(1.31)-PowerPoint Ch 12a商务统计课件

Chapter12,PartA

SimpleLinearRegressionSimpleLinearRegressionModelLeastSquaresMethodCoefficientofDeterminationModelAssumptionsTestingforSignificanceSimpleLinearRegression

Regressionanalysiscanbeusedtodevelopanequationshowinghowthevariablesarerelated.Managerialdecisionsoftenarebasedontherelationshipbetweentwoormorevariables.Thevariablesbeingusedtopredictthevalueofthedependentvariablearecalledtheindependent

variablesandaredenotedbyx.Thevariablebeingpredictediscalledthedependent

variableandisdenotedbyy.SimpleLinearRegressionTherelationshipbetweenthetwovariablesisapproximatedbyastraightline.

Simplelinearregressioninvolvesoneindependentvariableandonedependentvariable.Regressionanalysisinvolvingtwoormoreindependentvariablesiscalledmultipleregression.SimpleLinearRegressionModely=b0+b1x+ewhere:b0andb1arecalledparametersofthemodel,

eisarandomvariablecalledtheerrorterm.Thesimplelinearregressionmodelis:Theequationthatdescribeshowyisrelatedtoxandanerrortermiscalledtheregressionmodel.SimpleLinearRegressionEquationThesimplelinearregressionequationis:E(y)istheexpectedvalueofyforagivenxvalue.

b1istheslopeoftheregressionline.

b0istheyinterceptoftheregressionline.Graphoftheregressionequationisastraightline.E(y)=

0+

1xSimpleLinearRegressionEquationPositiveLinearRelationshipE(y)xSlopeb1ispositiveRegressionlineInterceptb0SimpleLinearRegressionEquationNegativeLinearRelationshipE(y)xSlopeb1isnegativeRegressionlineInterceptb0SimpleLinearRegressionEquationNoRelationshipE(y)xSlopeb1is0RegressionlineInterceptb0EstimatedSimpleLinearRegressionEquationTheestimatedsimplelinearregressionequation

istheestimatedvalueofyforagivenxvalue.

b1istheslopeoftheline.

b0istheyinterceptoftheline.Thegraphiscalledtheestimatedregressionline.EstimationProcessRegressionModely=b0+b1x+eRegressionEquationE(y)=b0+b1xUnknownParametersb0,b1SampleData:xyx1

y1....

xn

ynb0andb1provideestimatesofb0andb1EstimatedRegressionEquation

SampleStatisticsb0,b1LeastSquaresMethodLeastSquaresCriterionwhere:

yi=observedvalueofthedependentvariable fortheithobservation^yi=estimatedvalueofthedependentvariablefortheithobservationSlopefortheEstimatedRegressionEquationLeastSquaresMethodwhere:

xi=valueofindependentvariableforith observation_y=meanvaluefordependentvariable_x=meanvalueforindependentvariableyi=valueofdependentvariableforithobservationy-InterceptfortheEstimatedRegressionEquationLeastSquaresMethod ReedAutoperiodicallyhasaspecialweek-longsale.AspartoftheadvertisingcampaignReedrunsoneormoretelevisioncommercialsduringtheweekendprecedingthesale.Datafromasampleof5previoussalesareshownonthenextslide.SimpleLinearRegressionExample:ReedAutoSalesSimpleLinearRegressionExample:ReedAutoSalesNumberofTVAds(x)NumberofCarsSold(y)132131424181727Sx=10Sy=100EstimatedRegressionEquationSlopefortheEstimatedRegressionEquationy-InterceptfortheEstimatedRegressionEquationEstimatedRegressionEquationUsingExcel’sChartToolsforScatterDiagram&EstimatedRegressionEquationReedAutoSalesEstimatedRegressionLineCoefficientofDeterminationRelationshipAmongSST,SSR,SSEwhere:

SST=totalsumofsquares

SSR=sumofsquaresduetoregression

SSE=sumofsquaresduetoerrorSST=SSR+SSEThecoefficientofdeterminationis:CoefficientofDeterminationwhere:

SSR=sumofsquaresduetoregression SST=totalsumofsquaresr2=SSR/SSTCoefficientofDeterminationr2=SSR/SST=100/114=.8772Theregressionrelationshipisverystrong;87.72%ofthevariabilityinthenumberofcarssoldcanbeexplainedbythelinearrelationshipbetweenthenumberofTVadsandthenumberofcarssold.SampleCorrelationCoefficientwhere:

b1=theslopeoftheestimatedregression equationThesignofb1intheequation is“+”.SampleCorrelationCoefficientrxy=

+.9366AssumptionsAbouttheErrorTerme1.Theerror

isarandomvariablewithmeanofzero.2.Thevarianceof

,denotedby

2,isthesameforallvaluesoftheindependentvariable.3.Thevaluesof

areindependent.4.Theerror

isanormallydistributedrandomvariable.TestingforSignificanceTotestforasignificantregressionrelationship,wemustconductahypothesistesttodeterminewhetherthevalueofb1iszero.Twotestsarecommonlyused:tTestandFTestBoththettestandFtestrequireanestimateofs

2,thevarianceofe

intheregressionmodel.AnEstimateofs

2

TestingforSignificancewhere:s

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

2,andthenotations2isalsoused.TestingforSignificanceAnEstimateofsToestimateswetakethesquarerootofs2.Theresultingsiscalledthestandarderrorof

theestimate.Hypotheses

TestStatisticTestingforSignificance:tTestwhereRejectionRuleTestingforSignificance:tTestwhere:

t

isbasedonatdistribution withn-2degreesoffreedomRejectH0ifp-value<

a

ort

<-t

ort

>

t

1.Determinethehypotheses.2.Specifythelevelofsignificance.3.Selecttheteststatistic.a=.054.Statetherejectionrule.RejectH0ifp-value<.05or|t|>3.182(with3degreesoffreedom)TestingforSignificance:tTestTestingforSignificance:tTest5.Computethevalueoftheteststatistic.6.DeterminewhethertorejectH0.t=4.541providesanareaof.01intheuppertail.Hence,thep-valueislessthan.02.(Also,t=4.63>3.182.)WecanrejectH0.ConfidenceIntervalfor

1H0isrejectedifthehypothesizedvalueof

1isnotincludedintheconfidenceintervalfor

1.Wecanusea95%confidenceintervalfor

1totestthehypothesesjustusedinthettest.Theformofaconfidenceintervalfor

1is:ConfidenceIntervalfor

1where isthetvalueprovidinganareaofa/2intheuppertailofatdistributionwithn-2degreesoffreedomb1isthepointestimatoristhemarginoferrorConfidenceIntervalfor

1RejectH0if0isnotincludedintheconfidenceintervalfor

1.0isnotincludedintheconfidenceinterval.RejectH0=5+/-3.182(1.08)=5+/-3.44or1.56to8.44RejectionRule95%ConfidenceIntervalfor

1

Conclusion

Hypotheses

TestStatisticTestingforSignificance:FTestF=MSR/MSERejectionRuleTestingforSignificance:FTestwhere:

F

isbasedonanFdistributionwith 1degreeoffreedominthenumeratorand

n-2degreesoffreedominthedenominatorRejectH0if

p-value<

aorF

>

F

1.Determinethehypotheses.2.Specifythelevelofsignificance.3.Selecttheteststatistic.a=.054.Statetherejectionrule.RejectH0ifp-value<.05orF

>10.13(with1d.f.innumeratorand3d.f.indenominator)TestingforSignificance:FTestF=MSR/MSETestingforSignificance:FTest5.Computethevalueofthetests