Hypothesis tests for the presence of new sources of Poisson counts amidst background processes are frequently performed in high energy physics, gamma ray astronomy, and other branches of science. This talk briefly summarizes work in which we evaluate two classes of algorithms for dealing with uncertainty in the mean background in such tests. This talk briefly summarizes studies, performed with Robert Cousins and described in Ref. [1], of two methods for incorporating a systematic uncertainty

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... o a test of the background-only hypothesis for a Poisson process. In a situation common in both gamma-ray astronomy (GRA) and high-energy physics (HEP), n on events are observed from a Poisson process with mean µ s + µ b ; the signal mean µ s is of interest, while the background mean µ b is a nuisance parameter. In this work, we study tests of the background-only null hypothesis (µ s = 0) in two prototypical problems in GRA and HEP as follows.