An MD&DI September 1997 Column
To the Editor:
In the June 1997 article by Thom R. Nichols and Sheldon Dummer, "Assessing Pass/Fail Testing When There Are No Failures to Assess," the authors assume a two-sided interval in calculating the confidence interval. In many industrial situations, the upper boundary is the only boundary of importance. The authors note in Table I that the upper boundary is equal to /2. is defined as the probability of rejecting good material. In this application, the lower bound has no relevance, and the entire should be used to calculate the upper bound.
Based on the approach suggested by the authors, a 97.5% confidence interval corresponds to the acceptable quality level (AQL) value. From MIL-STD-105E, the AQL is defined as the maximum percent defective that would be accepted 95% of the time.
Despite my concerns, the authors do a nice job of explaining the concepts without delving into the statistical theory.
Statistical Methods Manager
Thom R. Nichols responds:
I appreciate Mr. Walfish's interest in our article. I believe, however, that he fails to recognize that a 100% (1) confidence interval for success is the interval PL and PU. These are the smallest and largest binomial probabilities for which occurrences of the observed proportion y/n have a probability that is at least equal to /2. Therefore they are the upper and lower limits of the interval. In other words, the binomial probability, PL or PU, is the chance of observing y or more successes or failures, depending on the measure of interest. In n trials it is /2, an intrinsic two-tailedness. The resultant interval is often referred to as an exact interval, outlined by Collet, who references Fisher and Yates (all cited in my article), that can be defined with Fnum, den df (/2).
In a more practical sense, if success is a probability, there is an interval that can contain success, and this interval has a liberal interpretation, defined as PU, and a conservative one, PL. Conversely, if failure is a probability, there is an interval that can contain failure, and this interval has a liberal interpretation defined as PL, and a conservative interpretation as PU. The choice of stating the limits, either upper or lower, depends on the risk one is willing to take.