A comparison of confidence intervals generated by the scheffé and bonferroni methods

Simultaneous confidence intervals for the p means of a multivariate normal random variable with known variances may be generated by the projection method of Scheffe and by the use of Bonferroni's inequality. It has been conjectured that the Bonferroni intervals are shorter than the Scheffe intervals, at least for the usual confidence levels. This conjecture is proved for all p≥2 and all confidence levels above 50%. It is shown, incidentally, that for all p≥2 Scheffe's intervals are shorter for sufficiently small confidence levels. The results are also applicable to the Bonferroni and Scheffe intervals generated for multinomial proportions.