require larger sample sizes to achieve the same length of bound. In other words, for the same sample size, an upper nonparametric tolerance bound will tend to be higher.

The procedure for calculating a nonparametric tolerance bound is as follows:

•  Order the sample data x1, . ., xn from smallest to largest. Denote the ordered set of data as x(1), …, x(n). One of the sample data points will be chosen as the nonparametric bound.

•  Find the smallest integer k so that imageimage, where X is a binomial( n, p) random variable. If k = n + 1, then use the x(n) as the order statistic. Otherwise, x(n) is the upper tolerance bound.

•  The actual confidence level is image. Note that it may not be possible to find a tolerance bound with the values of p and a that are desired. In particular, the smallest sample size needed to have 100(1 - a) percent confidence that the largest observation in the sample will exceed at least 100p percent of the population is n = log(α)/log(p).

In Figure H-1, the bell curve represents the population distribution and the solid vertical line is the 95th quantile of the population distribution. In practice, both of these are unknown. The dotted vertical line is the specification. Since the population distribution is unknown, we test a sample from the population. The small circles are 50-sample observations.

image

FIGURE H-1 Ninety-fifth quintile distribution.



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