A random sample of Americans showed that 4% believe their health care is excellent. If the conservative margin of error was 12.5%, what was the sample size?
A) 25
B) 36
C) 49
D) 64

Question

Quizplus · Accepted Answer

The margin of error (MOE) in a proportion is calculated using the formula MOE = Z * sqrt(p(1-p)/n), where Z is the Z-score corresponding to the confidence level (not provided, but not needed for this calculation), p is the proportion (0.04 in this case), and n is the sample size. Rearranging the formula to solve for n gives n = p(1-p) / (MOE/Z)^2. Given that the MOE is 12.5% or 0.125, and using the proportion of 0.04, we don't need the exact Z-score to solve for n because the question implies a conservative estimate, which typically means using a higher Z-score to ensure the margin of error encompasses the true proportion. Plugging the values into the rearranged formula without explicitly solving for Z, we aim to find a sample size (n) that fits the given MOE. Without direct calculation, the correct answer must be chosen from the options provided, and given the relationship between MOE and sample size (larger sample sizes give smaller MOEs), we infer that a larger sample size is needed for a relatively small MOE of 12.5% when the proportion is also small (4%). Among the provided options, 64 (D) is the largest sample size and thus the most likely to produce the given MOE under these conditions.

A Random Sample of Americans Showed That 4% Believe Their