1、Designation: E2334 09 (Reapproved 2018) An 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
2、issued under the fixed designation E2334; 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 chan
3、ge since the last 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 fornonconformities, in cases where the method of attributes isused and there is a zero r
4、esponse in a sample. Three cases areconsidered.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
5、-conforming itemsin 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 thispractice, but only when misclassification rates are well under-stood o
6、r known and can be approximated numerically.1.3 The values stated in inch-pound units are to be regardedas standard. No other units of measurement are included in thisstandard.1.4 This international standard was developed in accor-dance with internationally recognized principles on standard-ization
7、established in the Decision on Principles for theDevelopment of International Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade (TBT) Committee.2. Referenced Documents2.1 ASTM Standards:2E141 Practice for Acceptance of Evidence Based on theResul
8、ts of Probability SamplingE456 Terminology Relating to Quality and StatisticsE1402 Guide for Sampling DesignE1994 Practice for Use of Process Oriented AOQL andLTPD Sampling PlansE2586 Practice for Calculating and Using Basic Statistics2.2 ISO Standards:3ISO 3534-1 StatisticsVocabulary and Symbols, P
9、art 1:Probability and General Statistical TermsISO 3534-2 StatisticsVocabulary and Symbols, Part 2:Statistical Quality ControlNOTE 1Samples discussed in this practice should meet the require-ments (or approximately so) of a probability sample as defined in GuideE1402 or Terminology E456.3. Terminolo
10、gy3.1 DefinitionsUnless otherwise noted in this standard, allterms relating to quality and statistics are defined in Terminol-ogy E456.3.1.1 attributes, method of, nmeasurement of quality bythe method of attributes consists of noting the presence (orabsence) of some characteristic or attribute in ea
11、ch 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 confidence bound, nsee confidence limit. E25861This practice is under the jurisdiction ofASTM Committee E11 o
12、n Quality andStatistics and is the direct responsibility of Subcommittee E11.30 on StatisticalQuality Control.Current edition approved Sept. 1, 2018. Published September 2018. Originallyapproved in 2003. Last previous edition approved in 2013 as E2334 09 (2013)2.DOI: 10.1520/E2334-09R18.2For referen
13、ced ASTM Standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.3Available from American National Standards Institute (ANSI), 25 W. 43rd
14、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 StatesThis international standard was developed in accordance with internationally recognized principles on standardization established in
15、the Decision on Principles for theDevelopment of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.13.1.3 confidence coeffcient, nsee confidence level. E25863.1.4 confidence interval, nan interval estimate L, Uwith
16、the statistics L and U as limits for the parameter andwith confidence level 1 , where Pr(L U) 1.E25863.1.4.1 DiscussionThe confidence level, 1 , reflects theproportion of cases that the confidence interval L, U wouldcontain or cover the true parameter value in a series of repeatedrandom samples unde
17、r identical conditions. Once L and U aregiven values, the resulting confidence interval either does ordoes not contain it. In this sense, “confidence” applies not tothe particular interval but only to the long run proportion ofcases when repeating the procedure many times.3.1.5 confidence level, nth
18、e value 1-, of the probabilityassociated with a confidence interval, often expressed as apercentage. E25863.1.6 confidence limit, neach of the limits, L and U, of aconfidence interval, or the limit of a one-sided confidenceinterval. E25863.1.7 item, nan object or quantity of material on which aset o
19、f observations can be made.3.1.7.1 DiscussionAs used in this practice, “set” denotes asingle variable (the defined attribute). The term “samplingunit” is also used to denote an “item” (see Practice E141).3.1.8 non-conforming item, nan item containing at leastone non-conformity. ISO 3534-23.1.8.1 Dis
20、cussionThe 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 ofmaterial under consideration. E25863.1.11
21、probability sample, na sample in which the sam-pling units are selected by a chance process such that aspecified probability of selection can be attached to eachpossible sample that can be selected. E14023.1.12 sample, na group of observations or test resultstaken from a larger collection of observa
22、tions or test results,which serves to provide information that may be used as a basisfor making a decision concerning the larger collection. E25863.2 Definitions of Terms Specific to This Standard:3.2.1 zero response, nin the method of attributes, thephrase used to denote that zero non-conforming it
23、ems or zeronon-conformities were found (observed) in the item(s), unit,group, or area sampled.3.3 Symbols:3.3.1 Athe assurance index, as a percent or a probabilityvalue.3.3.2 Cconfidence coefficient as a percent or as a prob-ability value.3.3.3 Cdthe confidence coefficient calculated that a pa-ramet
24、er meets a certain requirement, that is, that p p0, thatD D0or that 0, 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 confidence coefficient for the st
25、atement, 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.3.3.10 pa process fraction non-conformin
26、g.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 the mean number of non-conformities (or events)over some area of interest
27、 for a Poisson process.3.3.14 0a specific value of for which a researcher willcalculate a confidence coefficient for the statement, 0,when there is a zero response in the sample.3.3.15 uthe upper confidence bound for the parameter .3.3.16 1the probability of classifying a conforming itemas non-confo
28、rming; or of finding a nonconformity where noneexists.3.3.17 2the 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 is selected from a process or avery large populatio
29、n 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 a group of n discrete items selected atrandom from
30、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 (probability) C. The binomial distribu-tion is the samp
31、ling 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 attribute. The sampledoes not exhibit the attribute.
32、 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.E2334 09 (2018)24.3 In Case 3, there is a process, but the ou
33、tput 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”sampled, and the defined attribute may oc
34、cur any number oftimes over the sampled portion. There is an unknown averagerate of occurrence, , 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 beblemishes per 100 ft2; for a volume o
35、f 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, , is proportional to the size of the interval ofinterest. Thus, if = 12 blemishes per 100 ft2of paper, this isequivalent to 1.2 blemishes
36、per 10 ft2or 30 blemishes per250 ft2. It is important to keep in mind the size of the intervalin the analysis and interpretation. The objective is to determinean upper confidence bound, u, for the unknown occurrencerate , whereby one can claim that uwith some confidencecoefficient (probability) C. T
37、he 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. Thismight be the case where the continuum is a process pr
38、oducingdiscrete 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, , might bedefined as scratches per 1000 parts or some similar metric.4.5 In each case, a sample of items or a portion of ac
39、ontinuum 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, D(Case 2), or an unknown rate of occurrence, (Case 3). I
40、n thispractice, 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 determine the sample size required to demonstrate a speci
41、ficp, D,or with some degree of confidence. They can also beused to determine the degree of confidence achieved indemonstrating a specified p, D,or.4.6 In this practice, allowance is made for misclassificationerror but only when misclassification rates are well understoodor known, and can be approxim
42、ated 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, and con-sumer quality level are not used in this practice. For moreinformation on these terms, see Practice E1994.
43、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, thehypergeometric and the Poisson probability distributi
44、ons.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 considered asdescribed in Section 4. Formulas and examples
45、 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 measure-ments have established the rates of misclassificati
46、on, 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 1be the probability of reporting a non-conforming item when the item is really conforming.5.2.2 Let 2be the probability of report
47、ing a conformingitem when the item is really non-conforming.5.2.3 Almost all applications of this practice require that 1be 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 completely discrete object andthe attribute is either present or not
48、 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 statistical control. The item having theattribute is often referred to
49、 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 parameters 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,
