1、Designation: E141 10An American National StandardStandard Practice forAcceptance of Evidence Based on the Results of ProbabilitySampling1This standard is issued under the fixed designation E141; the number immediately following the designation indicates the year oforiginal adoption or, in the case o
2、f 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 revision or reapproval.1. Scope1.1 This practice presents rules for accepting or rejectingevidence based on a sample. Statistical
3、 evidence for thispractice is in the form of an estimate of a proportion, anaverage, a total, or other numerical characteristic of a finitepopulation or lot. It is an estimate of the result which wouldhave been obtained by investigating the entire lot or populationunder the same rules and with the s
4、ame care as was used for thesample.1.2 One purpose of this practice is to describe straightfor-ward sample selection and data calculation procedures so thatcourts, commissions, etc. will be able to verify whether suchprocedures have been applied. The methods may not give leastuncertainty at least co
5、st, they should however furnish areasonable estimate with calculable uncertainty.1.3 This practice is primarily intended for one-of-a-kindstudies. Repetitive surveys allow estimates of sampling uncer-tainties to be pooled; the emphasis of this practice is onestimation of sampling uncertainty from th
6、e sample itself. Theparameter of interest for this practice is effectively a constant.Thus, the principal inference is a simple point estimate to beused as if it were the unknown constant, rather than, forexample, a forecast or prediction interval or distributiondevised to match a random quantity of
7、 interest.1.4 A system of units is not specified in this standard.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 establish appro-priate safety and health practices and determine the app
8、lica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E105 Practice for Probability Sampling of MaterialsE122 Practice for Calculating Sample Size to Estimate,With Specified Precision, the Average for a Characteristicof a Lot or ProcessE456 Terminology Relatin
9、g to Quality and StatisticsE1402 Guide for Sampling DesignE2586 Practice for Calculating and Using Basic Statistics3. Terminology3.1 DefinitionsRefer to Terminology E456 for definitionsof other statistical terms used in this practice.3.1.1 audit subsample, na small subsample of a sampleselected for
10、review of all sample selection and data collectionprocedures.3.1.2 equal complete coverage result, nthe numericalcharacteristic of interest calculated from observations made bydrawing randomly from the frame, all of the sampling unitscovered by the frame.3.1.2.1 DiscussionLocating the units and eval
11、uating themare supposed to be done in exactly the same way and at thesame time as was done for the sample. The quantity itself isdenoted u.The equal complete coverage result is never actuallycalculated. Its purpose is to serve as the objectively definedconcrete goal of the investigation. The quantit
12、y u may be thepopulation mean, (Y), total (Y), median (M), the proportion (P),or any other such quantity.3.1.3 frame, na list, compiled for sampling purposes,which designates all of the sampling units (items or groups) ofa population or universe to be considered in a specific study.E14023.1.4 probab
13、ility 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. E14023.1.5 replicate subsamples, na number of disjointsamples, each one separately drawn from the frame in
14、accordwith the same probability sampling plan.3.1.6 sample, na group of observations or test results,taken 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.E25861This practic
15、e is under the jurisdiction ofASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.10 on Sampling /Statistics.Current edition approved May 15, 2010. Published August 2010. Originallyapproved in 1959. Last previous edition approved in 2003 as E141 91 (2003)
16、1.DOI: 10.1520/E0141-10.2For 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.1Copyright ASTM International, 100 Ba
17、rr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.3.1.7 sampling unit, nan item, group of items, or segmentof material that can be selected as part of a probabilitysampling plan. E14024. Significance and Use4.1 This practice is designed to permit users of samplesurvey dat
18、a to judge the trustworthiness of results from suchsurveys. Practice E105 provides a statement of principles forguidance of ASTM technical committees and others in thepreparation of a sampling plan for a specific material. GuideE1402 describes the principal types of sampling designs.Practice E122 ai
19、ds in deciding on the required sample size.4.2 Section 5 gives extended definitions of the conceptsbasic to survey sampling and the user should verify that suchconcepts were indeed used and understood by those whoconducted the survey. What was the frame? How large (ex-actly) was the quantity N? How
20、was the parameter u estimatedand its standard error calculated? If replicate subsamples werenot used, why not? Adequate answers should be given for allquestions. There are many acceptable answers to the lastquestion.4.3 If the sample design was relatively simple, such assimple random or stratified,
21、then fully valid estimates ofsampling variance are easily available. If a more complexdesign was used then methods such as discussed in Ref (1)3orin Guide E1402 may be acceptable. Use of replicate sub-samples is the most straightforward way to estimate samplingvariances when the survey design is com
22、plex.4.4 Once the survey procedures that were used satisfySection 5, see if any increase in sample size is needed. Thecalculations for making it objectively are described in Section6.4.5 Refer to Section 7 to guide in the interpretation of theuncertainty in the reported value of the parameter estima
23、te, u,that is, the value of its standard error, se(u). The quantity se(u)should be reviewed to verify that the risks it entails arecommensurate with the size of the sample.4.6 When the audit subsample shows that there was reason-able conformity with prescribed procedures and when theknown instances
24、of departures from the survey plan can beshown to have no appreciable effect on the estimate, the valueof uis appropriate for use.5. Concepts and Procedures of Sampling5.1 Probability sampling is a procedure by which oneobtains a result from a selected set of sampling units that willagree, within ca
25、lculable limits of variation, with the equalcomplete coverage result. Probability sampling plans includeinstructions for using either (1) prepared tables of randomnumbers, (2) computer algorithms to generate pseudo-randomnumbers, or (3) certifiably honest physical devices to select thesample units s
26、o that inferences may be drawn from the testresults and decisions may be made with risks correctly calcu-lated by probability theory.5.1.1 Such plans are defined and their relative advantagesdiscussed in Guide E1402 and Refs (1-3).5.2 Procedures must be described in written form. Partiesinterested i
27、n collecting data should agree on the importance ofknowing u and its definition including measurement methods.The frame shall be carefully and explicitly constructed. Everysampling unit in the frame (1) has a unique serial number,which may be preassigned or determined by some definite ruleand (2) ha
28、s an addressa complete and clear instruction (orrules for its formulation) as to where and when to make theobservation or evaluation. Address instructions should refer toconcrete clerical materials such as directories, dials of clocksor of meters, ledgers, maps, aerial photographs, etc. Duplicatesin
29、 the frame shall be eliminated. N shall be well established.Random numbers (or a certifiably honest physical randomdevice) shall dictate selection of the sample. There shall be nosubstitution of one sampling unit for another. The method ofsample selection shall permit calculation of a standard error
30、 ofthe estimate. The use of replicate subsamples is recommended(see 5.4).An audit subsample should be selected and processedand any departures from prescribed measurement methods andlocation instructions noted (see 5.5). A report should list uandits standard error with the degrees of freedom in the
31、se(u).5.3 Parameter DefinitionThe equal complete coverageresult may or may not be acceptable evidence. Whether it isacceptable depends on many considerations such as definitions,method of test, care exercised in the testing, completeness ofthe frame, and on other points not to be settled by statisti
32、caltheory since these points belong to the subject matter, and arethe same whether one uses sampling or not. Mistakes, whetherin testing, counting, or weighing will affect the result of acomplete coverage just as such mistakes will affect the sampleresult. By a more expensive method of measurement o
33、r moreelaborate sampling frame, it may be possible to define aquantity, u8, as a target parameter or ideal goal of an investi-gation. Criticism that holds u to be an inappropriate goalshould demonstrate that the numerical difference between uand u8 is substantial. Measurements may be imprecise but s
34、olong as measurement errors are not too biased, a large size ofthe lot or population, N, insures that u and u8 are essentiallyequal.5.4 Replicate SubsamplesWhen appropriate, separatelaboratories should each work on separate replicate subsamplesand teams of investigators should be assigned to separat
35、ereplicate subsamples. This approach insures that the calculatedstandard error will not be a systematic underestimate. Suchsubsamples were called interpenetrating in Ref (4) where manyof their basic properties were described. See Ref (5) for furthertheory and applications.5.4.1 For some types of mat
36、erial, a sample selected withuniform spacing along the frame (systematic sample) hasincreased precision over a selection made with randomlyvarying spacings (simple random sample). Examples includesampling mineral ore or grain from a conveyor belt or samplingfrom a list of households along a street.
37、If the systematicsample is obtained by a single random start the plan is then aprobability sampling plan, but it does not permit calculating thestandard error as required by this practice. After dividing the3The boldface numbers in parentheses refer to a list of references at the end ofthis standard
38、.E141 102sample size by an integer k (such as k =4ork = 10) and usinga random start for each of k replicate subsamples, some of theincreased precision of systematic sampling (and a standarderror on k 1 degrees of freedom) can be achieved.5.5 An audit subsample of the survey sample should betaken for
39、 review of all procedures from use of the randomnumbers through locating and measurement, to editing, coding,data entry and tabulation. Selection of the audit subsample maybe done by putting the n sample observations in order as theyare collected, calculating the nearest integer to=n , or someother
40、convenient integer, and taking this number to be thespacing for systematic selection of the audit subsample.As fewas 10 observations may be adequate. The review shoulduncover any gross departures from prescribed practices or anyconceptual misunderstandings in the definitions. If the auditsubsample i
41、s large enough (say 30 observations or more) theregression of audited values on initial observations may beused to calibrate the estimate. This technique is the method oftwo-phase sampling as discussed in Ref (1). Helpful discussionof an audit appears in Ref (2).5.6 The estimate is a quantity calcul
42、ated on the n sampleobservations in the same way as the equal complete coverageresult u would have been calculated from the entire set of Npossible observations of the population; the symbol udenotesthe estimate. In calculating u, replicate subsample membershipis ignored.5.6.1 An estimate has a samp
43、ling distribution induced fromthe randomness in sample selection. The equal completecoverage result is effectively a constant while any estimate isonly the value from one particular sample. Thus, there is amean value of the sampling distribution and there is also astandard deviation of the sampling
44、distribution.5.7 The standard error is the quantity computed from theobservations as an estimate of the sampling standard deviationof the estimate; se(u) denotes the standard error.5.7.1 When u is the population average of the N quantitiesand a simple random sample of size n was drawn, then thesampl
45、e average ybecomes the usual estimate u, where:u5 y5(i 5 1nyi/ n (1)The quantities y1, y2, ., yndenote the observations. Thestandard error is calculated as:seu! 5 sey! 5(i 5 1nyi2 y!2/ nn 2 1! (2)There are n 1 degrees of freedom in this standard error.5.7.1.1 ExampleWhen the observations are:81.6, 7
46、8.7, 79.7, 78.3, 80.9, 79.5, 79.8, 80.3, 79.5, 80.7then y= 79.90 and se(y) = 0.32.5.7.2 Finite Population Correction (fpc)Multiplying se(y)by=1 2 n / N is always correct when the goal of the surveyis to estimate the finite population mean (u = Y). If randommeasurement error exists in the observation
47、s, then u8 based ona reference measurement method may be a more appropriatesurvey goal than u (see 5.3). If so, then se(y) would be furtheradjusted upward by an amount somewhat less than the down-ward adjustment of the fpc. Both of these adjustments are oftennumerically so small that these adjustmen
48、ts may be omittedleaving se(y) of Eq 2 as a slight overestimate.5.7.2.1 ExampleUsing the previous data and if N = 50,then se(y) becomes se(y) = 0.28 after applying the fpc.5.7.3 Proportions and Total CountsIf the quantity ofinterest is (a) a proportion or (b) a total and the sample issimple random t
49、hen the above formulas are still applicable. Aproportion is the mean of zeroes and ones, while the total is aconstant times the mean.5.7.3.1 When u is taken to be the population proportion(u = P) thenu5 p 5 (yi/n 5 a/n (3)where a is the number of units in the sample with theattribute, andsep!5=p1 2 p!/n 2 1! (4)5.7.3.2 When u = the population total (u = Y) thenu5 Np and seu! 5 N sep! (5)5.7.3.3 ExampleIf a simple random sample of sizen = 200 has a = 25 items with the attribute then the conclusionis u= 0.125 and se(u) =
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