Statistical Intervals, 2nd Edition

Appendix I Probability of Successful Demonstration

I.1 Demonstration Tests Based on a Normal Distribution Assumption

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 x₁, …, 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 ...