Given a sample ofnvector observations from a multivariate normal population, Anderson and Roy-Gnanadesikan have given for the variances a set of confidence intervals which are approximate in that a lower bound only is known for the joint confidence coefficient. In the present study, exact procedures are developed in terms of multivariate Chi-Square distributions, and more general approximate procedures are given via Bonferroni's inequality. A numerical investigation suggests that the Bonferroni lower bound is fairly sharp for a variety of parameter values, and it always is superior to the Roy-Gnanadesikan procedure in the bivariate case examined. A lower bound in terms of independent statistics further is examined for a special class of one-sided intervals.