CONFIDENCE-INTERVALS WITH MORE POWER TO DETERMINE THE SIGN - 2 ENDS CONSTRAIN THE MEANS

We present two new families of two-sided nonequivariant confidence int ervals for the mean theta of a continuous, unimodal, symmetric random variable. Compared with the conventional symmetric equivariant confide nce interval, they are shorter when the observation is small, and rest rict the sign of theta for smaller observations. One of the families, a modification of Pratt's construction of intervals with minimal expec ted length when theta = 0, is longer than the conventional symmetric i nterval when \X\ is large and has longer expected length when \theta\ is large. The other family gives the conventional symmetric interval w hen \X\ is large, with a change to the proximal endpoint when \X\ is s mall. Its expected length is smaller than that of the conventional sym metric interval when \theta\ is small, larger for an intermediate rang e of \theta\, and approaches that of the conventional interval for lar ge \theta\. This slight modification of the conventional two-sided int erval has most of the power advantage of a one-sided interval, but sho rt length. Neither procedure requires that a preferred direction be sp ecified in advance. The constants that determine the procedures can be found for symmetrically distributed statistics using any software pac kage that includes the cumulative distribution function and inverse cu mulative distribution function of the statistic, along with a root fin der. We present tables of constants needed to apply the procedures for normally and t-distributed test statistics, and give an application t o employment discrimination litigation.