Unit2高等数学英文 出国留学.docx

Unit2高等数学英文 出国留学.docx

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Unit2高等数学英文出国留学

Unit2:

IntroductiontoExperimentalDesign

LectureNotes（WrittenbyNathan）

I.SampleSurvey

•Thepopulationinastatisticalstudyistheentiregroupofindividualswewantinformationordrawaconclusionabout.

▪Anynumericalvaluethatcomesfromthepopulationisaparameter.

▪Acensuscollectsdatafromeveryindividualinthepopulation.

▪Intherealworld,timeandcostconsiderationsusuallymakeitimpossibletoanalyzeanentirepopulation–whose“truth”isessentiallyunknowable.

•Asampleisasubsetofthepopulationthatisactuallybeingexaminedandusedtorepresentthepopulation.

▪Anynumericalvaluethatcomesfromasampleiscalledastatistic.Theideaistousestatisticstodrawconclusionsaboutunknownparameters.Whileapopulationparameterisafixedquantity,statisticsvarydependingontheparticularsamplechosen.

▪SampleSurvey:

Thedistinctionbetweenpopulationandsampleisbasictostatistics.Toderiveconclusionsaboutthelargepopulation,weneedtobeconfidentthatthesamplewehavechosenrepresentsthatpopulationfairly.

II.Bias

Samplingmethodsthat,bytheirnature,tendtoover-orunderemphasizesomecharacteristicsofthepopulationaresaidtobebiased.Conclusionsbasedonsamplesdrawnwithbiasedmethodsareinherentlyflawed.Biasisnotjustbadluckinonesample.Itistheresultofabadstudydesignthatwillconsistentlymissthetruthaboutthepopulationinthesameway.

ØChoosingindividualsfromthepopulationwhoareeasytoreachresultsinaconveniencesample.Manysurveydesignssufferfromunder-coverage,inwhichsomeportionofthepopulationisnotsampledatall,orhasasmallerrepresentationinthesamplethanithasinthepopulation.

ØResponsebiasreferstoanythinginthesurveydesignthatinfluencestheresponses.oItisatypeofcognitivebiaswhichcanaffecttheresultsofastatisticalsurvey.（e.g.,respondentsanswerquestionsinthewaytheythinkthequestionerwantsthemtoanswerratherthanaccordingtotheirtruebeliefs.）

oSuchcircumstancesleadtoanonrandomdeviationoftheanswersfromtheirtruevalue.Becausethisdeviationtakesonaveragethesamedirectionamongrespondents,itcreatesasystematicerrorofthemeasure,orbias

oThesebiasesaremostprevalentinthetypesofstudiesandresearchthatinvolveparticipantself-report,suchasstructuredinterviewsorsurveys.Asaresult,responsebiasescanhavealargeimpactonthevalidityofthequestionnaireorsurveytowhichtheparticipantisresponding.

ØWordingbias:

Non-neutralorpoorlywordedquestionsmayleadtoanswersthatareveryunrepresentativeofthepopulation.

ØVoluntaryresponsebias:

samplesbasedonindividualswhooffertoparticipate,usuallybyrespondingtoageneralinvitation,typicallydisplaysexcessiveemphasizetopeoplewithstrongopinions,thusleadingtobias.

ØNon-responsebiasoccurswhenanindividualchosenforthesamplecan’tbecontactedorrefusestoparticipate.

oNon-responsetosurveysoftenexceeds50%,evenwithcarefulplanningandseveralfollow-upcalls.

oIfthepeoplewhoresponddifferfromthosewhodon’t,inawaythatisrelatedtotheresponse,biasresults.

oShort,easilyunderstoodsurveysgenerallyhavehigherresponserates

•Themaintechniquetoavoidbiasistoincorporaterandomnessintotheselectionprocess.Randomizingprotectsusfromtheinfluencesofallthefeaturesofourpopulationbymakingsurethat,onaverage,thesamplelooksliketherestofthepopulation.

•Thelargerthesample,thebettertheresults,butwhatiscriticalisthesamplesize,notthepercentageorfractionofthepopulation.Arandomsampleofsize500fromapopulationofsize100,000isjustasrepresentativeasarandomsampleofsize500fromapopulationofsize1,000,000.

III.Sampling1.RandomSampling

•Randomsamplinginvolvesusingachanceprocesstodeterminewhichmembersofapopulationareincludedinthesample.

•ASimplerandomsample（SRS）ofsizenischoseninsuchawaythateverypossiblegroupofnindividualsinthepopulationhasanequalchancetobeselectedasthesample.oAnSRSgiveseverypossiblesampleofthedesiredsizeanequalchancetobechosen.Italsogiveseachmemberofthepopulationanequalchancetobeincludedinthesample.

oAnSRSisthestandardagainstwhichwemeasureothersamplingmethods,andthesamplingmethodonwhichthetheoryofworkingwithsampleddataisbased.

ChoosinganSRSwithTechnology

Step1:

Label.Giveeachindividualinthepopulationadistinctnumericallabelfrom1toN

Step2:

Randomize.UseaRandomNumberGenerator（RNG）toobtainndifferentintegersfrom1toN.

•Samplesdrawnatrandomgenerallydifferonefromanother.Thesedifferencesleadtodifferentvaluesforthevariableswemeasure.Wecallthesesample-to-sampledifferencessamplingvariabilityorsamplingerror.

oDifferentsamplesgivedifferentsamplestatistics,allofwhichareestimatesofapopulationparameter.

oSamplingerrorrelatestonaturalvariationbetweensamples,canneverbeeliminated,canbedescribedusingprobability,andisgenerallysmallerifthesamplesizeislarger.

2.StratifiedSampling

•Instratifiedsamplingthepopulationisfirstdividedintohomogenous（meaningofsimilarifnotthesamefeatures）groups（knownasstrata）,thentakeanSRSoutofeachstratumandcombinetheseSRSstoformthefullsample.

o

Choosethestratabasedonfactsknownbeforethesampleistaken.Forexample,inastudyofsleephabitsonschoolnights,thepopulationofstudentsinalargeuniversitymightbedividedintofreshmen,sophomore,junior,andseniorstrata.

oSamplestakenwithinastratumvaryless,sotheresultingestimatescanbemoreprecise.Thisreducedsamplingvariabilityisthemostimportantbenefitofstratifying.Stratifiedsamplesgiveusefulinformationabouteachstratum.

•Stratifiedrandomsamplingworksbestwhentheindividualswithineachstratumaresimilarwithrespecttowhatisbeingmeasuredandwhentherearelargedifferencesbetweenstrata.

•Wecouldfurtherdoproportionalsampling,wherethesizesoftherandomsamplesfromeachstratumdependontheproportionofthetotalpopulationrepresentedbythestratum.

3.ClusterSampling

Inclustersamplingthepopulationisdividedintorepresentative,heterogeneousgroups,knownasclusters.Wethenrandomlyselectoneormoreclusterstobethesample.

•SomestatisticianstakeanSRSfromeachclusterratherthanincludingallmembersofthecluster.

•Clustersareinternallyheterogeneous,eachresemblingtheoverallpopulation.Eachclustershouldbesimilartoeveryothercluster.

•Clustersamplesareoftenusedforpracticalreasons–savingtimeandcost.Clustersamplingworksbestwhentheclusterslookjustlikethepopulationbutonasmallerscale.

4.MultistageSampling

Multistagesamplingreferstoaprocedureinvolvingtwoormoresteps,eachofwhich（beingofSRS）couldinvolveanyofthevarioussamplingtechniques.

TheGalluporganization,forexample,oftenfollowsaprocedureinwhichnationwidelocationsarerandomlyselected,thenneighborhoodsarerandomlyselectedineachoftheselocations,andfinallyhouseholdsarerandomlyselectedineachoftheseneighborhoods.

IV.Experiments1.ObservationalStudy（OS）

Anobservationalstudyobservesindividualsandmeasurevariablesofinterestsbutdoesnotinfluencetheresponses.

●InanOS,subjectschoosetheirownactionsandresearchersobservewhattheydo.

●Asamplesurveyisanobservationalstudyinwhichwedrawconclusionsaboutanentirepopulationbyconsideringanappropriatelychosensampletolookat.

2.Experiment

●Anexperimentdeliberatelyimposessome“treatment”onindividualstomeasuretheirresponses.

⏹Atreatmentisaspecificconditionappliedtotheindividualsinanexperiment.

●Anexperimentisperformedonobjectscalledexperimentalunits,andiftheunitsarepeople,theyarecalledsubjects.

●Theexperimentalunitsorsubjectsaretypicallydividedintotwogroups:

treatmentandcontrol.

⏹TreatmentGroup:

thegroupthatreceivesatreatment

⏹ControlGroup:

thegroupthatreceivesnotreatmentorold,establishedtreatment.Acontrolgroupmustbeusedforcomparison.Forexample,ifyouaretestingatreatmentforasprainedankle,youmusthaveagroupthatgetsnotreatmentbecausesprainedanklesnaturallygetbetterovertime.Youneedtoshowthatthetreatmentgroupgetsbetterfasterthanthecontrolgroup.

⏹PlaceboGroup:

Acontrolgroupthatreceivesaplacebo（fakedrug）inexperimentsinvolvingmedicines.

◆Placeboeffectreferstothefactthatmanypeoplerespondtoanykindof“perceived”treatment.Thepurposeoftheplaceboistoseparategenuinetreatmenteffectsfrompossiblesubjectresponsesduetosimplybeingpartofanexperiment.

●Experimentsinvolveexplanatoryvariables,calledfactors,whicharebelievedtohaveaneffectonresponsevariables,whichmeasuretheoutcomesofanexperiment.

⏹Thechoicesyouhaveforeachfactorareknownaslevels.

⏹Thedifferentfactor-levelcombinationsaretreatments.

●Inanexperiment,westudytheeffectsofthespecifictreatmentsweareinterestedin,whiletryingtocontrolfortheeffectsofothervariables.

⏹Thepurposeofanexperimentistodeterminewhetherthetreatmentcausesachangeintheresponse.

⏹Theexperimentcomparestheresponsesinthetreatmentgrouptotheresponsesinthecontrolgroup.

●Someexperimentaldesignsdon’tincludeacontrolgroup.That’