  Confidence intervals for the means of multiple normal populations are often
based on a hierarchical normal model. While commonly used interval procedures
based on such a model have the nominal coverage rate on average across a
population of groups, their actual coverage rate for a given group will be
above or below the nominal rate, depending on the value of the group mean.
Alternatively, a coverage rate that is constant as a function of a group's mean
can be simply achieved by using a standard $t$-interval, based on data only
from that group. The standard $t$-interval, however, fails to share information
across the groups and is therefore not adaptive to easily obtained information
about the distribution of group-specific means.
  In this article we construct confidence intervals that have a constant
frequentist coverage rate and that make use of information about across-group
heterogeneity, resulting in constant-coverage intervals that are narrower than
standard $t$-intervals on average across groups. Such intervals are constructed
by inverting biased tests for the mean of a normal population. Given a prior
distribution on the mean, Bayes-optimal biased tests can be inverted to form
Bayes-optimal confidence intervals with frequentist coverage that is constant
as a function of the mean. In the context of multiple groups, the prior
distribution is replaced by a model of across-group heterogeneity. The
parameters for this model can be estimated using data from all of the groups,
and used to obtain confidence intervals with constant group-specific coverage
that adapt to information about the distribution of group means.
