  Consider a two-by-two factorial experiment with more than 1 replicate.
Suppose that we have uncertain prior information that the two-factor
interaction is zero. We describe new simultaneous frequentist confidence
intervals for the 4 population cell means, with simultaneous confidence
coefficient 1-alpha, that utilize this prior information in the following
sense. These simultaneous confidence intervals define a cube with expected
volume that (a) is relatively small when the two-factor interaction is zero and
(b) has maximum value that is not too large. Also, these intervals coincide
with the standard simultaneous confidence intervals obtained by Tukey's method,
with simultaneous confidence coefficient 1-alpha, when the data strongly
contradict the prior information that the two-factor interaction is zero. We
illustrate the application of these new simultaneous confidence intervals to a
real data set.
