Confidence Level, Width, and Sample Size

A confidence interval for a proportion is p-hat plus or minus a margin of error, and that margin of error is z·times √(p-hat(1-p-hat)/n). Two things control its size. The critical value z·comes from how confident you want to be: 90% confidence uses z* = 1.645, 95% uses 1.96, 99% uses 2.576. Bigger confidence means a bigger z*, which stretches the margin of error. Sample size n sits in the denominator under a square root, so a bigger n shrinks the margin of error. Confidence level and sample size pull width in opposite directions: one comes from how sure you insist on being, the other from how much data backs up your estimate.

This is why you cannot get a narrow interval and high confidence for free. Demanding more confidence, with the same data, forces a wider net. The only way to narrow the net without giving up confidence is to collect more data, since n is under a square root, so width shrinks slower than n grows: quadrupling n only cuts the margin of error in half. Going from n = 100 to n = 900 (9 times as much data) shrinks the margin of error to one third of its original size.

This width-versus-confidence tradeoff is the bridge into why sample size planning matters before you ever collect data, and it sets up the idea that a confidence level describes the reliability of the method across many samples, not a probability statement about one interval. That distinction matters again when interpreting what a confidence interval actually captures.