1、Designation: E2709 10Standard Practice forDemonstrating Capability to Comply with a Lot AcceptanceProcedure1This standard is issued under the fixed designation E2709; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last
2、 revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice provides a general methodology for evalu-ating single-stage or multiple-stage lot acceptance procedureswhich
3、 involve a quality characteristic measured on a numericalscale. This methodology computes, at a prescribed confidencelevel, a lower bound on the probability of passing a lotacceptance procedure, using estimates of the parameters of thedistribution of test results from the lot.1.2 For a prescribed lo
4、wer probability bound, the method-ology can also generate an acceptance limit table, whichdefines a set of test method outcomes (for example, sampleaverages and standard deviations) that would pass the multiple-stage procedure at a prescribed confidence level.1.3 This approach may be used for demons
5、trating compli-ance with in-process, validation, or lot-release specifications.1.4 The system of units for this practice is not specified.1.5 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to
6、establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relating to Quality and StatisticsE2234 Practice for Sampling a Stream of Product byAttributes Indexed by AQLE2281 Pr
7、actice for Process and Measurement CapabilityIndicesE2282 Guide for Defining the Test Result of a Test MethodE2586 Practice for Calculating and Using Basic StatisticsE2587 Practice for Use of Control Charts in StatisticalProcess Control3. Terminology3.1 DefinitionsSee Terminology E456 for a more ext
8、en-sive listing of terms in ASTM Committee E11 standards.3.1.1 characteristic, na property of items in a sample orpopulation which, when measured, counted or otherwise ob-served, helps to distinguish between the items. E22823.1.2 mean, nof a population, , average or expectedvalue of a characteristic
9、 in a population, of a sample X , sum ofthe observed values in a sample divided by the sample size.E25863.1.3 multiple-stage lot acceptance procedure, na proce-dure for accepting a lot that involves more than one stage ofsampling and testing a given quality characteristic and one ormore acceptance c
10、riteria per stage.3.1.4 standard deviation, nof a population, s, the squareroot of the average or expected value of the squared deviationof a variable from its mean of a sample, s, the square root ofthe sum of the squared deviations of the observed values in thesample divided by the sample size minu
11、s 1. E25863.1.5 test method, na definitive procedure that produces atest result. E22823.2 Definitions of Terms Specific to This Standard:3.2.1 acceptable parameter region, nthe set of values ofparameters characterizing the distribution of test results forwhich the probability of passing the lot acce
12、ptance procedureis greater than a prescribed lower bound.3.2.2 acceptance region, nthe set of values of parameterestimates that will attain a prescribed lower bound on theprobability of passing a lot acceptance procedure at a pre-scribed level of confidence.3.2.3 acceptance limit, nthe boundary of t
13、he acceptanceregion, for example, the maximum sample standard deviationtest results for a given sample mean.4. Significance and Use4.1 Lot acceptance procedures are used in industry forinspecting quality characteristics of raw materials, in-processproduct, and finished product. These procedures, tog
14、ether with1This practice is under the jurisdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.20 on Test MethodEvaluation and Quality Control.Current edition approved May 1, 2010. Published October 2010. Originallyapproved in 2009. Last previo
15、us edition approved in 2009 as E2709 09. DOI:10.1520/E2709-10.2For referenced 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
16、.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.process controls, comprise a quality control program. Foradditional information on process control see Practice E2281dealing with process capability evaluation and Practice E2587dealin
17、g with the use of control charts in statistical processcontrol.4.1.1 Lot inspection procedures classify quality characteris-tics as either attributes (measured on discrete scales such aspercent defective) or variables (measured on continuous scalessuch as length, weight, or concentration).4.1.2 Oper
18、ating characteristic curves, which plot the rela-tionship of the lot acceptance probability versus the true lotpercent defective, are used to evaluate the discriminatorypower of a given lot inspection procedure, or acceptancesampling plan, and are discussed in Practice E2234.4.2 This practice consid
19、ers inspection procedures that mayinvolve multiple-stage sampling, where at each stage one candecide to accept the lot or to continue sampling, and thedecision to reject the lot is deferred until the last stage.4.2.1 At each stage there are one or more acceptance criteriaon the test results; for exa
20、mple, limits on each individual testresult, or limits on statistics based on the sample of test results,such as the average, standard deviation, or coefficient ofvariation (relative standard deviation).4.3 The methodology in this practice defines an acceptanceregion for a set of test results from th
21、e lot such that, at aprescribed confidence level, the probability that a sample fromthe lot will pass the original lot acceptance procedure is greaterthan or equal to a prespecified lower bound.4.3.1 Having test results fall in the acceptance region is notequivalent to passing the original lot accep
22、tance procedure, butprovides assurance that a sample would pass the lot acceptanceprocedure with a specified probability.4.3.2 This information can be used for process demonstra-tion or validation.4.3.3 This information can be used for lot release (accep-tance), but the lower bound may be conservati
23、ve in some cases.4.3.4 If the results are to be applied to test results fromfuture lots from the same process, then it is assumed that theprocess is in a state of statistical control (see 4.1). If this is notthe case then there can be no guarantee that the probabilityestimates would be valid predict
24、ions of future process perfor-mance.4.4 This methodology was originally developed by J. S.Bergum (1-4)3for use in two specific quality characteristics ofdrug products in the pharmaceutical industry: content unifor-mity and dissolution, as respectively defined in chapters and of the United States Pha
25、rmacopeia (5).4.5 Mathematical derivations would be required that arespecific to the individual criteria of each test.5. Methodology5.1 The process for defining the acceptance limits, startingfrom the definition of the original lot acceptance procedure, isoutlined. A computer program is normally req
26、uired to producethe acceptable parameter region and acceptance limits.5.1.1 An important class of procedures is for the case wherethe quality characteristic is normally distributed. Particularinstructions for that case are given in this section.5.2 Express the probability of passing the given lot ac
27、cep-tance procedure as a function of parameters characterizing thedistribution of the quality characteristic for items in the lot.5.2.1 When the characteristic is normally distributed, pa-rameters are the mean () and standard deviation (s) of the lot.5.2.2 An expression for the exact probability of
28、passing thelot acceptance procedure may be intractable.Alower bound forthe probability may be used. For multiple stage tests, thefollowing lower bounds on the probability of passing theprocedure as a function of probabilities of passing stages, andon the probability of passing a stage having multipl
29、e criteria asa function of the probabilities of passing the criteria, may beuseful (4).P pass k stage procedure! $max $PS1!, PS2!, . , PSk!% (1)where:P(Si) = is the probability of passing stage i, evaluatedregardless of whether previous stages pass or not.PSi! 5 PCi1and Ci2. and Cim! $1(mj511PCij! (
30、2)where:P(Cij) = is the probability of passing the j-th criterion of mwithin the i-th stage.5.3 Determine the contour of the region of parameter valuesfor which the expression for the probability of passing thegiven lot acceptance procedure is at least equal to the requiredlower bound (LB) on the pr
31、obability of acceptance (p). Thisdefines the region of acceptable parameters.5.3.1 For a normally distributed population, this will be aregion under a curve in the half-plane where is on thehorizontal axis, s on the vertical axis, such as that depicted inFig. 1.5.4 For each value of a statistic or s
32、et of statistics, derive ajoint confidence region (confidence coefficient 1-a) for thedistribution parameters. The size of sample to be taken, n, andthe statistics to be used, must be predetermined.5.4.1 For a normally distributed lot, the method of Lindgren(6) constructs a simultaneous confidence r
33、egion of (, s)values from the sample average X and the sample standarddeviation s from a set of n test results. Let Zpand xp2denotepercentiles of the standard normal distribution and of thechi-square distribution with n-1 degrees of freedom, respec-tively. Given a confidence level 100(1-a), choose d
34、 and e suchthat (1-a) = (1-2d )(1-e). The values:e51=1aandd51=1a!/2meet this condition. ThenPHSX s/=nD2#Z21dJPHn 1!s2s2# x21eJ5 12d!1e!5 1a! (3)3The boldface numbers in parentheses refer to a list of references at the end ofthis standard.E2709 102The region for (, s), two-sided for , one-sided for s
35、,isaninverted triangle with a minimum vertex at ( X , 0), as depictedin Fig. 1.5.5 Determine the contour of the acceptance region, whichconsists of values of the statistics ( X , s) for which theconfidence region (confidence level 1-a) is entirely containedin the acceptable parameter region. This is
36、 the acceptance limit.5.5.1 For a normally distributed characteristic, the accep-tance limit takes the form of a table giving, for each value ofthe sample mean, the maximum value of the standard deviation(or coefficient of variation) that would meet these require-ments.5.5.2 Using a computer program
37、 that calculates confidencelimits for and s given sample mean X and standard deviations, the acceptance limit for a normally distributed characteristiccan be derived using an iterative loop over increasing values ofthe sample standard deviation s (starting with s = 0) until theconfidence limits hit
38、the boundary of the acceptable parameterregion, for each potential value of the sample mean.5.5.3 To select the size of sample to be taken, n, theprobability that sample statistics ( X , s) will lie withinacceptance limits should be evaluated over a range of values ofn, for values of population para
39、meters (, s) of practicalinterest, and for which probabilities of passing the given lotacceptance procedure are well above the lower bound. Thelarger the sample size n that is chosen, the larger will be theacceptance region and the tighter the distribution of thestatistics. Choose n so that the prob
40、ability of passing accep-tance limits is greater than a desired value.5.6 To use the acceptance limit, sample randomly from thebatch or lot. Evaluate statistics for the sample. If statistics fallwithin the acceptance limit, then there is 1-a confidence thatthe probability of acceptance is at least p
41、.6. Examples6.1 An example of an evaluation of a single-stage lotacceptance procedure is given in Appendix X1. An acceptancelimit table is shown for a sample size of 30, but other samplesizes may be considered.6.2 An example of an evaluation of a two-stage lot accep-tance procedure with one or more
42、acceptance criterion at eachstage is given in Appendix X2. An acceptance limit table isshown for a sample size of 30.7. Keywords7.1 acceptance limits; acceptance sampling inspection;multiple-stage lot acceptance procedures; simultaneous confi-dence regions; specificationsFIG. 1 Example of Acceptance
43、 Limit Contour Showing a Simultaneous Confidence Interval With 95 % and 99 % Lower Bound ContoursE2709 103APPENDIXESX1. EXAMPLE OF A SINGLE STAGE ACCEPTANCE PROCEDUREX1.1 A single-stage lot acceptance procedure is stated asfollows: Sample five units at random from the lot and measurea numerical qual
44、ity characteristic (Xi) of each unit. Criterion:Pass if all 5 individual units are between 95 and 105;otherwise, fail.X1.2 Assume that the test results follow a normal distribu-tion with mean and standard deviation s. Let Z denote thestandard normal variate, that is, Z is normally distributed with=0
45、ands =1.X1.3 The criterion is 95#Xi#105 for i = 1, , 5. There-fore:Ppassing test! 5 P95 2 !/s,Z , 105 2 !/s !#5(X1.1)For any given values of and s, the probability of passingStage 1 can be determined.X1.4 A simultaneous confidence region for and s isgenerated using the methods of Lindgren (6). See 5
46、.4.1.X1.5 The acceptance limit table for this example wasgenerated by a computer program and is listed in Table X1.1.The table corresponds to a sample size of 30 using a 95 %confidence interval and a 95 % lower bound, and it lists theoutput showing the upper bound on the sample standarddeviation for
47、 sample means between 97 and 103.X1.6 A SAS program for the generation of the acceptancetable follows. See Fig. X1.1.TABLE X1.1 Acceptance Limit Table (95 % Confidence Interval/95 % Coverage)Mean Standard Deviation96.0 0.27397.0 0.54698.0 0.81999.0 1.092100.0 1.350101.0 1.092102.0 0.819103.0 0.54610
48、4.0 0.273E2709 104FIG. X1.1 SAS ProgramE2709 105X2. EXAMPLE OF A MULTIPLE-STAGE ACCEPTANCE PROCEDUREX2.1 A multiple-stage lot acceptance procedure is stated asfollows:Stage 1: Sample five units at random from the lot andmeasure a numerical quality characteristic (Xi) of each unit.Criterion: Pass if
49、all 5 individual units are between 95 and 105;otherwise go to Stage 2.Stage 2: Randomly sample five additional units from the lotand measure a numerical quality characteristic (Xi) of eachunit. Criteria: Pass if the average of the 10 test results isbetween 97 and 103 and all 10 individual results are between90 and 110; otherwise fail.X2.2 To obtain the lower bound on the probability ofacceptance, assume that the test results follow a normaldistribution with mean and standard deviation s. Let Zdenote the stand
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