  We consider the standard Neyman-Pearson hypothesis test of a
signal-plus-background hypothesis and background-only hypothesis in the
presence of uncertainty on the background-only prediction. Surprisingly, this
problem has not been addressed in the recent conferences on statistical
techniques in high-energy physics -- although the its confidence-interval
equivalent has been. We discuss the issues of power, similar tests, coverage,
and ordering rules. The method presented is compared to the Cousins-Highland
technique, the ratio of Poisson means, and ``profile'' method.
