ASTM E2334-09(2023)
(Practice)Standard Practice for Setting an Upper Confidence Bound for a Fraction or Number of Non-Conforming items, or a Rate of Occurrence for Non-Conformities, Using Attribute Data, When There is a Zero Response in the Sample
Standard Practice for Setting an Upper Confidence Bound for a Fraction or Number of Non-Conforming items, or a Rate of Occurrence for Non-Conformities, Using Attribute Data, When There is a Zero Response in the Sample
ABSTRACT
This practice presents methodology for the setting of an upper confidence bound regarding an unknown fraction or quantity non-conforming, or a rate of occurrence for nonconformities, in cases where the method of attributes is used and there is a zero response in a sample. Three cases are considered. In Case 1, the sample is selected from a process or a very large population of interest. In Case 2, a sample of n items is selected at random from a finite lot of N items. In Case 3, there is a process, but the output is a continuum, such as area (for example, a roll of paper or other material, a field of crop), volume (for example, a volume of liquid or gas), or time (for example, hours, days, quarterly, etc.) The sample size is defined as that portion of the �continuum� sampled, and the defined attribute may occur any number of times over the sampled portion.
SIGNIFICANCE AND USE
4.1 In Case 1, the sample is selected from a process or a very large population of interest. The population is essentially unlimited, and each item either has or has not the defined attribute. The population (process) has an unknown fraction of items p (long run average process non-conforming) having the attribute. The sample is a group of n discrete items selected at random from the process or population under consideration, and the attribute is not exhibited in the sample. The objective is to determine an upper confidence bound, pu, for the unknown fraction p whereby one can claim that p ≤ pu with some confidence coefficient (probability) C. The binomial distribution is the sampling distribution in this case.
4.2 In Case 2, a sample of n items is selected at random from a finite lot of N items. Like Case 1, each item either has or has not the defined attribute, and the population has an unknown number, D, of items having the attribute. The sample does not exhibit the attribute. The objective is to determine an upper confidence bound, Du, for the unknown number D, whereby one can claim that D ≤ Du with some confidence coefficient (probability) C. The hypergeometric distribution is the sampling distribution in this case.
4.3 In Case 3, there is a process, but the output is a continuum, such as area (for example, a roll of paper or other material, a field of crop), volume (for example, a volume of liquid or gas), or time (for example, hours, days, quarterly, etc.) The sample size is defined as that portion of the “continuum” sampled, and the defined attribute may occur any number of times over the sampled portion. There is an unknown average rate of occurrence, λ, for the defined attribute over the sampled interval of the continuum that is of interest. The sample does not exhibit the attribute. For a roll of paper, this might be blemishes per 100 ft2; for a volume of liquid, microbes per cubic litre; for a field of crop, spores per acre; for a time interval, ca...
SCOPE
1.1 This practice presents methodology for the setting of an upper confidence bound regarding a unknown fraction or quantity non-conforming, or a rate of occurrence for nonconformities, in cases where the method of attributes is used and there is a zero response in a sample. Three cases are considered.
1.1.1 The sample is selected from a process or a very large population of discrete items, and the number of non-conforming items in the sample is zero.
1.1.2 A sample of items is selected at random from a finite lot of discrete items, and the number of non-conforming items in the sample is zero.
1.1.3 The sample is a portion of a continuum (time, space, volume, area, etc.) and the number of non-conformities in the sample is zero.
1.2 Allowance is made for misclassification error in this practice, but only when misclassification rates are well understood or known and can be approximated numerically.
1.3 The values stated in inch-pound units are to be regarded as standard. No other units of measurement are included in this standard.
1.4 This ...
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This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E2334 − 09 (Reapproved 2023) An American National Standard
Standard Practice for
Setting an Upper Confidence Bound for a Fraction or
Number of Non-Conforming items, or a Rate of Occurrence
for Non-Conformities, Using Attribute Data, When There is a
Zero Response in the Sample
This standard is issued under the fixed designation E2334; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2. Referenced Documents
1.1 This practice presents methodology for the setting of an 2.1 ASTM Standards:
upper confidence bound regarding a unknown fraction or E141 Practice for Acceptance of Evidence Based on the
quantity non-conforming, or a rate of occurrence for Results of Probability Sampling
nonconformities, in cases where the method of attributes is E456 Terminology Relating to Quality and Statistics
used and there is a zero response in a sample. Three cases are E1402 Guide for Sampling Design
considered. E1994 Practice for Use of Process Oriented AOQL and
1.1.1 The sample is selected from a process or a very large LTPD Sampling Plans
population of discrete items, and the number of non- E2586 Practice for Calculating and Using Basic Statistics
conforming items in the sample is zero.
2.2 ISO Standards:
1.1.2 A sample of items is selected at random from a finite
ISO 3534-1 Statistics—Vocabulary and Symbols, Part 1:
lot of discrete items, and the number of non-conforming items
Probability and General Statistical Terms
in the sample is zero.
ISO 3534-2 Statistics—Vocabulary and Symbols, Part 2:
1.1.3 The sample is a portion of a continuum (time, space,
Statistical Quality Control
volume, area, etc.) and the number of non-conformities in the
NOTE 1—Samples discussed in this practice should meet the require-
sample is zero.
ments (or approximately so) of a probability sample as defined in Guide
E1402 or Terminology E456.
1.2 Allowance is made for misclassification error in this
practice, but only when misclassification rates are well under-
3. Terminology
stood or known and can be approximated numerically.
3.1 Definitions—Unless otherwise noted in this standard, all
1.3 The values stated in inch-pound units are to be regarded
terms relating to quality and statistics are defined in Terminol-
as standard. No other units of measurement are included in this
ogy E456.
standard.
3.1.1 attributes, method of, n—measurement of quality by
1.4 This standard does not purport to address all of the
the method of attributes consists of noting the presence (or
safety concerns, if any, associated with its use. It is the
absence) of some characteristic or attribute in each of the units
responsibility of the user of this standard to establish appro-
in the group under consideration, and counting how many of
priate safety, health, and environmental practices and deter-
the units do (or do not) possess the quality attribute, or how
mine the applicability of regulatory limitations prior to use.
many such events occur in the unit, group or area.
1.5 This international standard was developed in accor-
3.1.2 confidence bound, n—see confidence limit. E2586
dance with internationally recognized principles on standard-
3.1.3 confidence coeffıcient, n—see confidence level. E2586
ization established in the Decision on Principles for the
Development of International Standards, Guides and Recom-
3.1.4 confidence interval, n—an interval estimate [L, U]
mendations issued by the World Trade Organization Technical
with the statistics L and U as limits for the parameter θ and
Barriers to Trade (TBT) Committee.
1 2
This practice is under the jurisdiction of ASTM Committee E11 on Quality and For referenced ASTM Standards, visit the ASTM website, www.astm.org, or
Statistics and is the direct responsibility of Subcommittee E11.30 on Statistical contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Quality Control. Standards volume information, refer to the standard’s Document Summary page on
Current edition approved Jan. 1, 2023. Published February 2023. Originally the ASTM website.
approved in 2003. Last previous edition approved in 2018 as E2334 – 09(2018). Available from American National Standards Institute (ANSI), 25 W. 43rd St.,
DOI: 10.1520/E2334-09R23. 4th Floor, New York, NY 10036, http://www.ansi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2334 − 09 (2023)
with confidence level 1 – α, where Pr(L ≤ θ ≤ U) ≥ 1 – α. 3.3.5 D —a specified value of D for which a researcher will
E2586 calculate a confidence coefficient for the statement, D ≤ D ,
3.1.4.1 Discussion—The confidence level, 1 – α, reflects the when there is a zero response in the sample.
proportion of cases that the confidence interval [L, U] would
3.3.6 D —the upper confidence bound for the parameter D.
u
contain or cover the true parameter value in a series of repeated
3.3.7 N—the number of items in a finite population.
random samples under identical conditions. Once L and U are
3.3.8 n—the sample size, that is, the number of items in a
given values, the resulting confidence interval either does or
sample.
does not contain it. In this sense, “confidence” applies not to
the particular interval but only to the long run proportion of 3.3.9 n —the sample size required.
R
cases when repeating the procedure many times.
3.3.10 p—a process fraction non-conforming.
3.1.5 confidence level, n—the value 1-α, of the probability
3.3.11 p —a specified value of p for which a researcher will
associated with a confidence interval, often expressed as a
calculate a confidence coefficient, for the statement p ≤ p ,
percentage. E2586
when there is a zero response in the sample.
3.1.6 confidence limit, n—each of the limits, L and U, of a
3.3.12 p —the upper confidence bound for the parameter p.
u
confidence interval, or the limit of a one-sided confidence
3.3.13 λ—the mean number of non-conformities (or events)
interval. E2586
over some area of interest for a Poisson process.
3.1.7 item, n—an object or quantity of material on which a
3.3.14 λ —a specific value of λ for which a researcher will
set of observations can be made.
calculate a confidence coefficient for the statement, λ ≤ λ ,
3.1.7.1 Discussion—As used in this practice, “set” denotes a
when there is a zero response in the sample.
single variable (the defined attribute). The term “sampling
3.3.15 λ —the upper confidence bound for the parameter λ.
unit” is also used to denote an “item” (see Practice E141). u
3.3.16 θ —the probability of classifying a conforming item
3.1.8 non-conforming item, n—an item containing at least
as non-conforming; or of finding a nonconformity where none
one non-conformity. ISO 3534-2
exists.
3.1.8.1 Discussion—The term “defective item” is also used
in this context. 3.3.17 θ —the probability of classifying a non-conforming
item as conforming; or of failing to find a non-conformity
3.1.9 non-conformity, n—the non-fulfillment of a specified
where one should have been found.
requirement. ISO 3534-2
3.1.9.1 Discussion—The term “defect” is also used in this
4. Significance and Use
context.
4.1 In Case 1, the sample is selected from a process or a
3.1.10 population, n—the totality of items or units of
very large population of interest. The population is essentially
material under consideration. E2586
unlimited, and each item either has or has not the defined
3.1.11 probability sample, n—a sample in which the sam-
attribute. The population (process) has an unknown fraction of
pling units are selected by a chance process such that a
items p (long run average process non-conforming) having the
specified probability of selection can be attached to each
attribute. The sample is a group of n discrete items selected at
possible sample that can be selected. E1402
random from the process or population under consideration,
and the attribute is not exhibited in the sample. The objective
3.1.12 sample, n—a group of observations or test results
is to determine an upper confidence bound, p , for the unknown
taken from a larger collection of observations or test results,
u
fraction p whereby one can claim that p ≤ p with some
which serves to provide information that may be used as a basis u
confidence coefficient (probability) C. The binomial distribu-
for making a decision concerning the larger collection. E2586
tion is the sampling distribution in this case.
3.2 Definitions of Terms Specific to This Standard:
4.2 In Case 2, a sample of n items is selected at random
3.2.1 zero response, n—in the method of attributes, the
from a finite lot of N items. Like Case 1, each item either has
phrase used to denote that zero non-conforming items or zero
or has not the defined attribute, and the population has an
non-conformities were found (observed) in the item(s), unit,
unknown number, D, of items having the attribute. The sample
group, or area sampled.
does not exhibit the attribute. The objective is to determine an
3.3 Symbols:
upper confidence bound, D , for the unknown number D,
u
3.3.1 A—the assurance index, as a percent or a probability
whereby one can claim that D ≤ D with some confidence
u
value.
coefficient (probability) C. The hypergeometric distribution is
3.3.2 C—confidence coefficient as a percent or as a prob-
the sampling distribution in this case.
ability value.
4.3 In Case 3, there is a process, but the output is a
3.3.3 C —the confidence coefficient calculated that a pa-
d
continuum, such as area (for example, a roll of paper or other
rameter meets a certain requirement, that is, that p ≤ p , that
material, a field of crop), volume (for example, a volume of
D ≤ D or that λ ≤ λ , when there is a zero response in the
0 0
liquid or gas), or time (for example, hours, days, quarterly, etc.)
sample.
The sample size is defined as that portion of the “continuum”
3.3.4 D—the number of non-conforming items in a finite sampled, and the defined attribute may occur any number of
population containing N items. times over the sampled portion. There is an unknown average
E2334 − 09 (2023)
rate of occurrence, λ, for the defined attribute over the sampled described in Section 4. Formulas and examples for each case
interval of the continuum that is of interest. The sample does are given below. Mathematical notes are given in Appendix
not exhibit the attribute. For a roll of paper, this might be X1.
blemishes per 100 ft ; for a volume of liquid, microbes per
5.2 In some applications, the measurement method is
cubic litre; for a field of crop, spores per acre; for a time
known to be fallible to some extent resulting in a significant
interval, calls per hour, customers per day or accidents per
misclassification error. If experiments with repeated measure-
quarter. The rate, λ, is proportional to the size of the interval of
ments have established the rates of misclassification, and they
interest. Thus, if λ = 12 blemishes per 100 ft of paper, this is
are known to be constant, they should be included in the
equivalent to 1.2 blemishes per 10 ft or 30 blemishes per
calculating formulas. Two misclassification error probabilities
250 ft . It is important to keep in mind the size of the interval
are defined for this practice:
in the analysis and interpretation. The objective is to determine
5.2.1 Let θ be the probability of reporting a non-
an upper confidence bound, λ , for the unknown occurrence
u
conforming item when the item is really conforming.
rate λ, whereby one can claim that λ ≤ λ with some confidence
u
5.2.2 Let θ be the probability of reporting a conforming
coefficient (probability) C. The Poisson distribution is the
item when the item is really non-conforming.
sampling distribution in this case.
5.2.3 Almost all applications of this practice require that θ
be known to be 0 (see 6.1.2).
4.4 A variation on Case 3 is the situation where the sampled
“interval” is really a group of discrete items, and the defined
5.3 Formulas for upper confidence bounds in three cases:
attribute may occur any number of times within an item. This
5.3.1 Case 1—The item is a completely discrete object and
might be the case where the continuum is a process producing
the attribute is either present or not within the item. Only one
discrete items such as metal parts, and the attribute is defined
response is recorded per item (either go or no-go). The sample
as a scratch. Any number of scratches could occur on any
items originate from a process and hence the future population
single item. In such a case, the occurrence rate, λ, might be of interest is potentially unlimited in extent so long as the
defined as scratches per 1000 parts or some similar metric.
process remains in statistical control. The item having the
attribute is often referred to as a defective item or a non-
4.5 In each case, a sample of items or a portion of a
conforming item or unit. The sample consists of n randomly
continuum is examined for the presence of a defined attribute,
selected items from the population of interest. The n items are
and the attribute is not observed (that is, a zero response). The
inspected for the defined attribute. The sampling distribution is
objective is to determine an upper confidence bound for either
the binomial with parameters p equal to the process (popula-
an unknown proportion, p (Case 1), an unknown quantity, D
tion) fraction non-conforming and n the sample size. When
(Case 2), or an unknown rate of occurrence, λ (Case 3). In this
zero non-conforming items are observed in the sample (the
practice, confidence means the probability that the unknown
event “all_zeros”), and there are no misclassification errors, the
parameter is not more than the upper bound. More generally,
upper confidence bound, p , at confidence level C (0 < C <1),
u
these methods determine a relationship among sample size,
for the population proportion non-conforming is:
confidence and the upper confidence bound. They can be used
n
to determine the sample size required to demonstrate a specific
p 5 1 2 =1 2 C (1)
u
p, D, or λ with some degree of confidence. They can also be
5.3.1.1 Table 1 contains the calculated upper confidence
used to determine the degree of confidence achieved in
bound for the process fraction non-conforming when x = 0
demonstrating a specified p, D, or λ.
non-con
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