Statistics Definitions and Explanations

Definitions Population Theentirecollectionofelements(e.g.persons,items)thatwewanttostudyordrawconclusionsabout Sample Asubsetofthepopulationfromwhichwecollectdata Randomsample Asamplewhere…● Eachelementhasanequalchanceofbeingselected● Theselectionofeachelementisindependentoftheselectionofeveryotherelement Unbiasedestimate Anestimatesuchthattheaverageofsuchestimatesobtainedfromallpossiblesamplesofthesamesizegivesthetruevalueofthepopulationparameter Significancelevel TheprobabilityofrejectingH0 whenH0 isinfacttrue Criticalregion ThesetofvaluesofthesamplemeanforwhichH0 isrejected Criticalvalue Thevalue(s)thatdefinesthecriticalregion,i.e.thevalue(s)suchthatifthesamplemeanismoreextremethanthatvalue,H0 isrejected P-value [IfH1 :μ<μ0 ORH1 :μ>μ0 ]Theprobabilityofgettingasamplemeanless/greaterthantheobservedsamplemean [IfH1 :μ≠μ0 ]Theprobabilityofgettingasamplemeanmoreextremethantheobservedsamplemean Independentvariable Avariablewhosechangewillhaveaneffectonthedependentvariable Dependentvariable Avariablewhichisaffectedbychangesoftheindependentvariable(s) Residual Thedifferencebetweentheactualvalueofyandthepredictedvalueofybyamodelforagivenvalueofx Rememberthatonlyevents,notprobabilities,canbeindependent!

Explanations HypothesisTesting “Stateyourconclusionincontext” [Ifp-value<α%]Sincep-value<α,werejectH0 .Thereissufficientevidenceattheα%significancelevelthat… [Ifp-value>α%]Sincep-value>α,wedonotrejectH0 .Thereisinsufficientevidenceattheα%significancelevelthat… Whyisthedatasummarisedintheform(x-μ)? Sincethemeanofthepopulationparameterisμ,summarisingthedataintheform(x-μ)givesthedifferencebetweentheparameterandthemean,allowingthevaluestobesmallerandeasiertocalculate. Howshouldthesamplebeconducted?/Whyshouldthesampleberandom? Arandomsampleshouldbeconductedsuchthatthesampleisunbiased,i.e.representativeofthepopulation.● Forinstance,byassigningeachelementanaturalnumber,generatingalistofNrandomnumberswithacalculator,andselectingthecorrespondingNelements WhymustasampleofsizeN>30betaken? Thisisbecausethedistributionofthepopulationparameterisnotknown/thepopulationparameterisnotnormallydistributed,soalargesampleofN>30mustbetakensuchthattheCentralLimitTheoremappliesandthesamplemeanwillbeapproximatelynormallydistributed. Shouldaone-tailortwo-tailtestbeconducted? [IfH1 :μ<μ0 ORH1 :μ>μ0 ]Aone-tailtestshouldbeconducted,asweareonlyinterestedintestingifthepopulationmeanisless/greaterthanμ0 ,andnotwhetheritisgreater/lessthanμ0 . [IfH1 :μ≠μ0 ]Atwo-tailtestshouldbeconducted,asweareinterestedintestingifthepopulationmeandiffersfromμ0 .Todifferfromμ0 ,thepopulationmeancouldeitherbegreaterthanμ0 orlessthanμ0 ,makingatwo-tailtestappropriate. Whatassumptionsmustbemadeforthehypothesistest? [IfN<30]Itmustbeassumedthatthepopulationparameterisnormallydistributed. [Ifanewvarianceisnotgiven]Itmustbeassumedthatthepopulationvarianceremainsthesame. Uponchangingthesignificancelevel/populationvariance,willhisconclusionchange? [IfH0 isrejected,andα%increasesorthevariancedecreases]No,theconclusionwillnotchange.Thisisbecause…● H0 isalreadyrejectedattheα%significancelevel,andα%islessthanthenewsignificancelevel● Thep-valueisalreadylessthanα%,andthep-valuewillfurtherdecreaseifthevariancedecreases…soH0 willstillberejectedandtheconclusionwillremainthe

same. [IfH0 isnotrejected,andα%increasesorthevariancedecreases]Yes,theconclusionmaychange.Thisisbecause…● H0 isnotrejectedattheα%significancelevel,butthenewsignificancelevelisgreaterthanα%—andpossiblythep-value● Thep-valueisgreaterthanα%,butthep-valuewilldecrease—possiblybelowα%—ifthevariancedecreases…soH0 mayberejectedandtheconclusionmaychange. Permutefortheotherpossibilities… RandomVariables Whyisabinomialdistributionsuitableasamodel? Thisisbecause…● Thereareonlytwomutuallyexclusiveoutcomesof‘success’and‘failure’● Theprobabilityof‘success’remainsconstant● Eachtrialisindependentofeveryothertrial Whyisanormaldistributionunsuitableasamodel? Thiscouldbebecause…● Thedistributionisnotsymmetricalaboutthemean● Themedianisnotequivalenttothemean● Thevaluesarenotdistributedsuchthat68%/95%/99.7%ofthevaluesarewithin±σ/±2σ/±3σofthemean● Theprobabilityofgettinganimpossible/negativevalueis___,whichisfairlyhigh Whatassumptionshavetobemadeforyourcalculations? ItmustbeassumedthatX1 andX2 areindependent/XandYareindependent. WhyistheprobabilityofEventAsmallerthantheprobabilityofEventB? ThisisbecauseEventAisapropersubsetofEventB—ifEventAoccursEventBwilldefinitelyoccur,butEventBcouldoccurevenifEventAdoesnotoccur.● Forexample… CorrelationandRegression Basedonthescatterplot,whyisalinearmodelunsuitable? Alinearmodelisunsuitablebecauseasyincreases,xincreases/decreasesatanincreasing/decreasingrate,sothepointslieclosetoacurveandtherelationshipbetweenxandyisnon-linear. Basedonthevalueofr,whyisalinearmodelunsuitable? Alinearmodelisunsuitableas|r|isnotcloseto1/closeto0,sothereisnolinearrelationshipbetweenthexandy. IsModelAorModelBmoresuitable? UnderModelA,asxincreases,yincreases/decreasesatanincreasing/decreasing/constantrate.Conversely,underModelB,yincreases/decreasesatanincreasing/decreasing/constantrate.

Fromthescatterplot,sinceasxincreases,yincreases/decreasesatanincreasing/decreasing/constantrate,andthevalueof|r|underModelAisgreaterthanthevalueof|r|underModelB,ModelAismoresuitable. Whyisthelinearmodelunsuitableforlargevaluesofx? Thelinearmodelisunsuitableasforlargevaluesofx,thelinearmodelwillgivenegativevaluesofy,whichareclearlyimpossible. Whatisthesignificanceofminy=mx+c? Thevalueofmrepresentstheincreaseinyforeveryunitincreaseofx. Whyshouldtheregressionlineofyonxbeused? [Ifydependsonx]Thisisbecauseyistheindependentvariableandxisthedependentvariable. [Ifydoesnotdependonx]Thisisbecausethereisnoclearindependentordependentvariable,andweareestimatingthevalueofybasedonagivenvalueofx. Whyistheestimatereliable/unreliable? [Ifitisreliable]Theestimateisreliableas…● Thedataiswithinthegivendatarange,soweareinterpolating● Thevalueof|r|iscloseto1 [Ifitisunreliable]Theestimateisunreliableas…● Thedataisoutsidethegivendatarange,soweareextrapolating● Thevalueof|r|isnotcloseto1 Whyisitcalledthe“leastsquaresregressionline”? Thisisbecausetheleastsquaresregressionlineisthelineforwhichthesumofthesquaresoftheresidualsisminimised. Whyaretheresidualssquared? Thisisdonetoensuretheresidualswillnotcanceleachotheroutwhensummedandthesumoftheresidualswillnotbenegative,asresidualscanbeeitherpositive(whentheyareabovetheregressionline)ornegative(whentheyarebelowtheregressionline). Inaddition,squaringmakesthelargerresidualsevenlarger,increasingtheirimpactonthesumoftheresiduals. Willthescalingofunitsaffectthevalueofr? No,thescalingofunitswillnotaffectthevalueofr,astherelationshipbetweenthevariablesremainsunchanged. Whyshouldweremoveadatapoint? Thisisbecausethedatapointisanoutlierwhichmayskewordistorttherelationshipbetweenthetwovariables,negativelyaffectingtheaccuracyofpredictions.

[Ifapplicable]Thisisbecausethedatapointwastheresultofaninaccuratemeasurement,oritisnotpartofthepopulationwearestudying.