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,theconclu