How much confidence can we have in confidence intervals?

One of the standard techniques of teaching hypothesis testing is to outline how an appropriate test statistic should be selected and to compare its calculated value, based on the available sample data, with a critical value that is usually obtained from tables. In this way a single value of the population parameter is tested. For two-sided tests a common alternative, and increasingly popular, method of performing exactly the same tests is to use confidence intervals, since in a sense they perform a multitude of hypothesis tests all at the same time. It is the interpretation of just what the intervals tell us in relation to the location of the hypothesised mean that is the subject of this paper with some intriguing results.