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If all assumptions used in deriving a confidence interval are met, the nominal coverage probability will equal the coverage probability (termed "true" or "actual" coverage probability for emphasis). If any assumptions are not met, the actual coverage probability could be either be less than or greater than the nominal coverage probability. When the actual coverage probability is greater than the nominal coverage probability, the interval is termed "conservative", if it is less than the nominal coverage probability, the interval is termed "anticonservative", or "permissive."
A discrepancy between the coverage probability and the nominal coverage probability frequently occurs when approximating a discrete distribution with a continuous one. The construction of [[Binomial proportion confidence intervalbinomial confidence intervals]] is a classic example where coverage probabilities rarely equal nominal levels.<ref>{{cite journal  last = Agresti first = Alan  coauthors = Coull, Brent  year = 1998  title = Approximate Is Better than "Exact" for Interval Estimation of Binomial Proportions  journal = The American Statistician  volume = 52  number = 2  pages = 119–126  url=http://www.jstor.org/stable/2685469  doi = 10.2307/2685469  issue = 2}}</ref><ref>{{cite journal  last=Brown  first=Lawrence  coauthors=Cai, T. Tony; DasGupta, Anirban  title=Interval Estimation for a binomial proportion  journal=Statistical Science  year=2001  volume=16  number=2  pages=101–117  url=http://wwwstat.wharton.upenn.edu/~tcai/paper/BinomialStatSci.pdf  doi=10.1214/ss/1009213286}}</ref><ref>{{cite journal  last = Newcombe first = Robert  year = 1998  title = Twosided confidence intervals for the single proportion: Comparison of seven methods.  journal = Statistics in Medicine  volume = 17  number = 2  pages = 857–872  url=http://www3.interscience.wiley.com/journal/3156/abstract  doi = 10.1002/(SICI)10970258(19980430)17:8<857::AIDSIM777>3.0.CO;2E  pmid = 9595616  issue = 8}}</ref> For the binomial case, several techniques for constructing intervals have been created. The Wilson or Score confidence interval is one well known construction based on the normal distribution. Other constructions include the Wald, exact, AgrestiCoull, and likelihood intervals. While the Wilson interval may not be the most conservative estimate, it produces average coverage probabilities that are equal to nominal levels while still producing a comparatively narrow confidence interval.
The "probability" in ''coverage probability'' is interpreted with respect to a set of hypothetical repetitions of the entire data collection and analysis procedure. In these hypothetical repetitions, [[independence (probability theory)independent]] data sets following the same [[probability distribution]] as the actual data are considered, and a confidence interval is computed from each of these data sets.
