1、Designation: E 141 91 (Reapproved 2003)e1An American National StandardStandard Practice forAcceptance of Evidence Based on the Results of ProbabilitySampling1This standard is issued under the fixed designation E 141; the number immediately following the designation indicates the year oforiginal adop
2、tion 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 last revision or reapproval.e1NOTEEditorial changes were made throughout in November 2003.1. Scope1.1 This prac
3、tice presents rules for accepting or rejectingevidence based on a sample. Statistical 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 obta
4、ined by investigating the entire lot or populationunder the same rules and with the same 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 su
5、chprocedures have been applied. The methods may not give leastuncertainty at least cost, 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
6、pooled; the emphasis of this practice is onestimation of sampling uncertainty from the 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
7、 forecast or prediction interval or distributiondevised to match a random quantity of interest.1.4 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 pr
8、actices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:E 105 Practice for Probability Sampling of Materials2E 122 Practice for Choice of Sample Size to Estimate aMeasure of Quality for a Lot or Process2E 178 Practice for Dealing with
9、 Outlying Observations2E 456 Terminology for Statistical Methods2NOTE 1Practice E 105 provides a statement of principles for guidanceofASTM technical committees and others in the preparation of a samplingplan for a specific material. Practice E 122 aids in deciding on the requiredsample size. Practi
10、ce E 178 helps insure better behaved estimates.Terminology E 456 provides definitions of statistical terms used in thisstandard.3. Terminology3.1 Definitions:3.1.1 Equal Complete Coverage Result, nthe numericalcharacteristic (u) of interest calculated from observations madeby drawing randomly from t
11、he frame, all of the sampling unitscovered by the frame.3.1.1.1 DiscussionLocating the units and evaluating 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 actuallycalcula
12、ted. Its purpose is to serve as the objectively definedconcrete goal of the investigation. The quantity u may be thepopulation mean, (Y), total (Y), median (M), the proportion(P), or any other such quantity.3.1.2 frame, na list, compiled for sampling purposes,which designates all of the sampling uni
13、ts (items or groups) ofa population or universe to be considered in a specific study.3.1.2.1 DiscussionThe list may cover a specific shipmentor lot, all households in a county, a state, or country; forexample, any population of interest. Every sampling unit in theframe (1) has a unique serial number
14、, which may be preas-signed or determined by some definite rule, (2) has anaddressa complete and clear instruction (or rules for itsformulation) as to where and when to make the observation orevaluation, (3) is based on physically concrete clerical mate-rials such as directories, dials of clocks or
15、of meters, ledgers,maps, aerial photographs, etc., referred to in the addresses.1This practice is under the jurisdiction ofASTM Committee E11 on Quality andStatistical Methods and is the direct responsibility of Subcommittee E11.10 onSampling.Current edition approved August 15, 1991. Published Novem
16、ber 1991. Origi-nally published as E 141 59 T. Last previous edition E 141 69 (1975).2Annual Book of ASTM Standards, Vol 14.02.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.3.1.3 sample, na group of items, observations, test result
17、s,or portions of material, taken from a larger collection of suchitems; it provides information for decisions concerning thelarger collection.3.1.3.1 DiscussionA particular sample is identified by theset of serial numbers from the randomization device and by theaddresses on the frame generated by th
18、ose serial numbers.3.1.4 sampling unit, nan item, test specimen or portion ofmaterial that is to be subjected to evaluation as part of thesampling plan.3.1.4.1 DiscussionIf it is not feasible to select test speci-mens or laboratory samples individually, the sampling unitmay be a group of items, for
19、example, a row, an entire case ofitems, or a prescribed area (as in the examination of a finishingprocess).3.1.4.2 By a more expensive method of measurement (fu-ture time, more elaborate frame) it may be possible to define aquantity, u8, as a target parameter or ideal goal of an investi-gation. Crit
20、icism that holds u to be an inappropriate goalshould demonstrate that the numerical difference between uand u8 is substantial. Measurements may be imprecise but solong as measurement errors are not too biased, a large size ofthe lot or population, N, insures that u and u8 are essentiallyequal.4. Sig
21、nificance and Use4.1 This practice is designed to permit users of samplesurvey data to judge the trustworthiness of results from suchsurveys. Section 5 gives extended definitions of the conceptsbasic to survey sampling and the user should verify that suchconcepts were indeed used and understood by t
22、hose whoconducted the survey. What was the frame? How large (ex-actly) was the quantity N? How was the parameter u estimatedand its standard error calculated? If replicate subsamples werenot used, why not?4.2 Adequate answers should be given for all questions.There are many acceptable answers to the
23、 last question. If thesample design was relatively simple, such as simple random orstratified, then good estimates of sampling variance are easilyavailable. If a more complex design was used then methodssuch as discussed in 1 may be acceptable. Replicate sub-samples is the most straightforward way t
24、o estimate samplingvariances when the survey design is complex.4.3 Once the survey procedures that were used satisfySection 5, consult Section 4 to see if any increase in samplesize is needed. The calculations for making it are objectivelydescribed in Section 4.4.4 Refer to Section 6 to guide in the
25、 interpretation of theuncertainty in the reported value of the parameter estimate, u,i.e. 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.5. Descriptive Terms and Procedures5.1 Probability Sa
26、mpling Plansinclude instructions forusing either:5.1.1 carefully prepared tables of random number,5.1.2 computer algorithms, carefully programmed and runon a large computer, to generate pseudo-random numbers or,5.1.3 certifiably honest physical devices, such as coin flips,to select the sample units
27、so that inferences may be drawn fromthe test results and decisions may be made with risks correctlycalculated by probability theory.5.1.4 Such plans are defined and their relative advantagesdiscussed in 1, 2 and 6.5.2 Replicate Subsamplesa number of disjoint samples,each one separately drawn from th
28、e frame in accord with thesame probability sampling plan. When appropriate, separatelaboratories should each work on separate replicate subsamplesand teams of investigators should be assigned to separatereplicate subsamples. This approach insures that the calculatedstandard error will not be a syste
29、matic underestimate. Suchsubsamples were called interpenetrating in 5 where many oftheir basic properties were described. See 2 for further theoryand applications.5.2.1 DiscussionFor some types of material a sampleselected with uniform spacing along the frame (systematicsample) has increased precisi
30、on over a selection made withrandomly varying spacings (simple random sample). Examplesinclude sampling mineral ore or grain from a conveyor belt orsampling from a list of households along a street. If thesystematic sample is obtained by a single random start the planis then a probability sampling p
31、lan, but it does not permitcalculating the standard error as required by this practice.Afterdividing the sample size by an integer k (such as k =4ork= 10) and using a random start for each of k replicatesubsamples, some of the increased precision of systematicsampling (and a standard error on k 1 de
32、grees of freedom)can be achieved.5.2.2 Audit Subsamplea small subsample of the surveysample (as few as 10 observations may be adequate) for reviewof all procedures from use of the random numbers throughlocating and measurement, to editing, coding, data entry andtabulation. Selection of the audit sub
33、sample may be done byputting the n sample observations in order as they are collected,calculating the nearest integer to=n , or some other conve-nient integer, and taking this number to be the spacing forsystematic selection of the audit subsample. The review shoulduncover any gross departures from
34、prescribed practices or anyconceptual misunderstandings in the definitions. If the auditsubsample is 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
35、 in 1. Helpful discussion ofan audit appears in 2.5.2.3 Estimatea quantity calculated 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 u denotesthe estimate. (In c
36、alculating u, replicate subsample member-ship is ignored.)5.2.3.1 DiscussionAn estimate has a sampling distribu-tion induced from the randomness in sample selection. Theequal complete coverage result is effectively a constant whileany estimate is only the value from one particular sample.Thus, there
37、 is a mean value of the sampling distribution andthere is also a standard deviation of the sampling distribution.E 141 91 (2003)e125.2.4 Standard Errorthe quantity computed from theobservations as an estimate of the sampling standard deviationof the estimate; se (u) denotes the standard error.5.2.4.
38、1 Example 1When u is the population average of theN quantities and a simple random sample of size n was drawn,then the sample average y becomes the usual estimate u,whereu5y 5(i 5 1nyi/n. (1)The quantities y1, y2, ., yndenote the observations. Thestandard error is calculated as:se u! 5 se y! 5(i 5 1
39、nyi2 y!2/nn 2 1!. (2)There are n 1 degrees of freedom in this standard error.When the observations are:81.6, 78.7, 79.7, 78.3, 80.9, 79.5, 79.8, 80.3, 79.5, 80.7then y = 79.90 and se(y) = 0.32.As this example illustrates,formula (2) is correct when k replaces n and subsampleestimates are used in pla
40、ce of observations.5.2.4.2 Example 2 on the Finite Population Correction(fpc)Multiplying se (y) by=1 2 n/N is always correctwhen the goal of the survey is to estimate the finite populationmean (u = Y). Using the previous data and if N = 50, thense(y) becomes se(y) = 0.28 after applying the fpc. If r
41、andommeasurement error exists in the observations, then u8 based ona reference measurement method may be a more appropriatesurvey goal than u (see section 4.1.4.1). If so, then se(y) wouldbe further adjusted upward by an amount somewhat less thanthe downward adjustment of the fpc. Both of these adju
42、stmentsare often numerically so small that these adjustments may beomittedleaving se(y) of (2) as a slight overestimate.5.2.4.3 Example 3If the quantity of interest is (a) aproportion or (b) a total and the sample is simple random thenthe above formulas are still applicable. A proportion is themean
43、of zeroes and ones, while the total is a constant times themean. Thus:(a) when u is taken to be the population proportion (u = P)then;u5p 5 (yi/n 5 a/n (3)where:a is the number of units in the sample with the attribute, andsep!5=p1 2 p!/n 2 1! (4)(b) when u = the population total (u = Y) thenu5Ny an
44、d se u! 5 Nse y! (5)If a simple random sample of size n = 200 has a = 25 itemswith the attribute then the conclusion is u = 0.125 and se(u) = 0.023 on 199 degrees of freedom.5.2.4.4 Example 4.Ifu is a parameter other than a mean orif the sample design is complex, then replicate subsamplesshould be u
45、sed in the sample design. Denote the k separateestimates as ui,i= 1, 2, ., k and denote by u the estimate basedon the whole sample. The average of the uiwill be close to, butin general not equal to u. The standard error of u is calculatedas:se u! 5(i 5 1kui2u!2/kk 2 1! (6)where u is the average of t
46、he ui. The standard error isbased on k 1 degrees of freedom.The following estimates of percent “drug-in-suit” sales ofprescription drugs were based on 20 replicate subsamples; eachfollowed a stratified cluster sampling design. The separateestimates were: 6.8, 7.1, 8.4, 9.5, 8.6, 4.1, 3.7, 3.2, 3.8,
47、5.8, 8.8,5.0, 7.9, 8.8, 8.4, 8.1, 6.0, 6.3, 4.5, 5.8. The value of u was6.74 % and se(u) = 0.43 % on 19 degrees of freedom. Noticethat u = 6.58 does not equal u = 6.74. This is because u is aratio of two overall averages while u is the average of 20ratios. For an example with k = 2, average13 and35
48、andcompare to (1 + 3)/(3 + 5).5.2.5 Proceduresmust be described in written form andshould cover the following matters; (1) parties interested incollecting data should agree on the importance of knowing uand its definition including measurement methods, (2) theframe shall be carefully and explicitly
49、constructed; N shall bewell established, (3) random numbers (or a certifiably honestphysical random device) shall dictate selection of the sample.There will be no substitution of one sampling unit for another.The method of sample selection shall permit calculation of astandard error of the estimate (4) the use of replicate sub-samples is recommended (see section 5.24.2.2); an auditsubsample should be selected and processed and any depar-tures from prescribed measurement methods and locationinstructions noted (se
