On descriptive confidence intervals for the mean of asymmetric populations

Given a sample of n observations with sample mean (X) over bar and standard deviation S drawn independently from a population with unknown mean It, it is well known that the skewness of the statistic n(-1/2)((X) over bar - mu)/S is of the opposite sign to the skewness of the population. As a consequence, an equal-tailed confidence interval for It may be used for descriptive purposes, since the relative position of (X) over bar in the interval provides visual information about the skewness of the population. In this paper, we are interested in confidence intervals for the mean which share this descriptive property. We formally define two simple classes of intervals where the degree of asymmetry around (X) over bar monotonically depends on the sample skewness through a parameter lambda with values between 0 and 1. These classes contain the symmetric ordinary-z (or the ordinary-t) confidence interval as a special case. We show how to determine this parameter lambda in order to obtain an equal-tailed confidence interval for mu which is second order accurate. While our first solution has already been investigated in the literature and has serious drawbacks, our second solution appears to be new and sound. Moreover, our method provides the sample skewness with a new and concrete interpretation.