(The author is a stats professor at CMU.)

Quoting: "The plot shows the first 50 simulations. In the first simulation I picked some distribution {F_1}. Let {\theta_1} be the median of {F_1}. I generated {n=100} observations from {F_1} and then constructed the interval. The confidence interval is the first vertical line. The true value is the dot. For the second simulation, I chose a different distribution {F_2}. Then I generated the data and constructed the interval. I did this many times, each time using a different distribution with a different true median. The blue interval shows the one time that the confidence interval did not trap the median. I did this 10,000 times (only 50 are shown). The interval covered the true value 94.33 % of the time. I wanted to show this plot because, when some texts show confidence interval simulations like this they use the same distribution for each trial. This is unnecessary and it gives the false impression that you need to repeat the same experiment in order to discuss coverage."