I am confused about the difference between the nominal and the actual coverage probability.
Say we are trying to estimate the fraction of the current population of the US that has been diagnosed with cancer. We get a sample (e.g. 10,000 individuals) and then estimate this fraction ($f$) and a confidence interval around it $[a,b]$ with $\alpha$ (confidence level) $=0.95$.
At this point, if I understood things correctly, the nominal coverage probability is $\alpha$, i.e. $0.95$.
But what about the actual coverage probability? To me, there is only one true value of $f$, e.g. say $5\%$, and it doesn't make sense to talk about the actual probability that $f$ is within the interval $[a,b]$. It either is or it isn't within $[a,b]$.
What's wrong with this reasoning? And how would one obtain the actual coverage probability in practice?