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Chapter12SimpleLinearRegressionBusinessStatistics:AFirstCourse
FifthEdition1LearningObjectivesInthischapter,youlearn:
HowtouseregressionanalysistopredictthevalueofadependentvariablebasedonanindependentvariableThemeaningoftheregressioncoefficientsb0andb1HowtoevaluatetheassumptionsofregressionanalysisandknowwhattodoiftheassumptionsareviolatedTomakeinferencesabouttheslopeandcorrelationcoefficientToestimatemeanvaluesandpredictindividualvalues2Correlationvs.RegressionAscatterplotcanbeusedtoshowtherelationshipbetweentwovariablesCorrelationanalysisisusedtomeasurethestrengthoftheassociation(linearrelationship)betweentwovariablesCorrelationisonlyconcernedwithstrengthoftherelationshipNocausaleffectisimpliedwithcorrelationScatterplotswerefirstpresentedinCh.2CorrelationwasfirstpresentedinCh.33Introductionto
RegressionAnalysisRegressionanalysisisusedto:PredictthevalueofadependentvariablebasedonthevalueofatleastoneindependentvariableExplaintheimpactofchangesinanindependentvariableonthedependentvariableDependentvariable:thevariablewewishto predictorexplainIndependentvariable:thevariableusedtopredict orexplainthedependent variable4SimpleLinearRegressionModelOnlyoneindependentvariable,XRelationshipbetweenXandYisdescribedbyalinearfunctionChangesinYareassumedtoberelatedtochangesinX5TypesofRelationshipsYXYXYYXXLinearrelationshipsCurvilinearrelationships6TypesofRelationshipsYXYXYYXXStrongrelationshipsWeakrelationships(continued)7TypesofRelationshipsYXYXNorelationship(continued)8LinearcomponentSimpleLinearRegressionModelPopulation
YinterceptPopulationSlope
CoefficientRandomErrortermDependentVariableIndependentVariableRandomErrorcomponent9(continued)RandomErrorforthisXivalueYXObservedValueofYforXiPredictedValueofYforXi
XiSlope=β1Intercept=β0
εiSimpleLinearRegressionModel10ThesimplelinearregressionequationprovidesanestimateofthepopulationregressionlineSimpleLinearRegressionEquation(PredictionLine)Estimateoftheregression
interceptEstimateoftheregressionslope
Estimated(orpredicted)YvalueforobservationiValueofXforobservationi11TheLeastSquaresMethodb0andb1areobtainedbyfindingthevaluesofthatminimizethesumofthesquareddifferencesbetweenYand:12FindingtheLeastSquaresEquationThecoefficientsb0andb1,andotherregressionresultsinthischapter,willbefoundusingExcelorMinitabFormulasareshowninthetextforthosewhoareinterested13b0istheestimatedmeanvalueofYwhenthevalueofXiszerob1istheestimatedchangeinthemeanvalueofYasaresultofaone-unitchangeinXInterpretationofthe
SlopeandtheIntercept14SimpleLinearRegressionExampleArealestateagentwishestoexaminetherelationshipbetweenthesellingpriceofahomeanditssize(measuredinsquarefeet)Arandomsampleof10housesisselectedDependentvariable(Y)=housepricein$1000sIndependentvariable(X)=squarefeet15SimpleLinearRegressionExample:DataHousePricein$1000s(Y)SquareFeet(X)245140031216002791700308187519911002191550405235032424503191425255170016SimpleLinearRegressionExample:ScatterPlotHousepricemodel:ScatterPlot17SimpleLinearRegressionExample:UsingExcel18SimpleLinearRegressionExample:ExcelOutputRegressionStatisticsMultipleR0.76211RSquare0.58082AdjustedRSquare0.52842StandardError41.33032Observations10ANOVA
dfSSMSFSignificanceFRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000
CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.18580Theregressionequationis:19SimpleLinearRegressionExample:MinitabOutputTheregressionequationisPrice=98.2+0.110SquareFeet
Predictor
Coef
SECoef
T
PConstant
98.25
58.03
1.69
0.129SquareFeet
0.10977
0.03297
3.33
0.010
S=41.3303
R-Sq=58.1%
R-Sq(adj)=52.8%
AnalysisofVariance
Source
DF
SS
MS
F
PRegression
1
18935
18935
11.08
0.010ResidualError
8
13666
1708Total
9
32600Theregressionequationis:houseprice=98.24833+ 0.10977(squarefeet)20SimpleLinearRegressionExample:GraphicalRepresentationHousepricemodel:ScatterPlotandPredictionLineSlope=0.10977Intercept=98.24821SimpleLinearRegressionExample:Interpretationofbob0istheestimatedmeanvalueofYwhenthevalueofXiszero(ifX=0isintherangeofobservedXvalues)Becauseahousecannothaveasquarefootageof0,b0hasnopracticalapplication22SimpleLinearRegressionExample:Interpretingb1b1estimatesthechangeinthemeanvalueofYasaresultofaone-unitincreaseinXHere,b1=0.10977tellsusthatthemeanvalueofahouseincreasesby0.10977($1000)=$109.77,onaverage,foreachadditionalonesquarefootofsize23Predictthepriceforahousewith2000squarefeet:Thepredictedpriceforahousewith2000squarefeetis317.85($1,000s)=$317,850SimpleLinearRegressionExample:MakingPredictions24SimpleLinearRegressionExample:MakingPredictionsWhenusingaregressionmodelforprediction,onlypredictwithintherelevantrangeofdataRelevantrangeforinterpolationDonottrytoextrapolatebeyondtherangeofobservedX’s25MeasuresofVariationTotalvariationismadeupoftwoparts:TotalSumofSquaresRegressionSumofSquaresErrorSumofSquareswhere:
=Meanvalueofthedependentvariable
Yi=Observedvalueofthedependentvariable =PredictedvalueofYforthegivenXivalue26SST=totalsumofsquares(TotalVariation)MeasuresthevariationoftheYivaluesaroundtheirmeanYSSR=regressionsumofsquares(ExplainedVariation)VariationattributabletotherelationshipbetweenXandYSSE=errorsumofsquares(UnexplainedVariation)VariationinYattributabletofactorsotherthanX(continued)MeasuresofVariation27(continued)XiYXYiSST
=
(Yi
-
Y)2SSE
=
(Yi
-
Yi)2
SSR=
(Yi
-
Y)2
___Y
YY_Y
MeasuresofVariation28ThecoefficientofdeterminationistheportionofthetotalvariationinthedependentvariablethatisexplainedbyvariationintheindependentvariableThecoefficientofdeterminationisalsocalledr-squaredandisdenotedasr2CoefficientofDetermination,r2note:29r2=1Examplesofr2ValuesYXYXr2=1r2=1PerfectlinearrelationshipbetweenXandY:100%ofthevariationinYisexplainedbyvariationinX30Examplesofr2ValuesYXYX0<r2<1WeakerlinearrelationshipsbetweenXandY:SomebutnotallofthevariationinYisexplainedbyvariationinX31Examplesofr2Valuesr2=0NolinearrelationshipbetweenXandY:ThevalueofYdoesnotdependonX.(NoneofthevariationinYisexplainedbyvariationinX)YXr2=032SimpleLinearRegressionExample:CoefficientofDetermination,r2inExcelRegressionStatisticsMultipleR0.76211RSquare0.58082AdjustedRSquare0.52842StandardError41.33032Observations10ANOVA
dfSSMSFSignificanceFRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000
CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.1858058.08%ofthevariationinhousepricesisexplainedbyvariationinsquarefeet33SimpleLinearRegressionExample:CoefficientofDetermination,r2inMinitabTheregressionequationisPrice=98.2+0.110SquareFeet
Predictor
Coef
SECoef
T
PConstant
98.25
58.03
1.69
0.129SquareFeet
0.10977
0.03297
3.33
0.010
S=41.3303
R-Sq=58.1%
R-Sq(adj)=52.8%
AnalysisofVariance
Source
DF
SS
MS
F
PRegression
1
18935
18935
11.08
0.010ResidualError
8
13666
1708Total
9
3260058.08%ofthevariationinhousepricesisexplainedbyvariationinsquarefeet34StandardErrorofEstimateThestandarddeviationofthevariationofobservationsaroundtheregressionlineisestimatedbyWhere SSE=errorsumofsquares n=samplesize35SimpleLinearRegressionExample:
StandardErrorofEstimateinExcelRegressionStatisticsMultipleR0.76211RSquare0.58082AdjustedRSquare0.52842StandardError41.33032Observations10ANOVA
dfSSMSFSignificanceFRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000
CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.1858036SimpleLinearRegressionExample:
StandardErrorofEstimateinMinitabTheregressionequationisPrice=98.2+0.110SquareFeet
Predictor
Coef
SECoef
T
PConstant
98.25
58.03
1.69
0.129SquareFeet
0.10977
0.03297
3.33
0.010
S=41.3303
R-Sq=58.1%
R-Sq(adj)=52.8%
AnalysisofVariance
Source
DF
SS
MS
F
PRegression
1
18935
18935
11.08
0.010ResidualError
8
13666
1708Total
9
3260037ComparingStandardErrorsYYXXSYXisameasureofthevariationofobservedYvaluesfromtheregressionlineThemagnitudeofSYXshouldalwaysbejudgedrelativetothesizeoftheYvaluesinthesampledatai.e.,SYX=$41.33Kis
moderatelysmallrelativetohousepricesinthe$200K-$400Krange38LinearityTherelationshipbetweenXandYislinearIndependenceofErrorsErrorvaluesarestatisticallyindependentNormalityofErrorErrorvaluesarenormallydistributedforanygivenvalueofXEqualVariance(alsocalledhomoscedasticity)Theprobabilitydistributionoftheerrorshasconstantvariance39ResidualAnalysisTheresidualforobservationi,ei,isthedifferencebetweenitsobservedandpredictedvalueChecktheassumptionsofregressionbyexaminingtheresidualsExamineforlinearityassumptionEvaluateindependenceassumptionEvaluatenormaldistributionassumptionExamineforconstantvarianceforalllevelsofX(homoscedasticity)GraphicalAnalysisofResidualsCanplotresidualsvs.X40ResidualAnalysisforLinearityNotLinearLinear
xresidualsxYxYxresiduals41ResidualAnalysisforIndependenceNotIndependentIndependentXXresidualsresidualsXresiduals
42CheckingforNormalityExaminetheStem-and-LeafDisplayoftheResidualsExaminetheBoxplotoftheResidualsExaminetheHistogramoftheResidualsConstructaNormalProbabilityPlotoftheResiduals43ResidualAnalysisforNormalityPercentResidualWhenusinganormalprobabilityplot,normalerrorswillapproximatelydisplayinastraightline-3-2-10123010044ResidualAnalysisfor
EqualVarianceNon-constantvariance
ConstantvariancexxYxxYresidualsresiduals45SimpleLinearRegressionExample:ExcelResidualOutputRESIDUALOUTPUTPredictedHousePriceResiduals1251.92316-6.9231622273.8767138.123293284.85348-5.8534844304.062843.9371625218.99284-19.992846268.38832-49.388327356.2025148.797498367.17929-43.179299254.667464.3326410284.85348-29.85348Doesnotappeartoviolateanyregressionassumptions46InferencesAbouttheSlopeThestandarderroroftheregressionslopecoefficient(b1)isestimatedbywhere:
=Estimateofthestandarderroroftheslope =Standarderroroftheestimate47InferencesAbouttheSlope:
tTestttestforapopulationslopeIstherealinearrelationshipbetweenXandY?NullandalternativehypothesesH0:β1=0 (nolinearrelationship)H1:β1
≠0 (linearrelationshipdoesexist)Teststatistic
where:b1=regressionslopecoefficient
β1=hypothesizedslopeSb1=standarderroroftheslope48InferencesAbouttheSlope:
tTestExampleHousePricein$1000s(y)SquareFeet(x)2451400312160027917003081875199110021915504052350324245031914252551700EstimatedRegressionEquation:Theslopeofthismodelis0.1098Istherearelationshipbetweenthesquarefootageofthehouseanditssalesprice?49InferencesAbouttheSlope:
tTestExampleH0:β1=0H1:β1
≠0FromExceloutput:
CoefficientsStandardErrortStatP-valueIntercept98.2483358.033481.692960.12892SquareFeet0.109770.032973.329380.01039b1Predictor
Coef
SECoef
T
PConstant
98.25
58.03
1.69
0.129SquareFeet
0.10977
0.03297
3.33
0.010FromMinitaboutput:b150InferencesAbouttheSlope:
tTestExampleTestStatistic:tSTAT=3.329ThereissufficientevidencethatsquarefootageaffectshousepriceDecision:RejectH0RejectH0RejectH0a/2=.025-tα/2DonotrejectH00tα/2a/2=.025-2.30602.30603.329d.f.=10-2=8H0:β1=0H1:β1≠051InferencesAbouttheSlope:
tTestExampleH0:β1=0H1:β1≠0FromExceloutput:
CoefficientsStandardErrortStatP-valueIntercept98.2483358.033481.692960.12892SquareFeet0.109770.032973.329380.01039p-valueThereissufficientevidencethatsquarefootageaffectshouseprice.Decision:RejectH0,sincep-value<αPredictor
Coef
SECoef
T
PConstant
98.25
58.03
1.69
0.129SquareFeet
0.10977
0.03297
3.33
0.010FromMinitaboutput:52FTestforSignificanceFTeststatistic:
where
whereFSTATfollowsanFdistributionwith1numerator
and(n–2)denominatordegreesoffreedom
53F-TestforSignificance
ExcelOutputRegressionStatisticsMultipleR0.76211RSquare0.58082AdjustedRSquare0.52842StandardError41.33032Observations10ANOVA
dfSSMSFSignificanceFRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000
With1and8degreesoffreedomp-valuefortheF-Test54F-TestforSignificance
MinitabOutputAnalysisofVariance
Source
DF
SS
MS
F
PRegression
1
18935
18935
11.08
0.010ResidualError
8
13666
1708Total
9
32600With1and8degreesoffreedomp-valuefortheF-Test55H0:β1=0H1:β1≠0
=.05df1=1df2=8TestStatistic:Decision:Conclusion:RejectH0at
=0.05Thereissufficientevidencethathousesizeaffectssellingprice0
=.05F.05=5.32RejectH0DonotrejectH0CriticalValue:F
=5.32FTestforSignificance(continued)F56ConfidenceIntervalEstimate
fortheSlopeConfidenceIntervalEstimateoftheSlope:ExcelPrintoutforHousePrices:At95%levelofconfidence,theconfidenceintervalfortheslopeis(0.0337,0.1858)
CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.18580d.f.=n-257Sincetheunitsofthehousepricevariableis$1000s,weare95%confidentthattheaverageimpactonsalespriceisbetween$33.74and$185.80persquarefootofhousesize
CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.18580This95%confidenceintervaldoesnotinclude0.Conclusion:Thereisasignificantrelationshipbetweenhousepriceandsquarefeetatthe.05levelofsignificanceConfidenceIntervalEstimate
fortheSlope(continued)58tTestforaCorrelationCoefficientHypotheses H0:ρ=0 (nocorrelationbetweenXandY)
H1:ρ
≠0 (correlationexists)Teststatistic
(withn–2degreesoffreedom)59t-testForACorrelationCoefficientIsthereevidenceofalinearrelationshipbetweensquarefeetandhousepriceatthe.05levelofsignificance?H0:ρ
=0(Nocorrelation)H1:ρ
≠0(correlationexists)
=.05,df
=
10-2=8(continued)60t-testForACorrelationCoefficientConclusion:
Thereisevidenceofalinearassociationatthe5%levelofsignificanceDecision:
RejectH0RejectH0RejectH0a/2=.025-tα/2DonotrejectH00tα/2a/2=.025-2.30602.30603.329d.f.=10-2=8(continued)61EstimatingMeanValuesandPredictingIndividualValuesYX
XiY=b0+b1Xi
ConfidenceIntervalforthemeanofY,givenXiPredictionIntervalforanindividualY,givenXiGoal:FormintervalsaroundYtoexpressuncertaintyaboutthevalueofYforagivenXiY
62ConfidenceIntervalfor
theAverageY,GivenXConfidenceintervalestimateforthemeanvalueofYgivenaparticularXiSizeofintervalvariesaccordingtodistanceawayfrommean,
X63PredictionIntervalfor
anIndividualY,GivenXPredictionintervalestimateforanIndividualvalueofYgivenaparticularXiThisextratermaddstotheintervalwidthtoreflecttheaddeduncertaintyforanindividualcase64EstimationofMeanValues:ExampleFindthe95%confidenceintervalforthemeanpriceof2,000square-foothousesPredictedPriceYi=317.85($1,000s)
ConfidenceIntervalEstimateforμY|X=XTheconfidenceintervalendpointsare280.66and354.90,orfrom$280,660to$354,900i65EstimationofIndividualValues:ExampleFindthe95%predictionintervalforanindividualhousewith2,000squarefeetPredictedPriceYi=317.85($1,000s)
PredictionIntervalEstimateforYX=XThepredictionintervalendpointsare215.50and420.07,orfrom$215,500to$420,070i66FindingConfidenceand
PredictionIntervalsinExcelFromExcel,use PHStat|regression|simplelinearregression…Checkthe “confidenceandpredictionintervalforX=〞boxandentertheX-valueandconfidenceleveldesired67InputvaluesFindingConfidenceand
PredictionIntervalsinExcel(continued)ConfidenceInt
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