This section provides technical details for computing the probability of successful demonstration shown in Figures 9.1a–9.1d in Chapter 9.

### I.1.1 Probability of Successful Demonstration Based on a Normal Distribution One-Sided Confidence Bound on a Quantile

Confidence intervals for a quantile can be used to demonstrate with 100(1 − *α*)% confidence that , where is the *p*^{†} quantile of a NORM(*μ*, *σ*) distribution and *p*^{†} and *x*^{†} are specified. We need to introduce *p*^{†} here to distinguish this *specified* value from an actual probability *p*. Suppose that the available data are a random sample *x*_{1}, …, *x*_{n} from the NORM(*μ*, *σ*) distribution. The data are summarized by the sample mean and the sample standard deviation *s*, defined in Section 3.1.2.

The demonstration test is successful with 100(1 − *α*)% confidence if the one-sided upper 100(1 − *α*)% confidence bound is less than or equal to *x*^{†}. Thus, the probability of successful demonstration is

where the 100(1 − *α*)% one-sided upper confidence ...