Questions: Confidence Intervals for Proportions

Question 1

A poll of 400 voters finds that 52% prefer candidate A. Using z* = 1.96 for a 95% confidence interval, which is closest to the margin of error?

• A. About 1%
• B. About 2.5%
• C. About 5%
• D. About 10%

Answer: (C)

The margin of error is z* × √(p̂(1−p̂)/n) = 1.96 × √(0.52 × 0.48 / 400) = 1.96 × √(0.000624) ≈ 1.96 × 0.02498 ≈ 0.049, or about 5%. Option B (2.5%) would correspond to n = 1600, and option D (10%) to n = 100. This illustrates the key relationship: margin of error scales as 1/√n, so quadrupling n halves the margin of error.

Question 2

A researcher has a confidence interval with margin of error 4% (n = 625, 95% CI). To reduce the margin of error to 2%, approximately how many observations are needed?

• A. 1250 — double the sample
• B. 2500 — four times the sample
• C. 3125 — five times the sample
• D. 312 — half the sample, since error is inversely proportional to n

Answer: (B)

Margin of error scales as 1/√n. To cut the margin of error in half (from 4% to 2%), you need to multiply n by 4: √(4n) = 2√n, so the margin halves. Starting from n = 625, the new sample size is 625 × 4 = 2500. This 1/√n relationship is one of the most practically important facts in survey statistics — halving the margin of error is expensive, requiring four times as many observations.

Question 3

The margin of error for a confidence interval for a proportion is maximized (for a given sample size and confidence level) when the sample proportion p̂ = 0.5.

• T. True
• F. False

Answer: True

The standard error formula is √(p̂(1−p̂)/n). The expression p̂(1−p̂) is a downward-opening parabola in p̂, maximized at p̂ = 0.5, where it equals 0.25. This means the margin of error is largest — and the estimate most uncertain — when the sample is split 50/50. Conservative sample size planning therefore assumes p̂ = 0.5 when the true proportion is unknown, ensuring the sample size is adequate regardless of the actual value.

Question 4

A 95% confidence interval from this sample means there is a 95% probability that the true population proportion falls within this specific interval.

• T. True
• F. False

Answer: False

This is the most common misinterpretation of confidence intervals. Once the interval is computed from your sample, the true proportion either is or is not inside it — there is no probability involved for that fixed interval. The correct interpretation is frequentist: if you repeat the sampling procedure many times and compute a 95% CI each time, about 95% of those intervals will contain the true proportion. 'Confidence' describes the procedure, not the probability of any single realized interval.

Question 5

Why is the Wald interval unreliable when np̂ < 10, and what alternative should be used?

Model answer: The Wald interval relies on the normal approximation to the binomial: for large n, p̂ is approximately normally distributed. When np̂ < 10 (or n(1−p̂) < 10), there are too few successes (or failures) for this approximation to hold — the binomial distribution is skewed, and the symmetric normal interval has poor coverage, meaning it contains the true proportion less often than the stated confidence level. The alternative is the exact Clopper-Pearson interval, which inverts the exact binomial test and has guaranteed coverage properties, at the cost of being more conservative (wider).

The condition np̂ ≥ 10 and n(1−p̂) ≥ 10 is the standard check for when the normal approximation is adequate. Below this threshold, the asymmetry of the binomial distribution matters enough that symmetric normal-based intervals systematically miss the true proportion more often than advertised. This is a practical concern in clinical trials with rare events, polls about rare opinions, or any setting with extreme proportions.