[For the entire collection see Zbl 0638.00032.]
Let \(\pi_ 0,\pi_ 1,...,\pi_ k\) be \(k+1\) jointly distributed normal populations with unknown means \(\theta_ 0,\theta_ 1,...,\theta_ k\) and (known or unknown) covariance matrix \(\Sigma\). The population \(\pi_ 0\) is a control population while the others are treatment populations. Let \(\theta_{[1]}\leq...\leq \theta_{[k]}\) be the ordered \(\theta_ i\)-values. The goal is to select the treatment population associated with \(\theta_{[k]}\) (denoted by \(\pi_{(k)})\) provided that \(\theta_{[k]}>\theta_ 0\). The probability requirement is as follows: \[ P\{\pi_ 0\quad is\quad selected\}\geq P^*_ 0\quad if\quad \theta_{[k]}\leq \theta_ 0, \]
\[ P\{\pi_{(k)}\quad is\quad selected\}\geq P^*_ 1\quad if\quad \theta_{[k]}\geq \max \{\theta_{[k-1]},\theta_ 0\}+\Delta \] where \(P^*_ 0,P^*_ 1,\Delta\) are prespecified constants. Selection procedures are proposed for the cases where the control mean \(\theta_ 0\) is known or unknown. The procedures are single-stage if \(\Sigma\) is known and two-stage if \(\Sigma\) is unknown. It is shown how to determine the critical constants and the sample sizes required by the procedures to meet the specified probability requirement.