The problem of constructing a confidence region for simultaneously estimatingp,p≥ 2, linear regression parameters for which confidence statements can be made on the individual parameters is revisited. Here, an intercept may be included among thepparemeters. The technique is due to Working and Hotelling (1929) and Scheffè (1959) and uses thepseparate projections of the exact (1 – α) 100% confidence ellipsoid (ellipse ifp= 2) to give confidence intervals for each regression parameter. The Cartesian product of thesepconfidence intervals gives ap-dimensional rectangle that contains the confidence ellipsoid and hence has a joint confidence coefficient of at least (1–α). A simple calculus proof is given to determine these projections. The projection procedure is compared with the Bonferroni procedure for this case.