The Academy of Orthopedic Surgeons states that 80% of women wear shoes that are too small for their feet. A researcher wants to be 98% confident that this proportion is within
3 percentage points of the true proportion. How large a sample is necessary?

Question

Quizplus · Accepted Answer

To determine the sample size necessary for a given confidence level and margin of error, the formula for sample size in estimating proportions is used: $n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}}$, where $Z$ is the Z-score corresponding to the confidence level, $p$ is the estimated proportion, and $E$ is the margin of error. For a 98% confidence level, the Z-score is approximately 2.33 (from Z-tables). Given $p = 0.8$ and $E = 0.03$, substituting these values gives $n = \frac{{(2.33)^2 \cdot 0.8 \cdot (1-0.8)}}{{(0.03)^2}} \approx 966$.

The Academy of Orthopedic Surgeons States That 80% of Women