Interpretation of a confidence interval

In practice we only have a single sample (and a single confidence interval).

We do not know whether or not this interval actually includes the unknown population mean. However knowing that confidence intervals obtained in this way will usually include µ 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.

The method that we use to obtain the confidence interval has probability 0.95 of including µ. We cannot tell whether the single interval that we evaluate from our data set is one of these 'lucky' intervals, but knowing that the method works so often gives us 95% confidence in it.

Examples

The following data sets give examples of the calculations for obtaining 95% confidence intervals and the conclusions that we can draw from them.