Received: 18 June 2018 Revised: 13 October 2019 Accepted: 30 December 2019
DOI: 10.1002/qre.2623
RESEARCH ARTICLE
Confidence limits for compliance testing using mixed
acceptance criteria
Pasan M. Edirisinghe
1
Thomas Mathew
2
T. S. G. Peiris
1
1
Department of Mathematics, University
of Moratuwa, Moratuwa, Sri Lanka
2
Department of Mathematics and
Statistics, University of Maryland
Baltimore County, Baltimore, Maryland,
United States
Correspondence
Thomas Mathew, Department of
Mathematics and Statistics, University of
Maryland Baltimore County, 1000 Hilltop
Circle, Baltimore, MD 21250, USA.
Email: mathew@umbc.edu
Funding information
Faculty of Graduate Studies; University of
Moratuwa; Sri Lanka
Abstract
For manufactured items sold by weight or volume, this article considers mixed
acceptance criteria that put limits on the sample mean or on an upper confi-
dence limit based on the sample mean and on the number of individual sample
units that are nonconforming. For a normally distributed quality characteristic
of interest, this article develops lower confidence limits for the mixed acceptance
criteria applying the concept of a generalized pivotal quantity and applying a
bias-corrected and accelerated parametric bootstrap. The accuracy of the confi-
dence limits is assessed using estimated coverage probabilities, and the results
are illustrated with an example.
KEYWORDS
BC
a
bootstrap, generalized pivotal quantity, lower confidence limit
1 INTRODUCTION
This article considers sampling plans that are based on mixed acceptance criteria (or joint acceptance criteria) that put
limits on the sample mean or on an upper confidence limit based on the sample mean, along with limits on the number
of individual sample units that are nonconforming, where the latter is measured in terms of an observation being below
a threshold. Thus, we consider a quality characteristic of interest of a manufactured item (weight, volume content, etc),
denoted by X , and suppose X ∼ N(,
2
). For a random sample X
1
, X
2
,. … , X
n
, let
X and S
2
denote the sample mean and
sample variance, respectively. The mixed acceptance criterion that may be used to assess quality of the manufactured items
is based on two requirements: a requirement on the sample mean and another one on the individual items in the sample.
The requirement on the mean states that the sample mean should exceed a specified limit, say V
1
(where V
1
could be a
labeled content for the product). The requirement on the individual items states that the number of items having content
below a defective level (say, c) is no more than r. Here, c and r are both specified quantities. Let Y
c
denote the number of
items having content below c, among the n items in the random sample. Then Y
c
∼ Binomial(n; ), where =Φ
(
c−
)
,
Φ(.) being the standard normal cumulative distribution function (cdf). Thus, the requirement that the number of items
having content below c to be no more than r can be expressed as the binomial cdf evaluated at c, where the binomial
success probability is now a function of the parameters and
2
. The mixed acceptance criterion, say
1
, is given by
1
= P
(
X ≥ V
1
, Y
c
≤ r
)
= g(,), say. (1)
The higher the value of
1
, the better, both from the consumer's point of view and the producer's point of view.
It should be noted that
1
is an implicit function of and
2
, and hence, we have also used the notation g(,).
Work completed while the second author was visiting the Department of Mathematics, University of Moratuwa, Sri Lanka.
Qual Reliab Engng Int. 2020;1–8. wileyonlinelibrary.com/journal/qre © 2020 John Wiley & Sons, Ltd. 1