  We give a classical confidence belt construction which unifies the treatment
of upper confidence limits for null results and two-sided confidence intervals
for non-null results. The unified treatment solves a problem (apparently not
previously recognized) that the choice of upper limit or two-sided intervals
leads to intervals which are not confidence intervals if the choice is based on
the data. We apply the construction to two related problems which have recently
been a battle-ground between classical and Bayesian statistics: Poisson
processes with background, and Gaussian errors with a bounded physical region.
In contrast with the usual classical construction for upper limits, our
construction avoids unphysical confidence intervals. In contrast with some
popular Bayesian intervals, our intervals eliminate conservatism (frequentist
coverage greater than the stated confidence) in the Gaussian case and reduce it
to a level dictated by discreteness in the Poisson case. We generalize the
method in order to apply it to analysis of experiments searching for neutrino
oscillations. We show that this technique both gives correct coverage and is
powerful, while other classical techniques that have been used by neutrino
oscillation search experiments fail one or both of these criteria.
