1、Designation: E 2334 08An American National StandardStandard Practice forSetting an Upper Confidence Bound For a Fraction orNumber of Non-Conforming items, or a Rate of Occurrencefor Non-conformities, Using Attribute Data, When There is aZero Response in the Sample1This standard is issued under the f
2、ixed designation E 2334; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last
3、 revision or reapproval.1. Scope1.1 This practice presents methodology for the setting of anupper confidence bound regarding a unknown fraction orquantity non-conforming, or a rate of occurrence for noncon-formities, in cases where the method of attributes is used andthere is a zero response in a sa
4、mple. Three cases are consid-ered.1.1.1 The sample is selected from a process or a very largepopulation 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 finitelot of discrete items, and the number of non-conforming i
5、temsin 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 thesample is zero.1.2 Allowance is made for misclassification error in thisstandard, but only when misclassification rates are well under-stood or known and ca
6、n be approximated numerically.2. Referenced Documents2.1 ASTM Standards:2E 141 Practice for Acceptance of Evidence Based on theResults of Probability SamplingE 456 Terminology Relating to Quality and StatisticsE 1402 Terminology Relating to SamplingE 1994 Practice for Use of Process Oriented AOQL an
7、dLTPD Sampling PlansE 2586 Practice for Calculating and Using Basic Statistics2.2 ISO Standards:ISO 3534-1 StatisticsVocabulary and Symbols, Part 1:Probability and General Statistical Terms3ISO 3534-2 StatisticsVocabulary and Symbols, Part 2:Statistical Quality Control3NOTE 1Samples discussed in thi
8、s standard should meet the require-ments (or approximately so) of a probability sample as defined inTerminologies E 1402 or E 456.3. Terminology3.1 Definitions: Terminology E 456 provides a more exten-sive list of terms in E11 standards.3.1.1 attributes, method of, nmeasurement of quality bythe meth
9、od of attributes consists of noting the presence (orabsence) of some characteristic or attribute in each of the unitsin the group under consideration, and counting how many ofthe units do (or do not) possess the quality attribute, or howmany such events occur in the unit, group or area.3.1.2 confide
10、nce bound, nsee confidence limit.3.1.3 confidence coeffcient, nthe value, C, of the prob-ability associated with a confidence interval or statisticalcoverage interval. It is often expressed as a percentage.ISO 3534-13.1.4 confidence interval, nan interval estimate of apopulation parameter, calculate
11、d such that there is a givenlong-run probability that the parameter is included in theinterval.3.1.4.1 DiscussionA one-sided confidence interval is onefor which one of the limits is plus infinity, minus infinity, or anatural fixed limit (such as zero).3.1.5 confidence level, nsee confidence coeffcie
12、nt.3.1.6 confidence limit, nthe upper or lower limit of aconfidence interval.3.1.7 item, nan object or quantity of material on which aset of observations can be made.1This practice is under the jurisdiction ofASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommitte
13、e E11.30 on StatisticalQuality Control.Current edition approved Oct. 15, 2008. Published January 2009. Originallyapproved in 2003. Last previous edition approved in 2003 as E 2334031.2For referenced ASTM Standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.
14、org. For Annual Book of ASTMStandardsvolume information, refer to teh standards Document Summary page onthe ASTM website.3Available from American National Standards Institute (ANSI), 25 W. 43rd St.,4th Floor, New York, NY 10036, http:/www.ansi.org.1Copyright ASTM International, 100 Barr Harbor Drive
15、, PO Box C700, West Conshohocken, PA 19428-2959, United States.3.1.7.1 DiscussionAs used in this standard, “set” denotesa single variable (the defined attribute). The term “samplingunit” is also used to denote an “item” (see Practice E 141).3.1.8 non-conforming item, nan item containing at leastone
16、non-conformity. ISO 3534-23.1.8.1 DiscussionThe term “defective item” is also usedin this context.3.1.9 non-conformity, nthe non-fulfillment of a specifiedrequirement. ISO 3534-23.1.9.1 DiscussionThe term “defect” is also used in thiscontext.3.1.10 population, nthe totality of items or units ofmater
17、ial under consideration. E 25863.1.11 probability sample, na sample of which the sam-pling units have been selected by a chance process. At eachstep of selection, a specified probability of selection can beattached to each sampling unit available for selection.E 14023.1.12 sample, na group of observ
18、ations or test resultstaken from a larger collection of observations or test results ,which serves to provide information that may be used as a basisfor making a decision concerning the larger collection.E 25863.2 Definitions of Terms Specific to This Standard:3.2.1 zero response, nin the method of
19、attributes, thephrase used to denote that zero non-conforming items or zeronon-conformities were found (observed) in the item(s), unit,group or area sampled.3.3 Symbols:3.3.1 Athe assurance index.3.3.2 Cconfidence coefficient as a percent or as a prob-ability value.3.3.3 Cdthe confidence coefficient
20、 calculated that a pa-rameter meets a certain requirement, that is, that p#p0, that D# D0or that l # l0, when there is a zero response in thesample.3.3.4 Dthe number of non-conforming items in a finitepopulation containing N items.3.3.5 D0a specified value of D for which a researcher willcalculate a
21、 confidence coefficient for the statement, D # D0,when there is a zero response in the sample.3.3.6 Duthe upper confidence bound for the parameter D.3.3.7 Nthe number of items in a finite population.3.3.8 nthe sample size, that is, the number of items in asample.3.3.9 nRthe sample size required.3.3.
22、10 pa process fraction non-conforming.3.3.11 p0a specified value of p for which a researcher willcalculate a confidence coefficient, for the statement p # p0,when there is a zero response in the sample.3.3.12 puthe upper confidence bound for the parameter p.3.3.13 lthe mean number of non-conformitie
23、s (or events)over some area of interest for a Poisson process.3.3.14 l0a specific value of l for which a researcher willcalculate a confidence coefficient for the statement, l # l0,when there is a zero response in the sample.3.3.15 luthe upper confidence bound for the parameter l.3.3.16 u1the probab
24、ility of classifying a conforming itemas non-conforming; or of finding a nonconformity where noneexists.3.3.17 u2the probability of classifying a non-conformingitem as conforming; or of failing to find a non-conformitywhere one should have been found.4. Significance and Use4.1 In Case 1, the sample
25、is selected from a process or avery large population of interest. The population is essentiallyunlimited, and each item either has or has not the definedattribute. The population (process) has an unknown fraction ofitems p (long run average process non-conforming) having theattribute. The sample is
26、a group of n discrete items selected atrandom from the process or population under consideration,and the attribute is not exhibited in the sample. The objectiveis to determine an upper confidence bound, pu, for the unknownfraction p whereby one can claim that p # puwith someconfidence coefficient (p
27、robability) C. The binomial distribu-tion is the sampling distribution in this case.4.2 In Case 2, a sample of n items is selected at randomfrom a finite lot of N items. Like Case 1, each item either hasor has not the defined attribute, and the population has anunknown number, D, of items having the
28、 attribute. The sampledoes not exhibit the attribute. The objective is to determine anupper confidence bound, Du, for the unknown number D,whereby one can claim that D # Duwith some confidencecoefficient (probability) C. The hypergeometric distribution isthe sampling distribution in this case.4.3 In
29、 Case 3, there is a process, but the output is acontinuum, such as area (for example, a roll of paper or othermaterial, a field of crop), volume (for example, a volume ofliquid or gas), or time (for example, hours, days, quarterly, etc.)The sample size is defined as that portion of the “continuum”sa
30、mpled, and the defined attribute may occur any number oftimes over the sampled portion. There is an unknown averagerate of occurrence, l, for the defined attribute over the sampledinterval of the continuum that is of interest. The sample doesnot exhibit the attribute. For a roll of paper this might
31、beblemishes per 100 ft2; for a volume of liquid, microbes percubic litre; for a field of crop, spores per acre; for a timeinterval, calls per hour, customers per day or accidents perquarter. The rate, l, is proportional to the size of the interval ofinterest. Thus, if l = 12 blemishes per 100 ft2of
32、paper, this isequivalent to 1.2 blemishes per 10 ft2or 30 blemishes per 250ft2. It is important to keep in mind the size of the interval in theanalysis and interpretation. The objective is to determine anupper confidence bound, lu, for the unknown occurrence ratel, whereby one can claim that l # luw
33、ith some confidencecoefficient (probability) C. The Poisson distribution is thesampling distribution in this case.4.4 Avariation on Case 3 is the situation where the sampled“interval” is really a group of discrete items, and the definedattribute may occur any number of times within an item. Thismigh
34、t be the case where the continuum is a process producingdiscrete items such as metal parts, and the attribute is definedas a scratch. Any number of scratches could occur on anysingle item. In such a case the occurrence rate, l, might bedefined as scratches per 1000 parts or some similar metric.E2334
35、0824.5 In each case a sample of items or a portion of acontinuum is examined for the presence of a defined attribute,and the attribute is not observed (that is, a zero response). Theobjective is to determine an upper confidence bound for eitheran unknown proportion, p (Case 1), an unknown quantity,
36、D(Case 2), or an unknown rate of occurrence, l (Case 3). In thisstandard, confidence means the probability that the unknownparameter 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 usedto
37、 determine the sample size required to demonstrate a specificp, D or l with some degree of confidence. They can also beused to determine the degree of confidence achieved indemonstrating a specified p, D or l.4.6 In this standard allowance is made for misclassificationerror but only when misclassifi
38、cation rates are well understoodor known, and can be approximated numerically.4.7 It is possible to impose the language of classicalacceptance sampling theory on this method. Terms such as LotTolerance Percent Defective, Acceptable Quality Level, Con-sumer Quality Level are not used in this standard
39、. For moreinformation on these terms, see Practice E 1994.5. Procedure5.1 When a sample is inspected and a zero response isexhibited with respect to a defined attribute, we refer to thisevent as “all_zeros.” Formulas for calculating the probabilityof “all_zeros” in a sample are based on the binomial
40、, thehypergeometric and the Poisson probability distributions.When there is the possibility of misclassification error, adjust-ments to these distributions are used. This practice will clarifywhen each distribution is appropriate and how misclassificationerror is incorporated. Three basic cases are
41、considered asdescribed in Section 4. Formulas and examples for each caseare given below. Mathematical notes are given in AppendixX1.5.2 In some applications, the measurement method isknown to be fallible to some extent resulting in a significantmisclassification error. If experiments with repeated m
42、easure-ments have established the rates of misclassification, and theyare known to be constant, they should be included in thecalculating formulas. Two misclassification error probabilitiesare defined for this practice:5.2.1 Let u1be the probability of reporting a non-conforming item when the item i
43、s really conforming.5.2.2 Let u2be the probability of reporting a conformingitem when the item is really non-conforming.5.2.3 Almost all applications of this standard require that u1be known to be 0 (see 6.1.2).5.3 Formulas for upper confidence bounds in three cases:5.3.1 Case 1The item is a complet
44、ely discrete object andthe attribute is either present or not within the item. Only oneresponse is recorded per item (either go or no-go). The sampleitems originate from a process and hence the future populationof interest is potentially unlimited in extent so long as theprocess remains in statistic
45、al control. The item having theattribute is often referred to as a defective item or a non-conforming item or unit. The sample consists of n randomlyselected items from the population of interest. The n items areinspected for the defined attribute. The sampling distribution isthe binomial with param
46、eters p equal to the process (popula-tion) fraction non-conforming and n the sample size. Whenzero non-conforming items are observed in the sample (theevent “all_zeros”), and there are no misclassification errors, theupper confidence bound, pu, at confidence level C (0 C 1),for the population propor
47、tion non-conforming is:pu5 1 2 =n1 2 C (1)5.3.1.1 For the case with misclassification errors, when zeronon-conforming items are observed in the sample (all_zeros),the upper confidence bound, pu, at confidence level C is:pu51 2u12 =n1 2 C1 2u12u2!(2)5.3.1.2 Eq 2 reduces to Eq 1 when u1= u2= 0. To fin
48、d theminimum sample size required (nR) to state a confidence boundof puat confidence C if zero non-conforming items are to beobserved in the sample, solve Eq 2 for n. This is:nR5ln1 2 C!ln1 2 pu! 1 2u1! 1 puu2!(3)5.3.1.3 To find the confidence demonstrated (Cd)intheclaim that an unknown fraction non
49、-conforming p is no morethan a specified value, say p0, when zero non-conformances areobserved in a sample of n items solve Eq 2 for C. This is:Cd5 1 2 1 2 p0! 1 2u1! 1 p0u2!n(4)5.3.2 Case 2The item is a completely discrete object andthe attribute is either present or not within the item. Only oneresponse is recorded per item (either go or no-go). The sampleitems originate from a finite lot or population of N items. Thesample consists of n randomly selected items from among theN, without replacement. The population proportion defectiv
