Variances and Confidence Intervals
Determine the minimum sample size for a two-sided normal tolerance interval with specified population coverage, confidence level, and an allowable coverage exceedance margin. The calculation follows the Guenther approximation used in PASS's Tolerance Intervals for Normal Data procedure.
A normal-data two-sided tolerance interval is written as x̄ ± kS, where x̄ is the sample mean, S is the sample standard deviation, and k is the tolerance factor. The interval is constructed so that the specified population proportion P is covered with confidence 1 − α.
PASS bases its normal-data calculation on Guenther's approximation. For a sample of size N, the tolerance factor is computed from the normal quantile for the target coverage and a lower-tail chi-square quantile with N − 1 degrees of freedom, with Guenther's correction applied:
The sample size is increased one subject at a time until the Guenther precision criterion is satisfied for the upper coverage value P′ = P + δ. In the PASS formulation, α′ = Pr(p ≥ P + δ) controls the probability that the realized coverage exceeds the target by more than the specified margin. The implemented stopping rule is:
The calculator reports the resulting minimum N and the Guenther tolerance factor k that would be used to construct the corresponding sample tolerance interval, x̄ ± kS.
PASS Example 1 uses a two-sided 95% normal tolerance interval covering 90% of the population, with δ = 0.05 and α′ = 0.05. The published result is N = 179 and k = 1.8084. The same inputs are the calculator's built-in validation target.
NCSS, LLC. Tolerance Intervals for Normal Data, PASS Sample Size Software, Chapter 830. The documentation states that the procedure is based primarily on Guenther (1977), with additional references to Howe (1969), Hahn and Meeker (1991), Krishnamoorthy and Mathew (2009), and others.
Howe, W. G. (1969). “Two-Sided Tolerance Limits for Normal Populations—Some Improvements.” Journal of the American Statistical Association, 64(326), 610–620.
Guenther, W. C. (1977). Sampling Inspection in Statistical Quality Control. Griffin’s Statistical Monographs, No. 37.