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Source:J.Hollenbeck,SamplingDesigns,AveryandBurkhart,Chapter3,WhyNotMeasureEverything?,Measureeveryfeatureofinterest.Theresultisahighlyaccuratedescriptionofthepopulation.Drawbacks:Onlyviablewithsmallpopulations.Onlycost-effectivewithhigh-valuedfeatures.,FOR220AerialPhotoInterpretationandForestMeasurements,CompleteEnumeration(Census),WhyNotRandomlySelectSamples?,WewilldiscussseveralsamplingdesignsinthislectureMoststatisticalmethodsassumesimplerandomsamplingwasused.Samplingmethods,however,willvarydependingontheobjectivesofthesurvey,thenatureofthepopulationbeingsampled,andpriorinformationaboutthepopulationbeingsampled.,FOR220AerialPhotoInterpretationandForestMeasurements,SamplingDesigns,SimpleRandomSamplingSystematicSamplingStratifiedRandomSampling,FOR220AerialPhotoInterpretationandForestMeasurements,SamplingDesignsCoveredToday:,SamplingDesignI:SimpleRandomSampling,Everypossiblecombinationofsamplingunitshasanequalandindependentchanceofbeingselected.Theselectionofaparticularunittobesampledisnotinfluencedbytheotherunitsthathavebeenselectedorwillbeselected.Samplesareeitherchosenwithreplacementorwithoutreplacement.,FOR220AerialPhotoInterpretationandForestMeasurements,Assumptions:,SamplingDesignI:SimpleRandomSampling,Design:,FOR220AerialPhotoInterpretationandForestMeasurements,SamplingDesignI:SimpleRandomSampling,Estimatethepopulationmean:,Estimatethevarianceofindividualvalues:,Computethecoefficientofvariation:,FOR220AerialPhotoInterpretationandForestMeasurements,Weusethefamiliarequationsforestimatingcommonstatisticsforthepopulation,SamplingDesignI:SimpleRandomSampling,Computethestandarderror:,OR,(withreplacement,orinfinitepopulation),(withoutreplacementandfromafinitepopulation),Computeconfidencelimits:,FOR220AerialPhotoInterpretationandForestMeasurements,t=tstatisticfromtable,determinedbydegreesoffreedom(n-1).,Ingeneral,itissafetouse:t=2.0forsmalln(n30)ifyoudonothaveaccesstoat-table.,SamplingDesignI:SimpleRandomSampling,Samplesizeshouldbestatisticallyandpracticallyefficient.Enoughsampleunitsshouldbemeasuredtoobtainthedesiredlevelofprecision(nomore,noless).,FOR220AerialPhotoInterpretationandForestMeasurements,Howmanysamplestotake?,SamplingDesignI:SimpleRandomSampling,Determinesamplesize(withooutreplacement,orfinitepopulation):,Knowingthecoefficientofvariationandthaterrorisexpectedtobewithinx%ofthevalueofthemean,weusethefollowingforumula,n=,where,n=requiredsamplesize,A=allowableerrorpercent,t=t-value,CV=coefficientofvariation,andN=populationsize,SamplingDesignI:SimpleRandomSampling,Exampleusing0.10acfixedradiusplotsinatenacrestandAssume:WehavecalculatedtheCVandfounditwas30%Ourallowableerroris+/-10%ofthemeanAreusingt-valueof2Nisthetotalnumberofpotentialplotsthatcouldbeplacedinthestand(i.e.,population).Inthiscase,Nisstandsize/plotsizeor10/0.10=100,Determinesamplesize(withooutreplacement,orfinitepopulation):,n=,=26,n=,SamplingDesignII:SystematicSampling,Theinitialsamplingunitisrandomlyselectedorestablishedontheground.Allothersampleunitsarespacedatuniformintervalsthroughouttheareasampled.Samplingunitsareeasytolocate.Samplingunitsappeartoberepresentativeofanarea.,FOR220AerialPhotoInterpretationandForestMeasurements,Assumptions:,SamplingDesignII:SystematicSampling,Design:,FOR220AerialPhotoInterpretationandForestMeasurements,AGridSchemeismostcommon,SamplingDesignII:SystematicSampling,For:Regularspacingofsampleunitsmayyieldefficientestimatesofpopulationsundercertainconditions.*Against:Accuracyofpopulationestimatescanbelowifthereisperiodicorcyclicvariationinherentinthepopulation.,FOR220AerialPhotoInterpretationandForestMeasurements,Arguments:,SamplingDesignII:SystematicSampling,For:Thereisnopracticalalternativetoassumingthatpopulationsaredistributedinarandomorderacrossthelandscape.Against:Simplerandomsamplingstatisticaltechniquescantlogicallybeappliedtoasystematicdesignunlesspopulationsareassumedtoberandomlydistributedacrossthelandscape.,FOR220AerialPhotoInterpretationandForestMeasurements,Arguments:,SamplingDesignII:SystematicSampling,Wecan(andoftendo)usesystematicsamplingtoobtainestimatesaboutthemeanofpopulations.Whenanobjective,numericalstatementofprecisionisrequired,however,itshouldbeviewedasanapproximationoftheprecisionofthesamplingeffort.(i.e.95%confidenceintervals)Useformulaspresentedforsimplerandomsampling,andwhereappropriate,usethe“withoutreplacement”variationsofthoseequations(ifsamplingfromasmallpopulation),otherwiseusethenormalSRSstatisticaltechniques.,FOR220AerialPhotoInterpretationandForestMeasurements,Summary:,SamplingDesignIII:StratifiedRandomSampling,Apopulationissubdividedintosubpopulationsofknownsizes.Asimplerandomsampleofatleasttwounitsisdrawnfromeachsubpopulation.Why?Toobtainamorepreciseestimateofthepopulationmean.Ifthevariationwithinasubpopulationissmallinrelationtothetotalpopulationvariance,theestimateofthepopulationmeanwillbeconsiderablymoreprecisethanasimplerandomsampleofthesamesize.Why?Toobtainanestimateoftheresourceswithinthesubpopulations.,FOR220AerialPhotoInterpretationandForestMeasurements,Assumptions:,SamplingDesignIII:StratifiedRandomSampling,Design:,FOR220AerialPhotoInterpretationandForestMeasurements,Estimatetheoverallpopulationmean:,SamplingDesignIII:StratifiedRandomSampling,Where:L=numberofstrataNh=totalnumberofareaunitsinstratahN=totalnumberofareaunitsinallstrata,FOR220AerialPhotoInterpretationandForestMeasurements,Weoftenuseareaforstrataunits(acres,hectares)innaturalresourceapplications,SamplingDesignIII:StratifiedRandomSampling,FOR220AerialPhotoInterpretationandForestMeasurements,Estimatetheoverallpopulationmean:,Example:,Estimatetheoverallpopulationstandarderrorofthemean:,SamplingDesignIII:StratifiedRandomSampling,1.Firstcomputevariancewithineachstrata,Where:nh=totalnumberoftreeunitssampledinstratah,FOR220AerialPhotoInterpretationandForestMeasurements,SamplingDesignIII:StratifiedRandomSampling,Estimatetheoverallpopulationstandarderrorofthemean:,2.Second,computepopulationstandarderrorofthemean,(withreplacement)OR(withoutreplacement),FOR220AerialPhotoInterpretationandForestMeasurements,Remember:N=areaunits,n=treeunits,SamplingDesignIII:StratifiedRandomSampling,Estimatetheoverallpopulationstandarderrorofthemean:,3.Computeconfidenceintervalsifdesired,Estimatedconfidenceintervals:,FOR220Aeri