A university surveyed 36 randomly selected graduate students in a pilot study and found that 27 said they favored the new university health insurance plan and 9 said they opposed it.How many students must the university survey in order to make an estimate of how all graduate students feel about the plan with a margin of error to be no more than 5 percentage points? Use the 95 percent confidence level.
A) 187
B) 288
C) 366
D) 950

Question

Quizplus · Accepted Answer

To estimate the sample size needed for a proportion with a specified margin of error (E) at a certain confidence level, we use the formula: $n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}$, where $Z$ is the Z-score corresponding to the confidence level (for 95%, $Z = 1.96$), $p$ is the sample proportion (in this case, 27/36 favoring the plan, so $p = 0.75$), and $E$ is the margin of error expressed as a decimal (for 5%, $E = 0.05$). Plugging in the values: $n = \frac{1.96^2 \cdot 0.75 \cdot (1-0.75)}{0.05^2}$, which simplifies to approximately 288. Therefore, the university needs to survey 288 students to achieve a margin of error of no more than 5 percentage points at the 95% confidence level.

A University Surveyed 36 Randomly Selected Graduate Students in a Pilot