  We construct uncertainty intervals for weak Poisson signals in the presence
of background. We consider the case where a primary experiment yields a
realization of the signal plus background, and a second experiment yields a
realization of the background. The data acquisitions times for the
background-only experiment,T_bg, and the primary experiment,T, are selected so
that their ratio varies from 1 to 25. The expected number of background counts
in the primary experiment varies from 0.2 to 2. We construct 90 and 95 percent
confidence intervals based on a propagation-of-errors method as well as two
implementations of a Neyman procedure where acceptance regions are constructed
based on a likelihood-ratio criterion that automatically determines whether the
resulting confidence interval is one-sided or two-sided. The first Neyman
procedure (due to Feldman and Cousins) neglects uncertainty in the background.
In the other Neyman procedure, we account for uncertainty in the background
with a parametric bootstrap method. We also construct minimum length Bayesian
credibility intervals. For each method, we test for the presence of a signal
based on the value of the lower endpoint of the uncertainty interval. When
T_bg/T is 5 or more and the expected background is 2 or less, the Feldman
Cousins method outperforms the other methods considered.
