $n = \left(\frac{(2.576)^2 \cdot 0.59 \cdot (1-0.59)}{(0.03)^2}\right)$, which calculates to approximately 1,784. Therefore, the dean would need to sample about 1,784 students to estimate the proportion to within 3% with 99% reliability.

A University Dean Is Interested in Determining the Proportion of Students