On Two-Stage Comparisons with a Control Under Heteroscedastic Normal Distributions

New two-stage sampling methodologies are developed for both one-sided and two-sided comparisons between the means from treatment groups and a control. We suppose that we have k i ( ≥ 1) independent treatments with associated response variables $X_{i1},...,X_{ik_{i}},$ i = 1,...,r( ≥ 2). We let X ijl denote the lth observation recorded independently and assume that their common distribution is $N(\mu _{ij},\sigma _{i}^{2}),l=1,...,n_{i}(\geq 2),j=1,...,k_{i},i=1,...,r.$ Also, let X 0l denote the lth observation recorded independently from a control and we assume that their common distribution is $N(\mu _{0},\sigma _{0}^{2}),l=1,...,n_{0}(\geq 2).$ The parameters μ ij ’s, σ i ’s, μ 0, and σ 0 are assumed finite, unknown, and σ i ’s unequal, j = 1,...,k i , i = 1,...,r. Denote the treatment-control difference Δ ij = μ ij − μ 0 and our goal is to make simultaneous one-sided and two-sided fixed-precision confidence statements regarding all Δ ij ’s, j = 1,...,k i , i = 1,...,r. Specifically, given two preassigned numbers d( > 0) and 0