## Mathematics - Dissertations

#### Title

Selection and testing designs for selecting one among k normal populations, provided it is better than a standard

8-2002

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Pinyuen Chen

#### Keywords

Testing designs, Ranking, Two-stage design, Population selection

#### Subject Categories

Mathematics | Statistics and Probability

#### Abstract

We propose three separate procedures for selecting one among several normal populations, provided it is better than a specified standard. The three procedures stem from three different variance assumptions. The first procedure assumes the populations share a common known variance, the second assumes the populations share a common unknown variance, and the third assumes the population variances are unequal and unknown. Each procedure is a modification of the two-stage selection and testing procedures of Thall, Simon, and Ellenberg (1988, 1989), which were formulated to compare several binomial populations to a control population. In the first stage, ranking and selection techniques are used to screen out the one most promising population. In the second stage, hypothesis testing techniques are used to compare the chosen population to the standard. An option for terminating the procedure at stage one is offered, if none of the populations seem better than the standard.

In each of our procedures, we assume that we have k (≥2) normally distributed populations, π 1 , π 2 ,..., π k , having unknown means, μ 1 , μ 2 ,..., μ k , respectively. We also assume that we have a fixed known constant (standard) μ 0 to which the μ i are to be compared. The goal is to select one of the k populations, provided that it is better than the standard. If none of the k populations is better than the standard, then no population is to be chosen. We assume that a large population mean is desirable and hence, 'better than the standard' means that the population mean is sufficiently larger than the standard.

Tables of parameter values necessary to implement the procedures are provided, with a guarantee of appropriately defined size and power requirements. Sample size comparisons are performed between our procedures and the analogous procedures of Bechhofer and Turnbull (1978) for the case of common known and common unknown variance, and Taneja and Dudewicz (1992) for the case of unequal unknown variances. It is shown that total sample size requirements for our procedures are smaller than the corresponding procedures of Bechhofer and Turnbull. The sample size comparison with Taneja and Dudewicz's procedure indicates that our procedure is more efficient for small initial sample sizes and when none of the populations are better than the standard.

#### Access

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