S2.4a Sampling and hypothesis tests.ppt

theseiconsindicatethatteacher snotesorusefulwebaddressesareavailableinthenotespage thisiconindicatestheslidecontainsactivitiescreatedinflash theseactivitiesarenoteditable formoredetailedinstructions seethegettingstartedpresentation boardworksltd2006 1of58 a2 levelmaths statistics2forocr s2 4asamplingandhypothesistests contents boardworksltd2006 2of58 introductiontosampling introductiontosamplingsamplingfromanormaldistributioncalculatingfromsamplesunbiasedestimateshypothesistestingonbinomialdatachocolatetastingpracticalone sidedhypothesistestsonbinomialdataone sidedversustwo sidedtestscriticalregions thebritishgovernmentcarriesoutacensusoftheentirepopulationoftheunitedkingdomevery10years mostrecentlyinapril2001 thefirstcensusintheunitedkingdomwascarriedoutin1086withtheconstructionofthedoomesdaybook howevertheyhaveonlybeenconductedonaregularbasissince1801 thecensusprovidesthegovernmentwithadetailedpictureofthepopulationlivingineachpartofthecountry town cityorcountryside theresultsareusedtohelpplanpublicservices health housing transportandeducation forthefuture nationalcensus instatisticsweoftenwanttoobtaininformationfromagroupofindividualsoraboutagroupofobjects introductiontosampling asamplingframeisalistofallmembersofthepopulation acensusisaninvestigationinwhichinformationisobtainedfromeverymemberofthepopulation thepopulationisthesetofallindividualsorobjectsthatwewishtostudy introductiontosampling examples aheadteacherisinterestedinfindingouthowlonghersixthformstudentsspendinpart timeemploymenteachweek thepopulationisthesetofallsixthformstudentsinherschool apossiblesamplingframewouldbetheregistersofsixthformtutorgroups 2 anewspaperisinterestedinobtainingtheviewsofresidentslivingclosetothesiteofaproposednewairport thepopulationmightbealladultslivingwithina10mileradiusofthesite apossiblesamplingframecouldbethelocalelectoralroll examples 3 acarcompanyhasdiscoveredafaultthataffectsoneoftheirmodelsofcar thecompanymaywishtoknowhowwidespreadtheproblemmightbe thepopulationwouldbeallcarsproducedofthisparticularmodel apossiblesamplingframewouldbealistofallregisteredcarsofthismodelprovidedbythedvla introductiontosampling carryingoutacensusoftheentirepopulationisusuallynotfeasibleorsensible introductiontosampling moneytimeresources inaddition someinvestigationscouldresultinthedestructionoftheentirepopulation forexample ifalightbulbmanufacturerwishedtoinvestigatethelifetimeofitsbulbs acensuswouldresultinthedestructionofallthebulbsitproduced acensusisusuallycostlyintermsof insteadofsurveyingthewholepopulation informationcaninsteadbeobtainedfromasample thesamplingprocessshouldbeundertakencarefullytoensurethatthesampleisrepresentativeoftheentirepopulation biascanoccurifonesectionofthepopulationisover orunder represented introductiontosampling question alocalcouncilwishestoknowtheviewsoflocalpeopleonpublictransport criticizeeachofthefollowingsamplingregimes askthepeoplewaitingatthetowncentrebusstop leavequestionnairesinlocallibrariesforpeopletofillin askpeopleattheshoppingcentreonathursdaymorning onewaytoobtainafairsampleistouserandomsampling thismethodgiveseverymemberofthepopulationanequalchanceofbeingchosenforthesample amoreformaldefinitionofarandomsampleisasfollows thereareanumberofwaysinwhicharandomsamplecanbechosen onecommonlyusedtechniqueistouserandomnumbertables samplingmethods asampleofsizeniscalledarandomsampleifeverypossibleselectionofsizenhasthesameprobabilityofbeingchosen thetablebelowgivesalistofrandomdigits 259976452401234393053225197549628444212885355169905193439102356206753335713416584438085966235418626411469807561925290692923229288631523040940642775838281475 hereishowtouserandomdigitstoobtainasample randomnumbertables example asampleofsize15isrequiredfromapopulationofsize300 onepossibleapproachwouldbetoobtainasamplingframeforthepopulationandnumbereverymemberfrom001to300 youcouldthenobtainchainsof3randomdigitsfromtables ifthechaincorrespondstoanumberbetween001and300youcouldselectthatmemberofthepopulation otherwiseyoucoulddiscardthatchainandchooseanother example continued thismethodiswastefulofrandomdigitssincemostchainsof3digitswillbediscarded amoreefficientstrategywouldbetoassigneachmemberofthepopulationtoseveralchainsofrandomdigits randomnumbertables thisapproachleadstoonlychainsofdigitsbetween901and000beingdiscarded 259976452401234393053225197549628444212885355169905193439102356206753335713416584438085966235418626411469807561925290692923229288631523040940642775838281475 example continued supposethatweusethe2ndlineofrandomdigitsintheabovetable thenthesamplechosenwouldbe 834 234193 193439 139102 102356 56206 206753 153335 35 713 113416 116584 284438 138085 85966 cannotbeused 235 235418 118 randomnumbertables 259976452401234393053225197549628444212885355169905193439102356206753335713416584438085966235418626411469807561925290692923229288631523040940642775838281475 introductiontosamplingsamplingfromanormaldistributioncalculatingfromsamplesunbiasedestimateshypothesistestingonbinomialdatachocolatetastingpracticalone sidedhypothesistestsonbinomialdataone sidedversustwo sidedtestscriticalregions contents 13of58 boardworksltd2006 samplingfromanormaldistribution samplingfromanormaldistribution supposethatasampleofsizenistakenfroman 2 distributionandthatthesamplemeanis ifthesamplingprocessweretoberepeatedagain adifferentsamplewouldbeextractedandaslightlydifferentvalueforthesamplemeanwouldbeobtained thevalueofthesamplemeanisthereforesubjecttosamplingvariability thesamplemeanthereforehasadistribution knownasitssamplingdistribution itispossibletoshowthat whenasampleofsizenisdrawnfromanormaldistributionwithmean andstandarddeviation thesamplingdistributionofthesamplemeanis samplingfromanormaldistribution example ifasampleofsize40istakenfroman 15 24 distribution thenthesamplingdistributionofthesamplemeanis noticethatthevarianceofis thisshowsthatthesamplingvariabilitycanbedecreasedbytakinglargersamples i e increasingthevalueofn thestandarddeviationofis thisisusuallyreferredtoasthestandarderrorofthesamplemean samplingfromanormaldistribution introductiontosamplingsamplingfromanormaldistributioncalculatingfromsamplesunbiasedestimateshypothesistestingonbinomialdatachocolatetastingpracticalone sidedhypothesistestsonbinomialdataone sidedversustwo sidedtestscriticalregions contents boardworksltd2006 17of58 calculatingfromsamples recalltheformulawemetearlierforfindingthevarianceofasetofdata variance samplestandarddeviation theseformulaeareactuallyonlynormallyusedwhenwewishtocalculatethevarianceorstandarddeviationusingdatafromtheentirepopulation thestandarddeviationisthesquarerootofthis andissometimescalledtherootmeansquareddeviation rmsd rmsd thevarianceissometimescalledthemeansquareddeviation msd whenalargepopulationisbeingstudied datawillonlybecollectedforasample thesampledataisthenusedmakeinferencesaboutthepopulation sampledatamaybeusedtoestimatethemeanandvarianceofthewholepopulation samplestandarddeviation butthemostaccurateestimateofthepopulationvarianceisprovidedbythefollowingformula thisisreferredtoasthesamplevariance withthesquarerootbeingthesamplestandarddeviation s itcanbeshownthatthesamplemean givesthemostaccurateestimatepossibleofthepopulationmean example acrispmanufacturercarriesoutregularmonitoringofitspackingmachinesbytakingsamplesof20packetsofcrisps themasses xg obtainedinonesuchsamplewereasfollows findthemeanandthestandarddeviationofthemassesinthissampleofcrisppackets samplestandarddeviation note thequestionclearlymentionsthatthedataisfromasample wewillthereforeusetheformulaforthesamplestandarddeviation thesamplemeanisgivenby samplestandarddeviation thesamplestandarddeviation s isfoundasfollows introductiontosamplingsamplingfromanormaldistributioncalculatingfromsamplesunbiasedestimateshypothesistestingonbinomialdatachocolatetastingpracticalone sidedhypothesistestsonbinomialdataone sidedversustwo sidedtestscriticalregions contents 22of58 boardworksltd2006 unbiasedestimates astatisticisaquantitythatiscalculatedfromasampleofdata examplesinclude introductiontoestimation weareparticularlyinterestedinfindingestimatesofthepopulationmeanandstandarddeviation thequartiles thehighestvalue thesamplemean thesamplevariance notethatsamplevarianceusesn 1insteadofjustn itcanbeshownthatthesamplemeanprovidesanunbiasedestimateofthepopulationmean i e ifthesamplingprocesswascarriedoutoverandoveragain thesamplemeanwouldonaverageproducethepopulationmean likewisethesamplestandarddeviation s isanunbiasedestimateforthepopulationstandarddeviation unbiasedestimates notethattheformulagivesabiasedestimateofthepopulationvariance example anexaminertakesarandomsampleof12ofthestudentssittingaparticulara levelexamination theirpercentagemarkswere 55 64 76 48 73 51 67 31 55 85 60 62 calculateunbiasedestimatesofthemeanandthestandarddeviationofthemarksforallstudentssittingtheexam unbiasedestimates unbiasedestimates sothesamplestandarddeviation s 14 2 to3sf thesamplestandarddeviationgivesanunbiasedestimateofthepopulationstandarddeviation introductiontosamplingsamplingfromanormaldistributioncalculatingfromsamplesunbiasedestimateshypothesistestingonbinomialdatachocolatetastingpracticalone sidedhypothesistestsonbinomialdataone sidedversustwo sidedtestscriticalregions contents boardworksltd2006 27of58 hypothesistestingonbinomialdata considerthefollowingsimplesituation yoususpectthatadieisbiasedtowardsthenumbersix inordertotestthissuspicion youcouldperformanexperimentinwhichthedieisthrown20times ifthediewerefair youwouldexpectabout3sixes ifyouobtainedalotmorethan3sixesthenyoumightdecidethatthereisevidencetosupportyoursuspicions buthowdoyoudecideonwhatasuspiciousnumberofsixesis asimpleintroductoryexample considerthrowingafairdice20times theprobabilityofobtainingdifferentnumbersofsixesisshowninthegraph asimpleintroductoryexample so wenoticedfromthepreviousslidethat with20throwsofafairdie theprobabilityofgetting7ormoresixesisabout0 0371 thismeansthatifafairdiewerethrown20timesoverandoveragain thenyouwouldobtain7ormoresixeslessthanonceinevery20experiments thefigureof1in20 or5 isoftentakenasacut offpoint resultswithprobabilitiesbelowthislevelaresometimesregardedasbeingunlikelytohaveoccurredbychance however insituationswheremoreevidenceisrequired cut offvaluesof1 or0 1 aretypicallyused asimpleintroductoryexample inhypothesistestingweareessentiallypresentedwithtworivalhypotheses examplesmightinclude aformalintroductiontohypothesistests theserivalhypothesesarereferredtoasthenullandthealternativehypotheses thecoinisfair or thecoinisbiased theproportionoflocalpeopleinfavourofaby passis80 or theproportionissmallerthan80 thedrughasthesameeffectivenessasanexistingtreatment or thedrugismoreeffective thenullhypothesis h0 isoftenthoughtofasthecautioushypothesis itrepresentstheusualstateofaffairs thealternativehypothesis h1 isusuallytheonethatwesuspectorhopetobetrue hypothesistestingisconcernedwithexaminingthedatacollectedinexperiments anddecidinghowlikelytheresultistohaveoccurredifthenullhypothesisistrue thesignificancelevelofthetestisthechosencut offvaluebetweentheresultsthatmightplausiblyhavebeenobtainedbychanceifh0istrue andtheresultsthatareunlikelytohaveoccurred aformalintroductiontohypothesistests significancelevelsthataretypicallyusedare10 5 1 and0 1 thesesignificancelevelscorrespondtodifferentrigoursoftest thelowerthesignificancelevel thestrongertheevidencethetestwillprovide aformalintroductiontohypothesistests note itisimportanttoappreciatethatitisnotpossibletoprovethatahypothesisisdefinitelytrueinstatistics hypothesistestscanonlyprovidedifferentdegreesofevidenceinsupportofahypothesis a10 significancelevelcanonlyprovideweakevidenceinsupportofahypothesis a0 1 testismuchmorestringentandcanprovideverystrongevidence introductiontosamplingsamplingfromanormaldistributioncalculatingfromsamplesunbiasedestimateshypothesistestingonbinomialdatachocolatetastingpracticalone sidedhypothesistestsonbinomialdataone sidedversustwo sidedtestscriticalregions contents boardworksltd2006 34of58 chocolatetastingpractical doyouthinkyoucantastethedifferencebetweenbrandedchocolateandsupermarketown labelchocolate youaregoingtoperformanexperimenttofindout therewillbe2piecesofchocolatetotry onewillbeabrandedmakeofchocolate theotherwillbeasupermarket sown brand trytoidentifythebrandedmake chocolatetastingpractical chocolatetastingpractical introductiontosamplingsamplingfromanormaldistributioncalculatingfromsamplesunbiasedestimateshypothesistestingonbinomialdatachocolatetastingpracticalone sidedhypothesistestsonbinomialdataone sidedversustwo sidedtestscriticalregions contents boardworksltd2006 37of58 one sidedhypothesistestsonbinomialdata example mrjones acandidateinalocalelection claimstohavethesupportof40 oftheelectorate arivalcandidate misssmith believesthatmrjonesisexaggeratinghislevelofsupport sheasksarandomsampleof12localpeopleanddiscoversthat3ofthemsupportmrjones carryoutatestatthe5 significanceleveltoseewhetherthereisevidencethatmrjonesisexaggeratinghislevelofsupport one sidedhypothesistestsonbinomialdata solution webeginbywritingdownthe2rivalhypotheses letprepresenttheproportionoftheelectoratewhosupportmrjones h0 p 0 4h1 p 0 4 thishypothesisrepresentsourcautiousbelief i e thatmrjonesisnotlyingabouthissupport thishypothesisrepresentswhatissuspectedtobetrue i e thatmrjonesisexaggerating noticethatthehypotheseshavebeenwrittenmathematically intermsofaparameter p significancelevel 5 letxbethenumberofpeopleinthesamplewhosupportmrjones ifthenullhypothesisistrue thenx b 12 0 4 theobserveddatawasx 3 thisislessthanwewouldexpectifh0weretrue butisthisresultsoextremethatitisimplausible one sidedhypothesistestsonbinomialdata wecalculatep x 3 theprobabilityofresultsatleastasextremeasthoseobtained one sidedhypothesistestsonbinomialdata so p x 3 0 225 thesignificancelevelinthistestwaschosentobe5 theprobabilitycalculatedwasmuchhigherthanthis weconclude theevidenceisnotstrongenoughtorejecth0atthe5 significancelevel thedatadoesnotprovideevidencethatmrjoneswasexaggeratinghissupport thestepsrequiredtoanswerahypothesistestquestionusingbinomialdataare step1 writeouth0andh1inmathematicalterms step2 statethesignificancelevel ifnoneismentionedinthequestion itisusualtochoose5 step3 statethedistribution assumingthenullhypothesistobetrue step4 calculatetheprobability underh0 ofobtainingresultsasextremeasthosecollected step5 comparetheprobabilitywiththesignificancelevelandmakeconclusions canh0berejectedornot interpretyourresultsincontext one sidedhypothesistestsonbinomialdata examination stylequestion thestandardtreatmentforaparticularmedicalconditionhasasuccessrateof70 anewdrugislaunchedwhich itisclaimed treatsagreaterproportionofpatientssuccessfully adoctortriesthenewdrugon20patientsandfindsthatitsuccessfullytreats19ofthem testatthe1 significancelevelwhetherthereisevidencetosuggestthatthenewdrugtreatmentismoresuccessfulthanthestandardtreatment examination stylequestion solution letprepresenttheproportionofpatientsthataretreatedsuccessfully h0 p 0 7h1 p 0 7 thenewtreatmentisnomoresuccessfulthantheexistingtreatment thenewtreatmentisbetterthanthestandardtreatment significancelevel 1 letxbethenumberofpeoplesuccessfullytreatedbythenewdrug ifthenullhypothesisistrue thenx b 20 0 7 theobserveddataisx 19 usingtables p x 19 0 0076 1 werejectthenullhypothesisatthe1 level thereisquitestrongevidencethatthenewtreatmentismoresuccessful examination stylequestion introductiontosamplingsamplingfromanormaldistributioncalculatingfromsamplesunbiasedestimateshypothesistestingonbinomialdatachocolatetastingpracticalone sidedhypothesistestsonbinomialdataone sidedversustwo sidedtestscriticalregions contents boardworksltd2006 44of58 one sidedversustwo sidedtests theexamplesconsideredsofarcanallbeclassifiedasone sidedtests wehavebeentestingforeitheranincreaseoradecreaseinthevalueoftheparameter p sometimeswearenotlookingspecificallyforanincrease ordecrease inp butinsteadwemaywanttoexaminewhetherthevalueofphaschanged inthesesituationsweuseatwo sided oratwo tailed test atwo sidedhypothesistestcarriedoutatthe significancelevelisinasensetwoseparateone sidedtests thesignificancelevelisthereforesharedbetweenthesetwotests foreachtail one sidedversustwo sidedtests example arestauranthastraditionallyfoundthat60 ofitscustomershavebeenpleasedorverypleasedwiththequalityofthefoodserved anewchefisappointedandtherestaurantmanagementwishtofindoutwhetherthishaschangedtheproportionofcustomerswhoarehappywiththeirfood themanagementquestion16dinersanddiscoverthat14ofthemarepleasedorverypleasedwiththeirfood testatthe5 significancelevelwhethertherehasbeenachangeintheproportionofcontentedcustomers one sidedversustwo sidedtests solution letprepresenttheproportionofcustomerspleasedorverypleasedwiththequalityofthefoodserved thehypothesescanbestatedasfollows ho p 0 6 i e nochange h1 p 0 6 i e achangeintheproportion 5 significancelevel 2 5 foreachtail letxrepresentthenumberofcustomersthatarepleasedorverypleasedwiththeirfood thenunderthenullhypothesis x b 16 0 6 one sidedversustwo sidedtests ifh0weretrue wewouldexpect16 0 6 9 6customerstobepleasedwiththefoodquality theobservednumber 14 isonthehighside wecalculatep x 14 one sidedversustwo sidedtests sop x 14 0 0183 2 5 conclusion wecanrejectthenullhypothesisatthe5 significancelevel thereissomeevidencethattheproportionpleasedorverypleasedwiththeirfoodhaschanged examinationstylequestion adrivinginstructorknowsfrompastexperiencethat2outof3ofhisstudentspasstheirdrivingtestfirsttime anewdrivingexaminerisemployedatthetestcentre theinstructorwantstoknowwhetherthishaschangedtheproportionofhisstudentspassingtheirtestatthefirstattempt hemonitorsthenext12ofhisstudentstakingtheirtestsandfindsthat6passtheirtestfirsttimeround examination stylequestion a writedownasuitablenullandalternativehypothesisforthistest explainwhyyouralternativehypothesishastheformithas b carryoutthetestata10 significancelevel solution h0 p 2 3h1 p 2 3thealternativehypothesisistwo sidedsincetheinstructorislookingforachangeintheproportionofhisstudentspassingfirsttime examination stylequestion 10 significancelevel