The statistic y = (xₚ - x)/s_ν is studied where xₚ is the maximum of p normal independent chance variables with common mean and common unknown variance σ², x is another independent normal chance variable with the same mean and the same variance σ², and s²_ν (distributed as σ²χ²_ν/ν with ν degrees of freedom) is an estimate of the common variance which is independent of each one of the above $p + 1$ chance variables. Several different methods are proposed and studied for computing the probability integral of y and percentage points of y; in addition, a method for computing percentage points without first computing the probability integral of y is considered. A table of (upper) percentage points of y is given as Table I at the end of the paper. Applications of the statistic y to several ranking and selection problems are mentioned in Section 2. Moments of y are given in Section 3. In Section 7 it is shown that Table I can be used to obtain an approximation and bounds to the percentage points of a related statistic.
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Gupta et al. (1957) studied this question.