A local outdoor equipment store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will randomly sample among its 100,000 items in order to determine the proportion of merchandise that is outdated. The current owners have never determined their outdated percentage and can not help the buyers. Approximately how large a sample do the buyers need in order to insure that they are 98% confident that the margin of error is within 3%?
A) 1509
B) 648
C) 6033
D) 3017

Question

Quizplus · Accepted Answer

To ensure a 98% confidence level with a margin of error within 3%, the sample size can be calculated using the formula for sample size in estimating proportions: $n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}$, where $Z$ is the Z-score corresponding to the confidence level (approximately 2.33 for 98%), $p$ is the estimated proportion (0.5 is used when the proportion is unknown for maximum sample size), and $E$ is the margin of error (0.03 for 3%). Plugging in the values: $n = \frac{2.33^2 \cdot 0.5 \cdot (1-0.5)}{0.03^2}$, which simplifies to approximately 1509. This is the sample size needed to achieve the desired confidence level and margin of error.

A Local Outdoor Equipment Store Is Being Sold