What a Confidence Interval Is

A poll never hands you the true population proportion. It hands you a sample proportion, a single number that bounces around from sample to sample. Report just that number and you imply a precision you don't have. A confidence interval admits the uncertainty honestly: it's a range of values plausible enough to have produced your sample, built around the sample proportion using its standard error.

Picture taking 100 different random samples from the same population and building a 95% confidence interval from each one. The intervals will differ slightly, since each sample differs. About 95 of those 100 intervals will contain the true population proportion; about 5 will miss it entirely, usually because an unusually extreme sample pulled the interval away from the truth. That 95% is a property of the process, the swarm, not of any one interval you happen to be holding. Once you've calculated your actual interval, the true proportion is either in it or it isn't, no probability left to assign.

This sets up everything else in the unit. The margin of error that builds the interval's width comes straight from the standard error formula, and the same logic of long-run performance reappears when you judge p-values: both confidence intervals and significance tests describe how a method behaves over repeated sampling, not certainty about one dataset. Pushing confidence up to 99% widens each interval and widens the swarm's spread, trading precision for a higher long-run capture rate.