Supposex1andx2are observations on a bivariate normal distribution with known variance matrix Σ and unknown means vector (μ1, μ2). Suppose an interval estimate off(μ1, μ2), is desired with 1 – α confidence. Let an interval estimate be obtained as [minf(μ1, μ2), maxf(μ1, μ2)] subject to [(x1– μ1) (x2– μ2)] Σ−1[(x1– μ1)(x2– μ2)]′ ≤R2. It is shown that under certain conditions onf(μ1, μ2) and Σ,Rneed be taken only slightly larger thanx1(α), the value of a chi-variate on one degree of freedom exceeded with probabilityα.For example under the given conditions at a nominal .95 confidence one may useR= 2.06 rather than 1.96. Similar results are obtained when variances must be estimated for some special assumptions about Σ.