Confidence means repetition

How often do confidence intervals work?

We know the true population mean is 50 and the population standard deviation is 10. Each simulation draws 50 independent random samples from a normal population, and constructs an interval using the known population standard deviation. A gold interval misses the fixed true mean.

The confidence level is the long-run fraction of intervals that contain the fixed true parameter under repeated sampling. This is not a posterior probability that the true mean moves between endpoints. A single batch of 50 will fluctuate around its nominal level.