经济学Sampling 抽样技术统计学专业课课件

Sampling:

Design&AnalysisSharonL.LohrArizonaStateUniversitySampling:

Design&AnalysisShContentsCHAPTER1Introduction1.1ASampleControversy1.2RequirementsofaGoodSample1.3SelectionBias1.4MeasurementBias1.5QuestionnaireDesign1.6SamplingandNonsamplingErrorsCHAPTER2SimpleProbabilitySamples2.1TypesofProbabilitySamplesContents2.2FrameworkforProbabilitySampling2.3SimpleRandomSampling2.4ConfidenceIntervals2.5SampleSizeEstimation2.6SystematicSampling2.7RandomizationTheoryResultsforSimpleRandomSampling*2.8AModelforSimpleRandomSampling*2.9WhenShouldaSimpleRandom2.2FrameworkforProbabSampleBeUsed?CHAPTER3RatioandRegressionEstimation3.1RatioEstimation3.2RegressionEstimation3.3EstimationinDomains3.4ModelsforRatioandRegressionEstimation*3.5ComparisonSampleBeUsed?CHAPTER4StratifiedSampling4.1WhatIsStratifiedSampling?4.2TheoryofStratifiedSampling4.3SamplingWeights4.4AllocatingObservationstoStrata4.5DefiningStrata4.6AModelforStratifiedSampling*4.7Post-stratification4.8QuotaSamplingCHAPTER4StratifiedSamplingCHAPTER5ClusterSamplingwithEqualProbabilities5.1NotationforClusterSampling5.2One-StageClusterSampling5.3Two-StageClusterSampling5.4UsingWeightsinClusterSamples5.5DesigningaClusterSample5.6SystematicSampling5.7ModelsforClusterSampling*CHAPTER5ClusterSamplingwi

CHAPTER1IntroductionWhenstatisticsarenotbasedonstrictlyaccuratecalculations,theymisleadinsteadofguide.Themindeasilyletsitselfbetakeninbythefalseappearanceofexactitudewhichstatisticsretainintheirmistakes,andconfidentlyadeptserrorsclothedintheformofmathematicaltruth.

--AlexisdeTocqueville,DEMOCRACYINAMERICACHAPTER11.1ASamplingControversyShereHite'sbook“WomenandLove:ACulturalRevolutioninprogress”(1987):

84%ofwomenare"notsatisfiedemotionallywiththeirrelationships"(p804).70%ofallwomen"marriedfiveormoreyearsarehavingsexoutsideoftheirmarriages"(p856).1.1ASamplingControversy95%ofwomen"reportformsofemotionalandpsychologicalharassmentfrommenwithwhomtheyareinloverelationships(p810).84%ofwomenreportformsofcondescensionfromthemenintheirloverelationships(p809).*Harassment:toannoypersistentlysexualharassment:uninvitedandunwelcomeverbalorphysicalbehaviorofasexualnatureespeciallybyapersoninauthoritytowardasubordinate(asanemployeeorstudent)95%ofwomen"reportformsof*Condescension:1.voluntarydescentfromone'srankordignityinrelationswithaninferior;2.Theactofcondescendingoraninstanceofit.3.Patronizinglysuperiorbehaviororattitude.*Vignette:Adecorativedesignplacedatthebeginningorendofabookorchapterofabookoralongtheborderofapage.*Condescension:1.voluntaryThefollowingcharacteristicsofthesurveymakeHite’sresultunsuitableforgenerali-zation.Thesamplewasself-selected.Thequestionnairesweremailedtospecificgroups.Thequestionsinthesurveyaretoocomplicated.Manyofthequestionsarevague,usingwordssuchaslove.Manyofthequestionsareleading.

Thefollowingcharacteristics1.2RequirementsofaGoodSampleAperfectsampleshould:1.beascaled-downversionofthepopulation;2.canmirrorcharacteristicsofthewholepopulationSomedefinitionstomakethenotionofagoodsamplemoreprecise:ObservationunitAnobjectonwhichameasurementistaken.TargetpopulationThecompletecollectionofobservationswewanttostudy.Sample

Asubsetofapopulation.1.2RequirementsofaGoodSSampledpopulationThecollectionofallpossibleobservationunitsthatmighthavebeenchoseninasample.Thepopulationfromwhichthesamplewastaken.SamplingunitTheunitweactuallysample.Samplingframe

Thelistofsamplingunits.SampledpopulationThecolleTargetpopulationSamplingframepopulationSampledpopulationNotreachableRefusetorespondNotcapabletorespondNoteligibleforsurveyTargetpopulationInanidealsurvey,thesampledpopulationwillbeidenticaltothetargetpopulation,butthisidealisrarelymetexactly.IntheHitestudyTargetpopulation:alladultwomenintheUnitedStatesSampledpopulation:womenbelongingtowomen'sorganizationswhowouldreturnthequestionnaire.

Inanidealsurvey,thesampleIntheNationalCrimeVictimizationSurvey:Targetpopulation:allhouseholdsintheUnitedStatesSampledpopulation:householdsinthesamplingframethatare"athome"andagreetoanswerquestions.IntheNationalCrimeVictimizIntheNationalPesticideSurvey:Targetpopulation:allcommunitywatersystemsandruraldomesticwellsintheUnitedStates.Sampledpopulation:allcommunitywatersystemsandallidentifiabledomesticwellsoutsideofgovernmentreservationsthatbelongedtohouseholdswillingtocooperatewiththesurvey.IntheNationalPesticideSurvInpublicopinionpolls:Targetpopulation:personswhowillvoteinthenextelectionSampledpopulation:personswhocanbereachedbytelephoneandsaytheyarelikelytovoteinthenextelection

Inpublicopinionpolls:1.3SelectionBias

Thefollowingexamplesindicatesomewaysinwhichselectionbiascanoccur:Useasample-selectionprocedurethat,unknowntotheinvestigators,dependsonsomecharacteristicassociatedwiththepropertiesofinterest.Deliberatelyorpurposefullyselecta"representative"sample.–forinstance:”a

judgmentsample”Misspecifythetargetpopulation.1.3SelectionBiasFailtoincludeallthetargetpopulationinthesamplingframe,calledunder-coverage.Substituteaconvenientmemberofapopulationforadesignatedmemberwhoisnotreadilyavailable.Failtoobtainresponsesfromtheentirechosensample.Thisiscallednon-responseAllowthesampletoconsistentirelyofvolunteers

FailtoincludeallthetargetCASESTUDYLiteraryDigestAneververyfamousmagazineinUSAwhobegantakingpollstoforecasttheoutcomeoftheUSApresidentialelectionin1912.theirpollsattainedareputationforaccuracybecausetheyforecastthecorrectwinnerineveryelectionbetween1912and1932.CASESTUDYIn1932,forexample,thepollpredictedthatRooseveltwouldreceive56%ofthepopularvoteand474votesintheelectoralcollege;intheactualelection.Rooseveltreceived58%ofthepopularvoteand472votesintheelectoralcollege.

Electoralcollege:(intheU.S.)abodyofpeoplerepresentingthestatesoftheU.S.,whoformallycastvotesfortheelectionofthepresidentandvicepresident.In1932,forexample,thepOnOctober31,1936,thepollpredictedTheoutcomeis:RepublicanAlfLandon:55%PresidentRoosevelt:41%RepublicanAlfLandon:37%PresidentRoosevelt:61%OnOctober31,1936,thepollpTworeasonsthataccountedfortheoutcome:Oneproblemmayhavebeenundercoverageinthesamplingframe,whichreliedheavilyontelephonedirectoriesandautomobileregistrationlist;Thelowresponserate(lessthan25%)tothesurveywaslikelythesourceofmuchoftheerror.

OnelessontobelearntfromtheLiteraryDigestpollisthatthesheersizeofasampleisnoguaranteeofitaccuracyTworeasonsthataccountedfor1.4MeasurementBiasInfollowingcases,itismostlikelytohappen:Peoplesometimesdonottellthetruth.PeopledonotunderstandthequestionsPeopleforgetPeoplegivedifferentanswerstodifferentinterviewersPeoplecatertotheinterviewersTheinterviewermayhavehisowninclinationtothesurveyCertainwordsmayhavevaguemeaningThequestionnairedoesn’twordwellorisnotarrangedinagoodorder1.4MeasurementBias1.5QuestionnaireDesignDecidewhatyouwanttofindoutAlwaystestyourquestionsbeforetakingthesurveyKeepitsimpleandclearUsespecificquestionsinsteadofgeneralonesRelateyourquestionstotheconceptofinterest.Decidewhethertouseopenorclosedquestions(openquestions:therespondentsisnotpromptedwithcategoriesforresponse;closedones:multiplechoices)1.5QuestionnaireDesignReporttheactualquestionaskedAvoidquestionsthatpromptormotivatetherespondenttosaywhatyouwouldliketohearUseforced-choice,ratherthanagreeordisagreequestionsAskonlyoneconceptineachquestionPayattentiontoquestion-ordereffectsReporttheactualquestionask1.6samplingandnonsamplingerrorssamplingerrorsTheerrorthatresultsfromtakingonesampleinsteadofexaminingthewholepopulationnonsamplingerrorsTheerrorthatcannotbeattributedtothesample-to-samplevariability,causedchieflybyfollowingcauses:SelectionbiasIncorrectanswers1.6samplingandnonsamplingeIncompletevalueNonresponseSelectionbiasandmeasurementbiasareexamplesofnonsamplingerrorsInalotofcases,nonsamplingerrorsmayhavemuchworseeffectonaccuracyofthesamplethansamplingonesIncompletevalueWhydoweusesampling?Samplingcanprovidereliableinformationatfarlesscostthanacensus.Datacanbecollectedmorequickly,soestimatescanbepublishedinatimelyfashion.Finally,andlesswellknown,estimatesbasedonsamplesurveysareoftenmoreaccuratethanthosebasedonacensusbecauseinvestigatorscanbemorecarefulwhencollectingdataWhydoweusesampling?CHAPTER2SimpleProbabilitySamplesProbabilitySampling:inaprobabilitysample,eachunitinthepopulationhasaknown(butnotcertainlyequal)probabilityofselection,andachancemethodsuchasusingnumbersfromarandomnumbertableisusedtochoosethespecificunitstobeincludedinthesample.2.1TypesofProbabilitySamples1,Simplerandomsample2,Stratifiedsample3,ClustersampleCHAPTER2SimpleProbabilityAsimplerandomsample(SRS)isthesimplestformofprobabilitysample.AnSRSofsizenistakenwheneverypossiblesubsetofnunitsinthepopulationhasthesamechanceofbeingthesample.Inastratifiedrandomsample,thepopulationisdividedintosubgroupscalledstrata.ThenanSRSisselectedfromeachstratum,andtheSRSsinthestrataareselectedindependently.Asimplerandomsample(SRS)iInaclustersample,observationunitsinthepopulationareaggregatedintolargersamplingunits,calledclusters.ThenanSRSisdrawnundertheconditionthateachclusterisviewedasaunit.2.2FrameworkforProbabilitySamplingAspecialcaseforitisN=4,whichresultsin:Itspossiblesamples(n=2)are:Inaclustersample,observatiExample2.1:Example2.1:Example2.2:i1,2,3,4,5,6,7,81,2,4,4,7,7,7,8Example2.2:i1,2,3,4,5Theexpectedvalueof,isthemeanofthesamplingdistributionof:k2228303234363840424446485052581623746126473261TheexpectedvalueofThevarianceofthesamplingdistributionof,i.e,is:ThevarianceofthesamplingdTheMeanSquaredError(MSE)ratherthanvariancetomeasuretheaccuracyofanestimatoris:TheMeanSquaredError(MSE)rAnestimatorisunbiasedif:Anestimatorispreciseifthefollowingissmall:Anestimatorisaccurateifthefollowingissmall:Anestimatorisunbiasedif:Someindicatorsforthepopulation:Thepopulationtotalis:Themeanofthepopulationis:Thevarianceofthepopulationvaluesaboutthemeanis:SomeindicatorsforthepopulaThestandarddeviationofthepopulationvaluesaboutthemeanis:Thecoefficientofvariation(CV)is:Proportionisaspecialcaseofmean:ThestandarddeviationoftheThedistinctionbetweenmeanandproportionis:inthecaseofmean,thevariablecantakemorethantwovalues;whereasinproportioncase,itcantakeandcanonlytaketwovalues.ThedistinctionbetweenmeanWherethevariableis:Wherethevariableis:2.3SimpleRandomSamplingTherearetwotypesofSimpleRandomSample:1SimpleRandomSamplewithreplacement(SRSWR).Inthiscase,thereare

possiblesamplesandwemaygetduplicates;2.3SimpleRandomSampling2SimpleRandomSamplewithreplacement(SRS).Inthiscase,thereare

possiblesamplesandwemaynotgetduplicates.[经济学]Sampling抽样技术统计学专业课课件ForestimatingthepopulationmeaninanSRS,weusethesamplemean:Theisanunbiasedestimatorofthepopulationmean,andthevarianceofis:Inwhichiscalledthefinitepopulationcorrection(fpc):ForestimatingthepopulationForestimatingthepopulationvariance,weusethesamplevariance:Anunbiasedestimatorofisasfollow:ForestimatingthepopulationButtheestimatedvarianceofisusuallyreportedbyitsstandarderror(SE):Theestimatedcoefficientofvariationofanestimateis:ButtheestimatedvarianceofAllthisresultsapplytotheestimationofapopulationtotal,t,since:Theunbiasedestimatoroftis:Itsvarianceis:Buttheunbiasedestimatorofthisvarianceis:AllthisresultsapplytothesinceAsforproportionvariable,theparametersare:thussinceAsforproportionvariablTheestimatorsare:Where:Theestimatorsare:Where:2.4ConfidenceIntervals

Usedtoindicatehowaccurateourestimatesare.Usuallyappearsinthisway:Ifwetakeasanexample,thenwehave:2.4ConfidenceIntervalsIfThedistinctionbetweendistributionandsamplingdistributionfromit:A“distribution”referstotheoriginaldistributionofavariabley,whereasa“samplingdistribution”referstothedistributiongeneratedfromtheoriginalone,likethedistributionofand.Example2.111/4,14/4,15/4,16/4,17/4,,…,29/4Withauniformdistribution,only8cases.Withanon-uniformdistribution,upto15cases.ThedistinctionbetweendistriThedistinctionbetweenlawoflargenumbersandcentrallimittheorem:“lawoflargenumbers”saysthatthereisalmostnodifferencebetweensampleandpopulationmeanifnissufficientlylarge,bothdependentlyorindependently,withthesameordifferentdistribution;whereas“centrallimittheorem”saysthatthedistributionofanysamplemeanconvergestonormaldistributionifnissufficientlylarge,withthesameordifferentdistribution.ThedistinctionbetweenlawofBernoulli’slawoflargenumbersis:Linderbergandlevy’scentrallimittheoremis:Bernoulli’slawoflargenumb2.5SampleSizeEstimationAninvestigatoroftenmeasuresseveralvariablesandhasanumberofgoalsforasurvey.AnyonedesigninganSRSmustdecidewhatamountofsamplingerrorintheestimatesistolerableandmust

balancetheprecisionoftheestimateswiththecostofthesurvey.2.5SampleSizeEstimationFollowthesestepstoestimatethesamplesize:

Askquestionsas:A:Whatisexpectedofthesample?B:HowmuchprecisiondoIneed?C:Whataretheconsequencesofthesampleresults?D:Howmucherroristolerable?Findanequationrelatingthesamplesizenandourexpectationsofthesample

FollowthesestepstoestimateEstimateanyunknownquantitiesandsolveforn.Ifthesamplesizeyoucalculatedinlaststepismuchlargerthanyoucanafford.Gobackandadjustsomeoftheexpec-tationsforthesurveyandtryagain.Specifythetolerableerror

EstimateanyunknownquantitieFindanequationSolvingforn,wehave:Inrelativeprecisioncase,wehave

Findanequation2.7RandomizationTheoryResultsforSimpleRandomSampling*ToverifyandDefine,thenwehave2.7RandomizationTheoryRe

[经济学]Sampling抽样技术统计学专业课课件Asaconsequenceofequation(2.18)

inordertocalculatethevarianceof,notethat:

Asaconsequenceofequatio

Nowletusprove:

Proof:

Nowletusprove:

CHAPTER3RatioandRegressionEstimationExampleincensusbyLaplaceCHAPTER3RatioandRegressi3.1RatioEstimationEstimationmethodusingauxiliaryvariableDefine:

andThen,,,[经济学]Sampling抽样技术统计学专业课课件Ratioandregressionestimationbothtakeadvantageofthecorrelationcoefficientofxandyinthepopulation;thehigherthecorrelation,thebettertheywork.Definethepopulationcorrelationcoefficientofxandytobe:

Ratioandregressionestima3.1.1Whydoweuseratioestimation?Sometimeswesimplywanttoestimatearatio;AverageyieldperacrePercapitaincomeTheratioofliabilitiestoassetsTheratioofthefishcaughttothehoursspentfishing3.1.1WhydoweuseratioestiSometimeswewanttoestimateapopulationtotal,butthepopulationsizeNisunknown;SometimeswewanttoestimateRatioestimationisusedtoincreasetheprecisionofestimatedmeansandtotals.RatioestimationisusedtoinRatioestimationisusedtoadjustestimatesfromthesamplesothattheyreflectdemographicaltotals.Population:4000Sample:4002700females1300males240females160malesRatioestimationisusedtoad

tobecomeateacherSimpleestimation:Ratioestimation:240females160males84females40males240females84femalesRatioestimationisusedtoadjustnonresponse.Ratioestimationisusedtoad3.1.2BiasandMeanSquaredErrorofRatioEstimatorsRatioestimationisbiased,unlikeSRSestimation,butwithareducedvarianceasacompensatorforthepresenceofbias.Since:weget:3.1.2BiasandMeanSquaredThenwehave:Consequently,asshownbyHartleyandRoss(1954).Thenwehave:Anotherwaytoshowthisis:Thebiasofissmallif:1,thesamplesizenislarge;2,thesamplingfractionn/Nislarge;3,islarge4,issmall;5,thecorrelationRiscloseto1.Anotherwaytoshowthisis:Theproofforitis:Theproofforitis:[经济学]Sampling抽样技术统计学专业课课件IntheworksofHartley-Ross,anunbiasedestimatorfortheparameterBisgivenafterthefollowingisproved:Theunbiasedestimatoris:IntheworksofHartley-Ross,TheapproximatedMSEofis:TheapproximatedMSEofwillbesmallif:1,thesamplesizenislarge;2,thesamplingfractionn/Nislarge;3,thedeviationaboutthelineislarge;4,islarge;5,thecorrelationRiscloseto+1.TheapproximatedMSEofisTheproofforitis:And:Theproofforitis:And:[经济学]Sampling抽样技术统计学专业课课件Thereforeweget:[经济学]Sampling抽样技术统计学专业课课件Inpractice,Bisunknown,thenletItfollowsfromabove,wewillhave:Inpractice,Bisunknown,theTheCIscanbeconstructedas:Let’sjuststudytwoexamplesinthetextbook.TheCIscanbeconstructedas:AccuracyoftheMSEApproximationFor(3.6)tobeagoodapproximationofMSE,wewantalargesamplesizen(>30),andAdvantageofratioestimationAccuracyoftheMSEApThen,Sototheaccuracyoftheapproximation,holds,ifandonlyif:Then,3.2RegressionEstimationRatioestimationworksbestifthedataarewellfitbyastraightlinethroughtheorigin.Butsometimes,dataappeartobeevenlyscatteredaboutastraightlinethatdoesnotgothroughtheorigin,thatis,thedatalookasthoughtheusualstraight-lineregressionmodel

wouldprovideagoodfit.3.2RegressionEstimationwhere:Hereristhesamplecorrelationcoefficientofxandy.Liketheratioestimator,theregressionestimatorisbiased.[经济学]Sampling抽样技术统计学专业课课件Thebiasis:

LetThenThebiasis:[经济学]Sampling抽样技术统计学专业课课件Thatis:TheapproximationMSEissmallif:1,thesamplesizenislarge;2,thesamplingfractionn/Nislarge;3,issmall;4,thecorrelationRiscloseto-1or1.Thatis:Let,thentheStandardErroris:[经济学]Sampling抽样技术统计学专业课课件CHAPTER4StratifiedSampling4.1WhatIsStratifiedSampling?stratum:(1)ahorizontallayerofmaterial,especiallyoneofseveralparallellayersarrangedoneontopofanother;(2)alevelofsocietycomposedofpeoplewithsimilarsocial,cultural,oreconomicstatusstrata:thepluralformofstratumstratify:toform,arrange,ordepositinlayers.CHAPTER4StratifiedSamplingWhydoweuseStratifiedSampling?1,tobeprotectedfromthepossibilityofobtainingvery“bad”sample;Population:Sample:2500Females1500Males150Males250FemalesWhydoweuseStratifiedS2,tohavedataofknownprecisionforsubgroups;Population:Sample:3500Females500Males250Females150Males35005002501503,inordertoadministerasamplemoreconvenientlyortolowercostforthesurvey;MailsurveyPersonalinterviewTelephoneinterview……3,inordertoadministerasa4,toproducemorepreciseestimatesforthewholepopulation(ifdonecorrectly)BloodPressure(byage)ConcentrationofPlants(bytypeofterrain)

YoungsterstheMiddle-agedWoodlandMarshDotagesDrylandYoungsterstheWoodlandMarshDo4.2TheoryofStratifiedSamplingDefinition:Instratifiedrandomsampling,thesimplestformofstratifiedsampling．WeindependentlytakeanSRSfromeachstratumsothatnh

observationsarerandomlyselectedfromthepopulationunitsinstratumh.DefineShtobethesetofnh

unitsintheSRSforstratumh.4.2TheoryofStratifiedSaNotationforStratificationThepopulationsizeis:valueofjthunitinstratumh:

populationtotalinstratumh:

overallpopulationtotal:populationmeaninstratumh:NotationforStratification

overallpopulationmean:populationvarianceinstratumh:Thecorrespondingquantitiesforthesampleare:

(Curlicue:adecorativecurlortwistincalligraphyorinthedesignofanobject)

[经济学]Sampling抽样技术统计学专业课课件Thenwehavetheestimationforthepopulationas:andThepropertiesoftheseestimatorsfollowdirectlyfromthepropertiesofSRSestimators:Unbiasedness:Varianceoftheestimators:ThenwehavetheestimationfoVarianceestimatorsforstratifiedsamples:[经济学]Sampling抽样技术统计学专业课课件4.3SamplingWeightsDenote:(samplingweight)and(samplingfraction)Thenweget:

4.3SamplingWeightsandthus:

andthus:4.4AllocatingObservationstoStrata4.4.1ProportionalAllocationAnallocationmethodistoletthenumberofsampledunitineachstratumbeproportionaltothesizeofthecorrespondingstratum.Theprobabilityofselectionπhjisthesamen/Nforallstrata.

4.4AllocatingObsesinceAsforSRS,wehave

since

PopulationANOVAtable

SourcedfSumofsquaresBetweenstrataWithinstrataH-1

N-HTotal,aboutN-1SourcedfTherefore

Ifonly

ThereforeThefollowingwillhold:Virtually,witharelativelylargerstratumsize,wehaveAndthisresultsin:

Thefollowingwillhold:4.4.2OptimalAllocationLetand

4.4.2OptimalAllocationWeget:Wesampleheavilywithinastratumif:Thestratumaccountsforalargepartofthepopulation;Thevariancewithinthestratumislarge,wesamplemoreheavilytocompensatefortheheterogeneity.SamplinginthestratumisinexpensiveWeget:4.4.4DeterminethesamplesizeofthepopulationSince:Ifthefpc’sarenegligibleandifthenormalapproximationisvalid,theCIcanbe:

Supposetheabsoluteerroris,i.e.,wethenhave.Practically,theinthevwillbesubstitutedwith

4.4.4DeterminethesamplesizCHAPTER5ClusterSamplingwithEqualProbabilitiesRelativeConcepts:ClusterPrimarysamplingunits(psu’s)Secondarysamplingunits(ssu’s)Definition:seetheexampleonthetextbookWhydoweuseClusterSampling?1.Constructingsamplingframelistofobservationunitsmaybedifficult,expensive,orimpossible2.Thepopulationmaybewidelydistributedgeographicallyandmayoccurinnaturalclusterssuchashouseholdsorschools.CHAPTER5ClusterSamplingwiSimilaritiesanddifferencesbetweenstratumandclusterSimilaritiesbothareagroupingofthemembersofthepopulation;Differences(1)forstratasampling:selectanSRSsampleunitswithineverystratum;forstratasampling:selectanSRSclustersfromthepopulation,theunitsineachclusterconstitutingthewholesample.(2)Instratifiedsamplingcase:thesamplingmayincreasetheprecision;whereasinclustersampling,thesamplingmaydecreasetheprecision,butmayresultinmoreinformationperdollarspentSimilaritiesanddifferencesb5.1NotationforClusterSamplingyij:measurementforjthelementinithpsu

psulevel-populationquantities:N=numberofpsu’sinthepopulationMi=numberofssu’sinithpsu=totalnumberofpsu’sinthepopulation

=totalintheithpsu’s=population