Using an Interval to Judge a Claim

A confidence interval is built from one sample, but it's meant to capture the truth about the whole population, most of the time. So it works like a net of plausible values for p. When someone makes a claim about p, you just check: did the net catch it? If the claimed value sits outside the interval, that value is not one of the plausible ones given your data, so you have evidence against the claim. If it sits inside, your data don't contradict it.

Here's the mechanism. Suppose a 95% interval for p runs from 0.28 to 0.38, and someone claims p = 0.50. That claim is way outside the interval, so the sample gives good evidence the true proportion isn't 0.50 - it's probably lower. Compare that to an interval from 0.42 to 0.55 testing whether a majority (p > 0.50) supports a law: since 0.50 is inside the interval, you do not have convincing evidence of majority support, even though values above 0.50 are also in the interval and plausible.

The key catch: landing inside the interval never proves a claim true. Every value from 0.42 to 0.55 is equally plausible given the data - 0.50 isn't special just because it's there, and 0.43 is just as consistent with the sample as 0.50 is. Nearness to the center doesn't grade the evidence either; a value barely inside the boundary is still a plausible value, full stop, not a weaker one. This interval logic is the two-sided mirror of a hypothesis test's p-value: both use the same sample to decide if a claimed parameter value is believable, just packaged differently.