ChapterSimpleLinearRegression商务统计教学

会计学1ChapterSimpleLinearRegression商务统计教学SimpleLinearRegressionModely=b0+b1x+ewhere:b0andb1arecalledparametersofthemodel,

eisarandomvariablecalledtheerrorterm.

Thesimplelinearregressionmodelis:

Theequationthatdescribeshowyisrelatedtoxandanerrortermiscalledtheregressionmodel.第1页/共50页SimpleLinearRegressionEquationThesimplelinearregressionequationis:

E(y)istheexpectedvalueofyforagivenxvalue.

b1istheslopeoftheregressionline.

b0istheyinterceptoftheregressionline.

Graphoftheregressionequationisastraightline.E(y)=0+1x第2页/共50页SimpleLinearRegressionEquationPositiveLinearRelationshipE(y)xSlopeb1ispositiveRegressionlineInterceptb0第3页/共50页SimpleLinearRegressionEquationNegativeLinearRelationshipE(y)xSlopeb1isnegativeRegressionlineInterceptb0第4页/共50页SimpleLinearRegressionEquationNoRelationshipE(y)xSlopeb1is0RegressionlineInterceptb0第5页/共50页EstimatedSimpleLinearRegressionEquationTheestimatedsimplelinearregressionequation

istheestimatedvalueofyforagivenxvalue.

b1istheslopeoftheline.

b0istheyinterceptoftheline.

Thegraphiscalledtheestimatedregressionline.第6页/共50页EstimationProcessRegressionModely=b0+b1x+eRegressionEquationE(y)=b0+b1xUnknownParametersb0,b1SampleData:xyx1

y1....

xn

ynb0andb1provideestimatesofb0andb1EstimatedRegressionEquation

SampleStatisticsb0,b1第7页/共50页LeastSquaresMethodLeastSquaresCriterionwhere:

yi=observedvalueofthedependentvariable fortheithobservation^yi=estimatedvalueofthedependentvariablefortheithobservation第8页/共50页SlopefortheEstimatedRegressionEquationLeastSquaresMethod第9页/共50页y-InterceptfortheEstimatedRegressionEquation

LeastSquaresMethodwhere:

xi=valueofindependentvariableforith observationn=totalnumberofobservations_y=meanvaluefordependentvariable_x=meanvalueforindependentvariableyi=valueofdependentvariableforithobservation第10页/共50页

ReedAutoperiodicallyhasaspecialweek-longsale.AspartoftheadvertisingcampaignReedrunsoneormoretelevisioncommercialsduringtheweekendprecedingthesale.Datafromasampleof5previoussalesareshownonthenextslide.SimpleLinearRegressionExample:ReedAutoSales第11页/共50页SimpleLinearRegressionExample:ReedAutoSalesNumberofTVAdsNumberofCarsSold132131424181727第12页/共50页EstimatedRegressionEquationSlopefortheEstimatedRegressionEquationy-InterceptfortheEstimatedRegressionEquationEstimatedRegressionEquation第13页/共50页ScatterDiagramandTrendLine第14页/共50页CoefficientofDeterminationRelationshipAmongSST,SSR,SSEwhere:

SST=totalsumofsquares

SSR=sumofsquaresduetoregression

SSE=sumofsquaresduetoerrorSST=SSR+SSE第15页/共50页Thecoefficientofdeterminationis:CoefficientofDeterminationwhere:

SSR=sumofsquaresduetoregression SST=totalsumofsquaresr2=SSR/SST第16页/共50页CoefficientofDeterminationr2=SSR/SST=100/114=.8772

Theregressionrelationshipisverystrong;88%ofthevariabilityinthenumberofcarssoldcanbeexplainedbythelinearrelationshipbetweenthenumberofTVadsandthenumberofcarssold.第17页/共50页SampleCorrelationCoefficientwhere:

b1=theslopeoftheestimatedregression equation第18页/共50页Thesignofb1intheequation is“+”.SampleCorrelationCoefficientrxy=+.9366第19页/共50页AssumptionsAbouttheErrorTerme1.Theerrorisarandomvariablewithmeanofzero.2.Thevarianceof,denotedby2,isthesameforallvaluesoftheindependentvariable.3.Thevaluesofareindependent.4.Theerrorisanormallydistributedrandomvariable.第20页/共50页TestingforSignificance

Totestforasignificantregressionrelationship,wemustconductahypothesistesttodeterminewhetherthevalueofb1iszero.

Twotestsarecommonlyused:tTestandFTest

BoththettestandFtestrequireanestimateofs

2,thevarianceofe

intheregressionmodel.第21页/共50页AnEstimateofs

TestingforSignificancewhere:s

2=MSE=SSE/(n-2)Themeansquareerror(MSE)providestheestimateofs

2,andthenotations2isalsoused.第22页/共50页TestingforSignificanceAnEstimateofs

Toestimateswetakethesquarerootofs2.

Theresultingsiscalledthestandarderrorof

theestimate.第23页/共50页Hypotheses

TestStatisticTestingforSignificance:tTest第24页/共50页RejectionRuleTestingforSignificance:tTestwhere:

t

isbasedonatdistribution withn-2degreesoffreedomRejectH0ifp-value<

a

ort

<-tort

>

t第25页/共50页1.Determinethehypotheses.2.Specifythelevelofsignificance.3.Selecttheteststatistic.a=.054.Statetherejectionrule.RejectH0ifp-value<.05or|t|>3.182(with3degreesoffreedom)TestingforSignificance:tTest第26页/共50页TestingforSignificance:tTest5.Computethevalueoftheteststatistic.6.DeterminewhethertorejectH0.t=4.541providesanareaof.01intheuppertail.Hence,thep-valueislessthan.02.(Also,t=4.63>3.182.)WecanrejectH0.第27页/共50页ConfidenceIntervalfor1

H0isrejectedifthehypothesizedvalueof1isnotincludedintheconfidenceintervalfor1.

Wecanusea95%confidenceintervalfor1totestthehypothesesjustusedinthettest.第28页/共50页Theformofaconfidenceintervalfor1is:ConfidenceIntervalfor1where isthetvalueprovidinganareaofa/2intheuppertailofatdistributionwithn-2degreesoffreedomb1isthepointestimatoristhemarginoferror第29页/共50页ConfidenceIntervalfor1RejectH0if0isnotincludedintheconfidenceintervalfor1.0isnotincludedintheconfidenceinterval.RejectH0=5+/-3.182(1.08)=5+/-3.44or1.56to8.44RejectionRule95%ConfidenceIntervalfor1

Conclusion

第30页/共50页Hypotheses

TestStatisticTestingforSignificance:FTestF=MSR/MSE第31页/共50页RejectionRuleTestingforSignificance:FTestwhere:

FisbasedonanFdistributionwith 1degreeoffreedominthenumeratorand

n-2degreesoffreedominthedenominatorRejectH0if

p-value<

aorF

>

F第32页/共50页1.Determinethehypotheses.2.Specifythelevelofsignificance.3.Selecttheteststatistic.a=.054.Statetherejectionrule.RejectH0ifp-value<.05orF

>10.13(with1d.f.innumeratorand3d.f.indenominator)TestingforSignificance:FTestF=MSR/MSE第33页/共50页TestingforSignificance:FTest5.Computethevalueoftheteststatistic.6.DeterminewhethertorejectH0.

F=17.44providesanareaof.025intheuppertail.Thus,thep-valuecorrespondingtoF=21.43islessthan2(.025)=.05.Hence,werejectH0.F=MSR/MSE=100/4.667=21.43

ThestatisticalevidenceissufficienttoconcludethatwehaveasignificantrelationshipbetweenthenumberofTVadsairedandthenumberofcarssold.第34页/共50页SomeCautionsaboutthe

InterpretationofSignificanceTests

JustbecauseweareabletorejectH0:b1=0and demonstratestatisticalsignificancedoesnotenable ustoconcludethatthereisalinearrelationship betweenxandy.

RejectingH0:b1=0andconcludingthatthe relationshipbetweenxandyissignificantdoes notenableustoconcludethatacause-and-effect

relationshipispresentbetweenxandy.第35页/共50页UsingtheEstimatedRegressionEquation

forEstimationandPredictionwhere: confidencecoefficientis1-and

t/2isbasedonatdistribution withn-2degreesoffreedomConfidenceIntervalEstimateofE(yp)PredictionIntervalEstimateofyp第36页/共50页

If3TVadsarerunpriortoasale,weexpect themeannumberofcarssoldtobe:PointEstimation^y=10+5(3)=25cars第37页/共50页Excel’sConfidenceIntervalOutputConfidenceIntervalforE(yp)第38页/共50页

The95%confidenceintervalestimateofthemeannumberofcarssoldwhen3TVadsarerunis:ConfidenceIntervalforE(yp)25+4.61=20.39to29.61cars第39页/共50页Excel’sPredictionIntervalOutputPredictionIntervalforyp第40页/共50页

The95%predictionintervalestimateofthenumberofcarssoldinoneparticularweekwhen3TVadsarerunis:PredictionIntervalforyp25+8.28=16.72to33.28cars第41页/共50页ResidualAnalysis

Much