Interpreting a Confidence Interval

The parameter you're estimating, like the true proportion of voters who support a policy, is a fixed number. You don't know it, but it isn't random. What's random is your sample, and therefore your interval. Every time you draw a new sample, you get a different interval. So '95% confident' is not a claim about this one interval containing a moving target. It's a claim about your method: if you repeated the sampling process over and over, about 95% of the intervals you'd build would capture that fixed true value.

This is why the correct interpretation always has a specific shape: 'We are 95% confident that the interval from [lower] to [upper] captures the true population proportion.' The confidence lives in the process, earned by its long-run track record, and you're borrowing that track record to say something about this particular interval. It does not describe where individual data points or individual people in the population fall — the interval is about the population proportion, a single number, not about the spread of individual outcomes.

This sits right after you've learned to construct a confidence interval, and it's the half that keeps people honest about what they just built. Everything downstream, like comparing two proportions or reading a chi-square test, still rests on this same long-run frequency logic: intervals and p-values describe how a method behaves over repetition, not certainty about one fixed unknown number.