會計學1ChapterSimpleLinearRegression商務(wù)統(tǒng)計教學SimpleLinearRegressionModely=b0+b1x+ewhere:b0andb1arecalledparametersofthemodel,

eisarandomvariablecalledtheerrorterm.

Thesimplelinearregressionmodelis:

Theequationthatdescribeshowyisrelatedtoxandanerrortermiscalledtheregressionmodel.第1頁/共50頁SimpleLinearRegressionEquationThesimplelinearregressionequationis:

E(y)istheexpectedvalueofyforagivenxvalue.

b1istheslopeoftheregressionline.

b0istheyinterceptoftheregressionline.

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

istheestimatedvalueofyforagivenxvalue.

b1istheslopeoftheline.

b0istheyinterceptoftheline.

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

y1....

xn

ynb0andb1provideestimatesofb0andb1EstimatedRegressionEquation

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

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

LeastSquaresMethodwhere:

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

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

SST=totalsumofsquares

SSR=sumofsquaresduetoregression

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

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

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

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

Totestforasignificantregressionrelationship,wemustconductahypothesistesttodeterminewhetherthevalueofb1iszero.

Twotestsarecommonlyused:tTestandFTest

BoththettestandFtestrequireanestimateofs

2,thevarianceofe

intheregressionmodel.第21頁/共50頁AnEstimateofs

TestingforSignificancewhere:s

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

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

Toestimateswetakethesquarerootofs2.

Theresultingsiscalledthestandarderrorof

theestimate.第23頁/共50頁Hypotheses

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

t

isbasedonatdistribution withn-2degreesoffreedomRejectH0ifp-value<

a

ort

<-tort

>

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

H0isrejectedifthehypothesizedvalueof1isnotincludedintheconfidenceintervalfor1.

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

Conclusion

第30頁/共50頁Hypotheses

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

FisbasedonanFdistributionwith 1degreeoffreedominthenumeratorand

n-2degreesoffreedominthedenominatorRejectH0if

p-value<

aorF

>

F第32頁/共50頁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頁/共50頁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頁/共50頁SomeCautionsaboutthe

InterpretationofSignificanceTests

JustbecauseweareabletorejectH0:b1=0and demonstratestatisticalsignificancedoesnotenable ustoconcludethatthereisalinearrelationship betweenxandy.

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

relationshipispresentbetweenxandy.第35頁/共50頁UsingtheEstimatedRegressionEquation

forEstimationandPredictionwhere: confidencecoefficientis1-and

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

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

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

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

Much