1、Designation: E2334 09 (Reapproved 2013)2An 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.1NOTESection 3 was editorially corrected in August 2013.2NOTETerms were editorially corrected in April 2016.1. Scope1.1 This practice presents methodology for the setting of anupper confidence bound regarding a unknown fraction orquantity non-conforming, or a
4、 rate of occurrence fornonconformities, in cases where the method of attributes isused and there is a zero response 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
5、zero.1.1.2 A sample of items is selected at random from a finitelot of discrete items, and the number of non-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 mad
6、e for misclassification error in thisstandard, but only when misclassification rates are well under-stood or known and can be approximated numerically.2. Referenced Documents2.1 ASTM Standards:2E141 Practice for Acceptance of Evidence Based on theResults of Probability SamplingE456 Terminology Relat
7、ing 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, Part 1:Probability and General Statistical TermsI
8、SO 3534-2 StatisticsVocabulary and Symbols, Part 2:Statistical Quality ControlNOTE 1Samples discussed in this standard should meet the require-ments (or approximately so) of a probability sample as defined inTerminologies E1402 or E456.3. Terminology3.1 DefinitionsUnless otherwise noted in this stan
9、dard, 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 each of the unitsin the group under consideration, and
10、 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. E25863.1.3 confidence coeffcient, nsee confidence level. E25863.1.4 confidence interval, nan interval estimate L, Uwith
11、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
12、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 to the1This practice is under the jurisdiction ofASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommit
13、tee E11.30 on StatisticalQuality Control.Current edition approved April 1, 2013. Published April 2013. Originallyapproved in 2003. Last previous edition approved in 2009 as E2334 09. DOI:10.1520/E2334-09R13E02.2For referenced ASTM Standards, visit the ASTM website, www.astm.org, orcontact ASTM Custo
14、mer Service at serviceastm.org. For Annual Book of ASTMStandardsvolume information, refer to thestandards 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.Copyright ASTM Internation
15、al, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1particular interval but only to the long run proportion of caseswhen repeating the procedure many times.3.1.5 confidence level, nthe value 1-, of the probabilityassociated with a confidence interval, often expres
16、sed 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 of observations can be made.3.1.7.1 DiscussionAs used in this standard, “set” deno
17、tesa single 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 DiscussionThe term “defective item” is also usedin this context.3.1.9 non-conformity
18、, 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 probability sample, na sample in which the sam-pling units are selected by a chan
19、ce 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 observations or test results,which serves to provide information that may be used as a b
20、asisfor 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 items or zeronon-conformities were found (observed) in the item(s), unit,group, or
21、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-rameter meets a certain requirement, that is, that p p0, that D D0or that 0, when ther
22、e is a zero response in the sample.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 statement, D D0,when there is a zero response in the sample.3.3.6 Duthe upper con
23、fidence 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-conforming.3.3.11 p0a specified value of p for which a researcher willcalculate a confid
24、ence 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 for a Poisson process.3.3.14 0a specific value of for which a researcher willc
25、alculate 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-conforming; or of finding a nonconformity where noneexists.3.3.17 2the probability o
26、f 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 population of interest. The population is essentiallyunlimited, and each item either has
27、 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 the process or population under consideration,and the attribute is not exhibite
28、d 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 sampling distribution in this case.4.2 In Case 2, a sample of n items is selected a
29、t 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. The objective is to determine anupper confidence bound, Du, for the unknown nu
30、mber D,whereby one can claim that D Duwith some confidencecoefficient (probability) C. The hypergeometric distribution isthe sampling distribution in this case.4.3 In 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 cro
31、p), 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 occur any number oftimes over the sampled portion. There is an unknown averagerate of occurrence,
32、 , 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 of liquid, microbes percubic litre; for a field of crop, spores per acre; for a timeinterval, cal
33、ls 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 isE2334 09 (2013)22equivalent to 1.2 blemishes per 10 ft2or 30 blemishes per 250ft2. It is important to keep in mind the size
34、of the interval in theanalysis and interpretation. The objective is to determine anupper confidence bound, u, for the unknown occurrence rate ,whereby one can claim that uwith some confidencecoefficient (probability) C. The Poisson distribution is thesampling distribution in this case.4.4 Avariation
35、 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 producingdiscrete items such as metal parts, and the attribute is definedas a scr
36、atch. 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 acontinuum is examined for the presence of a defined attribute,and the attribute is
37、 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). In thisstandard, confidence means the probability that the unknownparameter is not
38、 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 specificp, D or with some degree of confidence. They can also beused to determine the
39、degree of confidence achieved indemonstrating a specified p, D or .4.6 In this standard allowance is made for misclassificationerror but only when misclassification rates are well understoodor known, and can be approximated numerically.4.7 It is possible to impose the language of classicalacceptance
40、 sampling theory on this method. Terms such as LotTolerance Percent Defective, Acceptable Quality Level, Con-sumer Quality Level are not used in this standard. For moreinformation on these terms, see Practice E1994.5. Procedure5.1 When a sample is inspected and a zero response isexhibited with respe
41、ct 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 distributions.When there is the possibility of misclassification error, adjust-ments to these d
42、istributions 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 for each caseare given below. Mathematical notes are given in AppendixX1.5.2 In some
43、 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 misclassification, and theyare known to be constant, they should be included in thecalculating formu
44、las. 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 reporting a conformingitem when the item is really non-conforming.5.2.3 Almost all applicat
45、ions of this standard 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 within the item. Only oneresponse is recorded per item (either go or no-go). The sam
46、pleitems 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 as a defective item or a non-conforming item or unit. The sample consists of n rando
47、mlyselected 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
48、 (theevent “all_zeros”), and there are no misclassification errors, theupper confidence bound, pu, at confidence level C (0 C 1),for the population proportion non-conforming is:pu5 1 2 =n1 2 C (1)5.3.1.1 Table 1 contains the calculated upper confidenceTABLE 1 Upper 100C% Confidence Bound, pu, for th
49、e ProcessFraction Non-Conforming, p, When Zero non-conforming Unitsappear in a sample of Size, nn C=0.90 C=0.95 C=0.995 0.369043 0.450720 0.60189310 0.205672 0.258866 0.36904315 0.142304 0.181036 0.26435820 0.108749 0.139108 0.20567230 0.073881 0.095034 0.14230440 0.055939 0.072158 0.10874950 0.045007 0.058155 0.08798960 0.037649 0.048703 0.07388170 0.032359 0.041893 0.06367180 0.028372 0.036754 0.05593990 0.025260 0.032738 0.049881100 0.022763 0.029513 0.045007150 0.015233
