ASTM E2334-09(2013)e2
(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
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, cal...
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 standard, but only when misclassification rates are well understood or known and can be approximated numerically.
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´2
Designation:E2334 −09 (Reapproved 2013) 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.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Section 3 was editorially corrected in August 2013.
ε NOTE—Terms were editorially corrected in April 2016.
1. Scope LTPD Sampling Plans
E2586Practice for Calculating and Using Basic Statistics
1.1 This practice presents methodology for the setting of an
2.2 ISO Standards:
upper confidence bound regarding a unknown fraction or
ISO 3534-1Statistics—Vocabulary and Symbols, Part 1:
quantity non-conforming, or a rate of occurrence for
Probability and General Statistical Terms
nonconformities, in cases where the method of attributes is
ISO 3534-2Statistics—Vocabulary and Symbols, Part 2:
used and there is a zero response in a sample. Three cases are
Statistical Quality Control
considered.
1.1.1 The sample is selected from a process or a very large
NOTE 1—Samples discussed in this standard should meet the require-
population of discrete items, and the number of non- ments (or approximately so) of a probability sample as defined in
Terminologies E1402 or E456.
conforming items in the sample is zero.
1.1.2 Asample of items is selected at random from a finite
3. Terminology
lot of discrete items, and the number of non-conforming items
3.1 Definitions—Unlessotherwisenotedinthisstandard,all
in the sample is zero.
1.1.3 The sample is a portion of a continuum (time, space, terms relating to quality and statistics are defined in Terminol-
ogy E456.
volume, area etc.) and the number of non-conformities in the
sample is zero. 3.1.1 attributes, method of, n—measurement of quality by
the method of attributes consists of noting the presence (or
1.2 Allowance is made for misclassification error in this
absence) of some characteristic or attribute in each of the units
standard, but only when misclassification rates are well under-
in the group under consideration, and counting how many of
stood or known and can be approximated numerically.
the units do (or do not) possess the quality attribute, or how
many such events occur in the unit, group or area.
2. Referenced Documents
3.1.2 confidence bound, n—see confidence limit. E2586
2.1 ASTM Standards:
E141Practice for Acceptance of Evidence Based on the 3.1.3 confidence coeffıcient, n—see confidence level. E2586
Results of Probability Sampling
3.1.4 confidence interval, n—an interval estimate [L, U]
E456Terminology Relating to Quality and Statistics
with the statistics L and U as limits for the parameter θ and
E1402Guide for Sampling Design
with confidence level 1– α, where Pr(L ≤θ≤ U) ≥ 1– α.
E1994Practice for Use of Process Oriented AOQL and
E2586
3.1.4.1 Discussion—Theconfidencelevel,1– α,reflectsthe
proportion of cases that the confidence interval [L, U] would
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
containorcoverthetrueparametervalueinaseriesofrepeated
Statistics and is the direct responsibility of Subcommittee E11.30 on Statistical
random samples under identical conditions. Once L and U are
Quality Control.
Current edition approved April 1, 2013. Published April 2013. Originally
given values, the resulting confidence interval either does or
approved in 2003. Last previous edition approved in 2009 as E2334–09. DOI:
doesnotcontainit.Inthissense "confidence"appliesnottothe
10.1520/E2334-09R13E02.
For referenced ASTM Standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Standardsvolume information, refer to thestandard’s Document Summary page on Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
the ASTM website. 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
´2
E2334−09 (2013)
particular interval but only to the long run proportion of cases 3.3.9 n —the sample size required.
R
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 —aspecifiedvalueof pforwhicharesearcherwill
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 standard, “set” denotes
when there is a zero response in the sample.
a single variable (the defined attribute). The term “sampling
3.3.15 λ —the upper confidence bound for the parameter λ.
u
unit” is also used to denote an “item” (see Practice E141).
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
3.3.17 θ —the probability of classifying a non-conforming
in this context.
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
attribute.The population (process) has an unknown fraction of
3.1.11 probability sample, n—a sample in which the sam-
items p (long run average process non-conforming) having the
pling units are selected by a chance process such that a
attribute. The sample is a group of n discrete items selected at
specified probability of selection can be attached to each
random from the process or population under consideration,
possible sample that can be selected. E1402
and the attribute is not exhibited in the sample. The objective
3.1.12 sample, n—a group of observations or test results
istodetermineanupperconfidencebound,p ,fortheunknown
u
taken from a larger collection of observations or test results,
fraction p whereby one can claim that p ≤ p with some
u
whichservestoprovideinformationthatmaybeusedasabasis
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,
unknownnumber, D,ofitemshavingtheattribute.Thesample
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
whereby one can claim that D ≤ D with some confidence
3.3.1 A—the assurance index, as a percent or a probability
u
coefficient (probability) C. The hypergeometric distribution is
value.
the sampling distribution in this case.
3.3.2 C—confidence coefficient as a percent or as a prob-
4.3 In Case 3, there is a process, but the output is a
ability value.
continuum, such as area (for example, a roll of paper or other
3.3.3 C —the confidence coefficient calculated that a pa-
d
material, a field of crop), volume (for example, a volume of
rameter meets a certain requirement, that is, that p ≤ p , that D
liquidorgas),ortime(forexample,hours,days,quarterly,etc.)
≤ D orthatλ≤λ ,whenthereisazeroresponseinthesample.
0 0
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
3.3.5 D —aspecifiedvalueof Dforwhicharesearcherwill rateofoccurrence, λ,forthedefinedattributeoverthesampled
calculate a confidence coefficient for the statement, D ≤ D , interval of the continuum that is of interest. The sample does
when there is a zero response in the sample. not exhibit the attribute. For a roll of paper this might be
blemishes per 100 ft ; for a volume of liquid, microbes per
3.3.6 D —the upper confidence bound for the parameter D.
u
cubic litre; for a field of crop, spores per acre; for a time
3.3.7 N—the number of items in a finite population.
interval, calls per hour, customers per day or accidents per
3.3.8 n—the sample size, that is, the number of items in a quarter.Therate, λ,isproportionaltothesizeoftheintervalof
sample. interest. Thus, if λ = 12 blemishes per 100 ft of paper, this is
´2
E2334−09 (2013)
equivalent to 1.2 blemishes per 10 ft or 30 blemishes per 250 calculating formulas. Two misclassification error probabilities
ft .Itisimportanttokeepinmindthesizeoftheintervalinthe are defined for this practice:
analysis and interpretation. The objective is to determine an 5.2.1 Let θ be the probability of reporting a non-
upperconfidencebound, λ ,fortheunknownoccurrencerate λ, conforming item when the item is really conforming.
u
whereby one can claim that λ≤λ with some confidence 5.2.2 Let θ be the probability of reporting a conforming
u 2
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 standard require that θ
be known to be 0 (see 6.1.2).
4.4 AvariationonCase3isthesituationwherethesampled
5.3 Formulas for upper confidence bounds in three cases:
“interval” is really a group of discrete items, and the defined
5.3.1 Case 1—The item is a completely discrete object and
attribute may occur any number of times within an item. This
the attribute is either present or not within the item. Only one
might be the case where the continuum is a process producing
response is recorded per item (either go or no-go).The sample
discrete items such as metal parts, and the attribute is defined
items originate from a process and hence the future population
as a scratch. Any number of scratches could occur on any
of interest is potentially unlimited in extent so long as the
single item. In such a case the occurrence rate, λ, might be
process remains in statistical control. The item having the
defined as scratches per 1000 parts or some similar metric.
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
inspectedforthedefinedattribute.Thesamplingdistributionis
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
standard, confidence means the probability that the unknown
event“all_zeros”),andtherearenomisclassificationerrors,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
todeterminethesamplesizerequiredtodemonstrateaspecific p 51 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
demonstrating a specified p, D or λ.
TABLE 1 Upper 100C% Confidence Bound, p , for the Process
u
4.6 In this standard allowance is made for misclassification
Fraction Non-Conforming, p, When Zero non-conforming Units
appear in a sample of Size, n
errorbutonlywhenmisclassificationratesarewellunderstood
or known, and can be approximated numerically.
n C=0.90 C=0.95 C=0.99
5 0.369043 0.450720 0.601893
4.7 It is possible to impose the language of classical
10 0.205672 0.258866 0.369043
15 0.142304 0.181036 0.264358
acceptance sampling theory on this method.Terms such as Lot
20 0.108749 0.139108 0.205672
Tolerance Percent Defective, Acceptable Quality Level, Con-
30 0.073881 0.095034 0.142304
sumer Quality Level are not used in this standard. For more
40 0.055939 0.072158 0.108749
50 0.045007 0.058155 0.087989
information on these terms, see Practice E1994.
60 0.037649 0.048703 0.073881
70 0.032359 0.041893 0.063671
5. Procedure 80 0.028372 0.036754 0.055939
90 0.025260 0.032738 0.049881
5.1 When a sample is inspected and a zero response is
100 0.022763 0.029513 0.045007
150 0.015233 0.019773 0.030235
exhibited with respect to a defined attribute, we refer to this
175 0.013071 0.016973 0.025972
event as “all_zeros.” Formulas for calculating the probability
200 0.011447 0.014867 0.022763
of “all_zeros” in a sample are based on the binomial, the 225 0.010182 0.013226 0.020259
250 0.09168 0.011911 0.018252
hypergeometric and the Poisson probability distributions.
275 0.008338 0.010834 0.016607
When there is the possibility of misclassification error, adjust-
300 0.007646 0.009936 0.015233
ments to these distributions are used.
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
´2 ´2
Designation: E2334 − 09 (Reapproved 2013) E2334 − 09 (Reapproved 2013)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.
ε NOTE—Section 3 was editorially corrected in August 2013.
ε NOTE—Terms were editorially corrected in April 2016.
1. 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 standard, but only when misclassification rates are well understood or
known and can be approximated numerically.
2. Referenced Documents
2.1 ASTM Standards:
E141 Practice for Acceptance of Evidence Based on the Results of Probability Sampling
E456 Terminology Relating to Quality and Statistics
E1402 Guide for Sampling Design
E1994 Practice for Use of Process Oriented AOQL and LTPD Sampling Plans
E2586 Practice for Calculating and Using Basic Statistics
2.2 ISO Standards:
ISO 3534-1 Statistics—Vocabulary and Symbols, Part 1: Probability and General Statistical Terms
ISO 3534-2 Statistics—Vocabulary and Symbols, Part 2: Statistical Quality Control
NOTE 1—Samples discussed in this standard should meet the requirements (or approximately so) of a probability sample as defined in Terminologies
E1402 or E456.
3. Terminology
3.1 Definitions:Definitions
3.1.1 Terminology E456 provides a more extensive list of terms in E11 standards.—Unless otherwise noted in this standard, all
terms relating to quality and statistics are defined in Terminology E456.
This practice is under the jurisdiction of ASTM Committee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.30 on Statistical Quality
Control.
Current edition approved April 1, 2013. Published April 2013. Originally approved in 2003. Last previous edition approved in 2009 as E2334 – 09. DOI:
10.1520/E2334-09R13E01.10.1520/E2334-09R13E02.
For referenced ASTM Standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Standardsvolume information, refer to thestandard’s Document Summary page on the ASTM website.
Available from American National Standards Institute (ANSI), 25 W. 43rd St., 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
´2
E2334 − 09 (2013)
3.1.1 attributes, method of, n—measurement of quality by the method of attributes consists of noting the presence (or absence)
of some characteristic or attribute in each of the units in the group under consideration, and counting how many of the units do
(or do not) possess the quality attribute, or how many such events occur in the unit, group or area.
3.1.2 confidence bound, n—see confidence limit. E2586
3.1.3 confidence coeffıcient, n—the value, see C,confidence of level.the probability associated with a confidence interval or
statistical coverage interval. It is often expressed as a percentage.
ISO 3534-1, E2586
3.1.4 confidence interval, n—an interval estimate of a population parameter, calculated such that there is a given long-run
probability that the parameter is included in the interval.[L, U] with the statistics L and U as limits for the parameter θ and with
confidence level 1 – α, where Pr(L ≤ θ ≤ U) ≥ 1 – α. E2586
3.1.4.1 Discussion—
A one-sided confidence interval is one for which one of the limits is plus infinity, minus infinity, or a natural fixed limit (such as
zero).The confidence level, 1 – α, reflects the proportion of cases that the confidence interval [L, U] would contain or cover the
true parameter value in a series of repeated random samples under identical conditions. Once L and U are given values, the
resulting confidence interval either does or does not contain it. In this sense "confidence" applies not to the particular interval but
only to the long run proportion of cases when repeating the procedure many times.
3.1.5 confidence level, n—seethe confidence coefficient.value 1-α, of the probability associated with a confidence interval, often
expressed as a percentage. E2586
3.1.6 confidence limit, n—the upper or lowereach of the limits, L and U, of a confidence interval, or the limit of a one-sided
confidence interval. E2586
3.1.7 item, n—an object or quantity of material on which a set of observations can be made.
3.1.7.1 Discussion—
As used in this standard, “set” denotes a single variable (the defined attribute). The term “sampling unit” is also used to denote
an “item” (see Practice E141).
3.1.8 non-conforming item, n—an item containing at least one non-conformity. ISO 3534-2
3.1.8.1 Discussion—
The term “defective item” is also used in this context.
3.1.9 non-conformity, n—the non-fulfillment of a specified requirement. ISO 3534-2
3.1.9.1 Discussion—
The term “defect” is also used in this context.
3.1.10 population, n—the totality of items or units of material under consideration. E2586
3.1.11 probability sample, n—a sample in which the sampling units are selected by a chance process such that a specified
probability of selection can be attached to each possible sample that can be selected. E1402
3.1.12 sample, n—a group of observations or test results taken from a larger collection of observations or test results, which
serves to provide information that may be used as a basis for making a decision concerning the larger collection. E2586
3.2 Definitions of Terms Specific to This Standard:
3.2.1 probability sample, n—a sample of which the sampling units have been selected by a chance process. At each step of
selection, a specified probability of selection can be attached to each sampling unit available for selection. E1402
3.2.1 zero response, n—in the method of attributes, the phrase used to denote that zero non-conforming items or zero
non-conformities were found (observed) in the item(s), unit, group, or area sampled.
3.3 Symbols:
3.3.1 A—the assurance index, as a percent or a probability value.
3.3.2 C—confidence coefficient as a percent or as a probability value.
3.3.3 C —the confidence coefficient calculated that a parameter meets a certain requirement, that is, that p ≤ p , that D ≤ D
d 0 0
or that λ ≤ λ , when there is a zero response in the sample.
´2
E2334 − 09 (2013)
3.3.4 D—the number of non-conforming items in a finite population containing N items.
3.3.5 D —a specified value of D for which a researcher will calculate a confidence coefficient for the statement, D ≤ D , when
0 0
there is a zero response in the sample.
3.3.6 D —the upper confidence bound for the parameter D.
u
3.3.7 N—the number of items in a finite population.
3.3.8 n—the sample size, that is, the number of items in a sample.
3.3.9 n —the sample size required.
R
3.3.10 p—a process fraction non-conforming.
3.3.11 p —a specified value of p for which a researcher will calculate a confidence coefficient, for the statement p ≤ p , when
0 0
there is a zero response in the sample.
3.3.12 p —the upper confidence bound for the parameter p.
u
3.3.13 λ—the mean number of non-conformities (or events) over some area of interest for a Poisson process.
3.3.14 λ —a specific value of λ for which a researcher will calculate a confidence coefficient for the statement, λ ≤ λ , when
0 0
there is a zero response in the sample.
3.3.15 λ —the upper confidence bound for the parameter λ.
u
3.3.16 θ —the probability of classifying a conforming item as non-conforming; or of finding a nonconformity where none
exists.
3.3.17 θ —the probability of classifying a non-conforming item as conforming; or of failing to find a non-conformity where one
should have been found.
4. 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, p , for the unknown fraction p whereby one can claim that p ≤ p with some confidence coefficient
u u
(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, D , for the unknown number D, whereby one can claim
u
that D ≤ D with some confidence coefficient (probability) C. The hypergeometric distribution is the sampling distribution in this
u
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 ft ; for a volume
of liquid, microbes per cubic litre; for a field of crop, spores per acre; for a time interval, calls per hour, customers per day or
accidents per quarter. The rate, λ, is proportional to the size of the interval of interest. Thus, if λ = 12 blemishes per 100 ft of paper,
2 2
this is equivalent to 1.2 blemishes per 10 ft or 30 blemishes per 250 ft . It is important to keep in mind the size of the interval
in the analysis and interpretation. The objective is to determine an upper confidence bound, λ , for the unknown occurrence rate
u
λ, whereby one can claim that λ ≤ λ with some confidence coefficient (probability) C. The Poisson distribution is the sampling
u
distribution in this case.
4.4 A variation on Case 3 is the situation where the sampled “interval” is really a group of discrete items, and the defined
attribute may occur any number of times within an item. This might be the case where the continuum is a process producing
discrete items such as metal parts, and the attribute is defined as a scratch. Any number of scratches could occur on any single item.
In such a case the occurrence rate, λ, might be defined as scratches per 1000 parts or some similar metric.
4.5 In each case a sample of items or a portion of a continuum is examined for the presence of a defined attribute, and the
attribute is not observed (that is, a zero response). The objective is to determine an upper confidence bound for either an unknown
proportion, p (Case 1), an unknown quantity, D (Case 2), or an unknown rate of occurrence, λ (Case 3). In this standard, confidence
means the probability that the unknown parameter is not more than the upper bound. More generally, these methods determine a
relationship among sample size, confidence and the upper confidence bound. They can be used to determine the sample size
required to demonstrate a specific p,D or λ with some degree of confidence. They can also be used to determine the degree of
confidence achieved in demonstrating a specified p,D or λ.
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4.6 In this standard allowance is made for misclassification error but only when misclassification rates are well understood or
known, and can be approximated numerically.
4.7 It is possible to impose the language of classical acceptance sampling theory on this method. Terms such as Lot Tolerance
Percent Defective, Acceptable Quality Level, Consumer Quality Level are not used in this standard. For more information on these
terms, see Practice E1994.
5. Procedure
5.1 When a sample is inspected and a zero response is exhibited with respect to a defined attribute, we refer to this event as
“all_zeros.” Formulas for calculating the probability of “all_zeros” in a sample are based on the binomial, the hypergeometric and
the Poisson probability distributions. When there is the possibility of misclassification error, adjustments to these distributions are
used. This practice will clarify when each distribution is appropriate and how misclassification error is incorporated. Three basic
cases are considered as described in Section 4. Formulas and examples for each case are given below. Mathematical notes are given
in Appendix X1.
5.2 In some applications, the measurement method is known to be fallible to some extent resulting in a significant
misclassification error. If experiments with repeated measurements have established the rates of misclassification, and they are
known to be constant, they should be included in the calculating formulas. Two misclassification error probabilities are defined for
this practice:
5.2.1 Let θ be the probability of reporting a non-conforming item when the item is really conforming.
5.2.2 Let θ be the probability of reporting a conforming item when t
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