  We consider the power to reject false values of the parameter in Frequentist
methods for the calculation of confidence intervals. We connect the power with
the physical significance (reliability) of confidence intervals for a parameter
bounded to be non-negative. We show that the confidence intervals (upper
limits) obtained with a (biased) method that near the boundary has large power
in testing the parameter against larger alternatives and small power in testing
the parameter against smaller alternatives are physically more significant.
Considering the recently proposed methods with correct coverage, we show that
the physical significance of upper limits is smallest in the Unified Approach
and highest in the Maximum Likelihood Estimator method. We illustrate our
arguments in the specific cases of a bounded Gaussian distribution and a
Poisson distribution with known background.
