Interpretation of a confidence interval

95% confidence intervals are found from sample data and are therefore random, so they do not always include the parameter that is being estimated.

The diagram below is based on a simulation of random samples of n = 20 values from a normal population. The 95% confidence intervals for µ were calculated using the known population standard deviation, σ = 2.

A few random samples resulted in CIs that did not include µ = 12 (the population mean used to generate the simulated samples). If we had continued the simulation with more samples, eventually 95% of the CIs would have included the true parameter value.

In practice, we only have a single sample, and we do not know whether or not it is one of the 'lucky 95%' whose confidence intervals include µ, but we are 95% confident that it is.

Knowing that confidence intervals obtained in this way will usually include it is very helpful. In practice,...

Being right most of the time is the best one can hope for, since there is always the possibility of being misled by an unlucky sample.