For the case in which two independent samples arc to be compared using a nonparametric test for location shift, we propose a bootstrap technique for estimating the sample sizes required to achieve a specified power. The estimator (called BOOT) uses information from a small pilot experiment. For the special case of the Wilcoxon test, a simulation study is conducted to compare BOOT to two other sample-size estimators. One method (called ANPV) is based on the assumption that the underlying distribution is normal with a variance estimated from the pilot data. The other method (called NOETHER) adapts the sample size formula of Noether for use with a location-shift alternative. The BOOT and NOETHER sample-size estimators are particularly appropriate for this nonparametric setting because they do not require assumptions about the shape of the underlying continuous probability distribution. The simulation study shows that (a) sample size estimates can have large uncertainty, (b) BOOT is at least as accurate as and can be much more accurate than ANPV, and (c) BOOT and NOETHER achieve similar accuracy, although NOETHER is prone to underestimation.