AbstractSuppose that observations from populations π1, …, πk(k≥ 1) are normally distributed with unknown means μ1., μk, respectively, and a common known variance σ2. Let μ[1]μ … ≤ μ[k]denote the ranked means. We takenindependent observations from each population, denote the sample mean of thenobservation from π1byXi(i= 1, …,k), and define the ranked sample meansX[1]≤ … ≤X[k]. The problem of confidence interval estimation of μ(1), …,μ[k]is stated and related to previous work (Section 1). The following results are obtained (Section 2). Fori= 1, …,kand any γ(0<γ<1) an upper confidence interval for μ[i]with minimal probability of coverage γ is (− ∞,X[i]+h i*) withh i*= (σ/n1/2) Φ−1(γ1/k‐i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ[i]with minimal probability of coverage γ is (Xi[i]–g i*, + ∞) withg i*= (σ/n1/2) Φ−1(γ1/i). For the upper confidence interval on μ[i]the maximal probability of coverage is 1– [1 – γ1/k‐i+1]i, while for the lower confidence interval on μ[i]the maximal probability of coverage is 1–[1– γ1/i]k‐i+1. Thus the maximal overprotection can always be calculated. The overprotection is tabled fork= 2, 3. These results extend to certain translation parameter families. It is proven that, under a bounded completeness condition, a monotone upper confidence intervalh(X1, …,Xk) for