  We consider a linear regression model with regression parameter
beta=(beta_1,...,beta_p) and independent and identically N(0,sigma^2)
distributed errors. Suppose that the parameter of interest is theta = a^T beta
where a is a specified vector. Define the parameter tau=c^T beta-t where the
vector c and the number t are specified and a and c are linearly independent.
Also suppose that we have uncertain prior information that tau = 0. We present
a new frequentist 1-alpha confidence interval for theta that utilizes this
prior information. We require this confidence interval to (a) have endpoints
that are continuous functions of the data and (b) coincide with the standard
1-alpha confidence interval when the data strongly contradicts this prior
information. This interval is optimal in the sense that it has minimum weighted
average expected length where the largest weight is given to this expected
length when tau=0. This minimization leads to an interval that has the
following desirable properties. This interval has expected length that (a) is
relatively small when the prior information about tau is correct and (b) has a
maximum value that is not too large. The following problem will be used to
illustrate the application of this new confidence interval. Consider a 2-by 2
factorial experiment with 20 replicates. Suppose that the parameter of interest
theta is a specified simple effect and that we have uncertain prior information
that the two-factor interaction is zero. Our aim is to find a frequentist 0.95
confidence interval for theta that utilizes this prior information.
