1、Designation: E 2334 03e1An 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
2、 fixed 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 (e) indicates an editorial change since the l
3、ast revision or reapproval.e1NOTEEq 4 was corrected editorially in May 2007.1. 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 at
4、tributes is used andthere is a zero response in a sample. 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 o
5、f 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 made for misclassification error in thisstandard, but only when misclass
6、ification rates are well under-stood or known and can 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 Samplin
7、gE 1994 Practice for Use of Process Oriented AOQL andLTPD Sampling Plans2.2 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 this stan
8、dard should meet the require-ments (or approximately so) of a probability sample as defined inTerminologies E 1402 or E 456.3. Terminology3.1 Definitions:3.1.1 attributes, method of, nmeasurement of quality bythe method of attributes consists of noting the presence (orabsence) of some characteristic
9、 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. E 4563.1.2 confidence bound, nsee confidence limit.3.1.3 confidence coeffcient, nthe value, C,
10、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 level, nsee confidence coeffcient.3.1.5 confidence limit, neach of the limits, T1and T2,ofthe two sided confidence interval, or the limit T of t
11、he onesided confidence interval. ISO 3534-13.1.6 one sided confidence interval, nwhen T is a functionof the observed values such that, u being a populationparameter to be estimated, the probability P (T $ u)ortheprobability P (T # u) is at least equal to C where C is a fixedpositive number less than
12、 1. The interval from the smallestvalue of u up to T or the interval from T to the largest possiblevalue of u is a one sided, C, confidence interval for u.ISO 3534-11This practice is under the jurisdiction ofASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee
13、 E11.30 on DataAnalysis.Current edition approved Oct. 1, 2003. Published February 2004.2For referenced ASTM Standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandardsvolume information, refer to teh standards Document Summary
14、 page onthe ASTM website.3Available from American National Standards Institute, 11 W. 42nd Street, 13thFloor, New York, NY 10036.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.3.1.7 non-conformity, nthe non-fulfillment of a specifie
15、drequirement. ISO 3534-23.1.7.1 DiscussionThe term “defect” is also used in thiscontext.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 population, nthe totality of items or units of ma
16、te-rial under consideration. E 4563.1.10 sample, na group of items, observations or testresults, or portion of material taken from a large collection ofitems or quantity of material, which serves to provide infor-mation that may be used as a basis for making a decisionconcerning the larger collectio
17、n or quantity. E 4563.1.11 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 item, nan object or quantity of mater
18、ial on which aset of observations can be made. E 4563.1.12.1 DiscussionAs used in this standard, “set” de-notes a single variable (the defined attribute). The term“sampling unit” is also used to denote an “item” (see PracticeE 141).3.2 Definitions of Terms Specific to This Standard:3.2.1 zero respon
19、se, 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 area sampled.Symbols:A = the assurance indexC = confidence coefficient as a percent or as a probabilityvalueCd= the confidence coeffic
20、ient calculated that a parametermeets a certain requirement, that is, that p # p0, thatD # D0or that l # l0, when there is a zero responsein the sampleD = the number of non-conforming items in a finitepopulation containing N itemsD0= a specified value of D for which a researcher willcalculate a conf
21、idence coefficient for the statement, D# D0, when there is a zero response in the sampleDu= the upper confidence bound for the parameter DN = the number of items in a finite populationn = the sample size, that is, the number of items in asamplenR= the sample size requiredp = a process fraction non-c
22、onformingp0= a 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 samplepu= the upper confidence bound for the parameter pl = the mean number of non-conformities (or events) oversome area of interest for a
23、 Poisson processl0= a 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 samplelu= the upper confidence bound for the parameter lu1= the probability of classifying a conforming item asnon-conforming; or of
24、finding a nonconformity wherenone existsu2= the probability of classifying a non-conforming itemas conforming; or of failing to find a non-conformitywhere one should have been found4. Significance and Use4.1 In Case 1, the sample is selected from a process or avery large population of interest. The
25、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 the process or popu
26、lation 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 sampling distribution
27、 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. The objective is
28、 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 Case 3, there is a process, but the output is acontinuum, such as are
29、a (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 occur any number oftimes over the
30、 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 beblemishes per 100 ft2; for a volume of liquid, microbes percubic lit
31、re; 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 paper, this isequivalent to 1.2 blemishes per 10 ft2or 30 blemishes pe
32、r 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 # luwith some confidencecoefficient (probability) C. The Poisson distributi
33、on 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 producingE233403e12disc
34、rete 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.4.5 In each case a sample of items or a portion of acontinuum is
35、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, l (Case 3). In thisstan
36、dard, 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 specificp, D or
37、 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 misclassification rates are well understoodor known, and can be approximated nu
38、merically.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. For moreinformation on these terms, see Practice E 1994.5. Procedu
39、re5.1 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 distributions.When t
40、here 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 for each
41、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 misclassification, and th
42、eyare 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 is really conforming.5.2.2 Let u2be the probability of reporting a co
43、nformingitem 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 completely discrete object andthe attribute is either present or not within
44、 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 as a d
45、efective 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 a
46、nd 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 proportion non-conforming is:pu5 1 2 =n1 2 C (1)5.3.1.1 For the case with
47、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 find theminimum sample size required (nR) to state a confidence boundof
48、 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) in theclaim that an unknown fraction non-conforming p is no morethan a specified value, say p0, when zero
49、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 defective isp = D/N where the unknown D is the integer number ofnon-conforming (defective) item
