We develop new simultaneous confidence intervals for the components of a multivariate mean. The intervals determine the signs of the parameters more frequently than standard intervals do: the set of data values for which each interval includes parameter values with only one sign is larger. When one or more estimated means are small, the new intervals sacrifice some length to avoid crossing zero. But when all the estimated means are large, the new intervals coincide with standard simultaneous confidence intervals, so there is no sacrifice of precision. The improved ability to determine signs is remarkable. For example, if two means are to be estimated and the intervals are allowed to be at most 80 % longer than standard intervals, when only one mean is small its sign is determined almost as well as by a one-sided test that ignores multiplicity and has a pre-specified direction. When both are small the sign is determined better than by two-sided tests that ignore multiplicity. The intervals are constructed by inverting level-α tests to form a 1−α confidence set, then projecting that set onto the coordinate axes to get confidence intervals. The tests have hyperrectangular acceptance regions that minimize the maximum amount by which the acceptance region protrudes from the orthant that contains the hypothesized parameter value, subject to a constraint on the maximum side length of the hyperrectangle. R and SAS scripts are available online. Key Words: Non-equivariant hypothesis test, hyperrectangular acceptance region
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