Statistics for Business: Decision Making and Analysis, 3rd Edition

... pot.

Compare that situation to the same bet after observing the sample and calculating the interval. Do we really want to bet in this case? Our friend knows μ, so she knows whether μ lies inside our confidence interval. If she takes this bet, don’t expect to win!

The difference lies in the timing. The formula $\overline{X}\pm t_{0.025,n-1}S/\sqrt{n}$ produces an interval that covers μ in 95% of samples. The observed sample is either one of the lucky ones, in which the confidence interval covers μ, or it’s not. When we say “We are 95% confident,” we’re describing a procedure that works for 95% of samples. If we lined up the intervals from many, many samples, 95% of these intervals would contain μ and 5% would not. Our sample and confidence interval are a random draw from ...