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Chapter12SimpleLinearRegressionBusinessStatistics:AFirstCourseFifthEditionChap12-1BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.Chapter12BusinessStatistics:LearningObjectivesInthischapter,youlearn:nHowtouseregressionanalysistopredictthevalueofadependentvariablebasedonanindependentvariablenThemeaningoftheregressioncoefficientsb0andb1nHowtoevaluatetheassumptionsofregressionanalysisandknowwhattodoiftheassumptionsareviolatednTomakeinferencesabouttheslopeandcorrelationcoefficientnToestimatemeanvaluesandpredictindividualvalues2BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.LearningObjectivesInthischaCorrelationvs.RegressionnAscatterplotcanbeusedtoshowtherelationshipbetweentwovariablesnCorrelationanalysisisusedtomeasurethestrengthoftheassociation(linearrelationship)betweentwovariablesnCorrelationisonlyconcernedwithstrengthoftherelationshipnNocausaleffectisimpliedwithcorrelationnScatterplotswerefirstpresentedinCh.2nCorrelationwasfirstpresentedinCh.33BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.Correlationvs.RegressionAscIntroductiontoRegressionAnalysisnRegressionanalysisisusedto:nPredictthevalueofadependentvariablebasedonthevalueofatleastoneindependentvariablenExplaintheimpactofchangesinanindependentvariableonthedependentvariableDependentvariable:thevariablewewishtopredictorexplainIndependentvariable:thevariableusedtopredictorexplainthedependentvariable4BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.IntroductiontoRegressionAnSimpleLinearRegressionModelnOnlyoneindependentvariable,XnRelationshipbetweenXandYisdescribedbyalinearfunctionnChangesinYareassumedtoberelatedtochangesinX5BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionModelTypesofRelationshipsYXYXYYXXLinearrelationshipsCurvilinearrelationships6BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.TypesofRelationshipsYXYXYYXXTypesofRelationshipsYXYXYYXXStrongrelationshipsWeakrelationships(continued)7BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.TypesofRelationshipsYXYXYYXXTypesofRelationshipsYXYXNorelationship(continued)8BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.TypesofRelationshipsYXYXNorLinearcomponentSimpleLinearRegressionModelPopulationYinterceptPopulationSlopeCoefficientRandomErrortermDependentVariableIndependentVariableRandomErrorcomponent9BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.LinearcomponentSimpleLinear(continued)RandomErrorforthisXivalueYXObservedValueofYforXiPredictedValueofYforXiXiSlope=1Intercept=0iSimpleLinearRegressionModel10BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.(continued)RandomErrorforthThesimplelinearregressionequationprovidesanestimateofthepopulationregressionlineSimpleLinearRegressionEquation(PredictionLine)EstimateoftheregressioninterceptEstimateoftheregressionslopeEstimated(orpredicted)YvalueforobservationiValueofXforobservationi11BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ThesimplelinearregressioneTheLeastSquaresMethodb0andb1areobtainedbyfindingthevaluesofthatminimizethesumofthesquareddifferencesbetweenYand:12BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.TheLeastSquaresMethodb0anFindingtheLeastSquaresEquationnThecoefficientsb0andb1,andotherregressionresultsinthischapter,willbefoundusingExcelorMinitabFormulasareshowninthetextforthosewhoareinterested13BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.FindingtheLeastSquaresEquanb0istheestimatedmeanvalueofYwhenthevalueofXiszeronb1istheestimatedchangeinthemeanvalueofYasaresultofaone-unitchangeinXInterpretationoftheSlopeandtheIntercept14BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.b0istheestimatedmeanvalueSimpleLinearRegressionExamplenArealestateagentwishestoexaminetherelationshipbetweenthesellingpriceofahomeanditssize(measuredinsquarefeet)nArandomsampleof10housesisselectednDependentvariable(Y)=housepricein$1000snIndependentvariable(X)=squarefeet15BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:DataHousePricein$1000s(Y)SquareFeet(X)245140031216002791700308187519911002191550405235032424503191425255170016BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:ScatterPlotHousepricemodel:ScatterPlot17BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:UsingExcel18BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:ExcelOutputRegressionStatisticsMultipleR0.76211RSquare0.58082AdjustedRSquare0.52842StandardError41.33032Observations10ANOVAdfSSMSFSignificanceFRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.18580Theregressionequationis:19BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:MinitabOutputTheregressionequationisPrice=98.2+0.110SquareFeetPredictorCoefSECoefTPConstant98.2558.031.690.129SquareFeet0.109770.032973.330.010S=41.3303R-Sq=58.1%R-Sq(adj)=52.8%AnalysisofVarianceSourceDFSSMSFPRegression1189351893511.080.010ResidualError8136661708Total932600Theregressionequationis:houseprice=98.24833+0.10977(squarefeet)20BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:GraphicalRepresentationHousepricemodel:ScatterPlotandPredictionLineSlope=0.10977Intercept=98.24821BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:Interpretationofbonb0istheestimatedmeanvalueofYwhenthevalueofXiszero(ifX=0isintherangeofobservedXvalues)nBecauseahousecannothaveasquarefootageof0,b0hasnopracticalapplication22BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:Interpretingb1nb1estimatesthechangeinthemeanvalueofYasaresultofaone-unitincreaseinXnHere,b1=0.10977tellsusthatthemeanvalueofahouseincreasesby0.10977($1000)=$109.77,onaverage,foreachadditionalonesquarefootofsize23BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampPredictthepriceforahousewith2000squarefeet:Thepredictedpriceforahousewith2000squarefeetis317.85($1,000s)=$317,850SimpleLinearRegressionExample:MakingPredictions24BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.PredictthepriceforahouseSimpleLinearRegressionExample:MakingPredictionsnWhenusingaregressionmodelforprediction,onlypredictwithintherelevantrangeofdataRelevantrangeforinterpolationDonottrytoextrapolatebeyondtherangeofobservedXs25BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampMeasuresofVariationnTotalvariationismadeupoftwoparts:TotalSumofSquaresRegressionSumofSquaresErrorSumofSquareswhere:=MeanvalueofthedependentvariableYi=Observedvalueofthedependentvariable=PredictedvalueofYforthegivenXivalue26BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.MeasuresofVariationTotalvarnSST=totalsumofsquares(TotalVariation)nMeasuresthevariationoftheYivaluesaroundtheirmeanYnSSR=regressionsumofsquares(ExplainedVariation)nVariationattributabletotherelationshipbetweenXandYnSSE=errorsumofsquares(UnexplainedVariation)nVariationinYattributabletofactorsotherthanX(continued)MeasuresofVariation27BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SST=totalsumofsquares(continued)XiYXYiSST=(Yi-Y)2SSE=(Yi-Yi)2 SSR=(Yi-Y)2 _Y YY_Y MeasuresofVariation28BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.(continued)XiYXYiSST=(Yi-nThecoefficientofdeterminationistheportionofthetotalvariationinthedependentvariablethatisexplainedbyvariationintheindependentvariablenThecoefficientofdeterminationisalsocalledr-squaredandisdenotedasr2CoefficientofDetermination,r2note:29BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.Thecoefficientofdeterminatir2=1Examplesofr2ValuesYXYXr2=1r2=1PerfectlinearrelationshipbetweenXandY:100%ofthevariationinYisexplainedbyvariationinX30BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.r2=1Examplesofr2ValuesYXYExamplesofr2ValuesYXYX0r21WeakerlinearrelationshipsbetweenXandY:SomebutnotallofthevariationinYisexplainedbyvariationinX31BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.Examplesofr2ValuesYXYX0rExamplesofr2Valuesr2=0NolinearrelationshipbetweenXandY:ThevalueofYdoesnotdependonX.(NoneofthevariationinYisexplainedbyvariationinX)YXr2=032BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.Examplesofr2Valuesr2=0NoSimpleLinearRegressionExample:CoefficientofDetermination,r2inExcelRegressionStatisticsMultipleR0.76211RSquare0.58082AdjustedRSquare0.52842StandardError41.33032Observations10ANOVAdfSSMSFSignificanceFRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.1858058.08%ofthevariationinhousepricesisexplainedbyvariationinsquarefeet33BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:CoefficientofDetermination,r2inMinitabTheregressionequationisPrice=98.2+0.110SquareFeetPredictorCoefSECoefTPConstant98.2558.031.690.129SquareFeet0.109770.032973.330.010S=41.3303R-Sq=58.1%R-Sq(adj)=52.8%AnalysisofVarianceSourceDFSSMSFPRegression1189351893511.080.010ResidualError8136661708Total93260058.08%ofthevariationinhousepricesisexplainedbyvariationinsquarefeet34BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampStandardErrorofEstimatenThestandarddeviationofthevariationofobservationsaroundtheregressionlineisestimatedbyWhereSSE=errorsumofsquaresn=samplesize35BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.StandardErrorofEstimateTheSimpleLinearRegressionExample:StandardErrorofEstimateinExcelRegressionStatisticsMultipleR0.76211RSquare0.58082AdjustedRSquare0.52842StandardError41.33032Observations10ANOVAdfSSMSFSignificanceFRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.1858036BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampSimpleLinearRegressionExample:StandardErrorofEstimateinMinitabTheregressionequationisPrice=98.2+0.110SquareFeetPredictorCoefSECoefTPConstant98.2558.031.690.129SquareFeet0.109770.032973.330.010S=41.3303R-Sq=58.1%R-Sq(adj)=52.8%AnalysisofVarianceSourceDFSSMSFPRegression1189351893511.080.010ResidualError8136661708Total93260037BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampComparingStandardErrorsYYXXSYXisameasureofthevariationofobservedYvaluesfromtheregressionlineThemagnitudeofSYXshouldalwaysbejudgedrelativetothesizeoftheYvaluesinthesampledatai.e.,SYX=$41.33Kismoderatelysmallrelativetohousepricesinthe$200K-$400Krange38BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ComparingStandardErrorsYYXXSAssumptionsofRegressionL.I.N.EnLinearitynTherelationshipbetweenXandYislinearnIndependenceofErrorsnErrorvaluesarestatisticallyindependentnNormalityofErrornErrorvaluesarenormallydistributedforanygivenvalueofXnEqualVariance(alsocalledhomoscedasticity)nTheprobabilitydistributionoftheerrorshasconstantvariance39BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.AssumptionsofRegressionL.I.ResidualAnalysisnTheresidualforobservationi,ei,isthedifferencebetweenitsobservedandpredictedvaluenChecktheassumptionsofregressionbyexaminingtheresidualsnExamineforlinearityassumptionnEvaluateindependenceassumptionnEvaluatenormaldistributionassumptionnExamineforconstantvarianceforalllevelsofX(homoscedasticity)nGraphicalAnalysisofResidualsnCanplotresidualsvs.X40BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ResidualAnalysisTheresidualResidualAnalysisforLinearityNotLinearLinearxresidualsxYxYxresiduals41BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ResidualAnalysisforLinearitResidualAnalysisforIndependenceNotIndependentIndependentXXresidualsresidualsXresiduals42BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ResidualAnalysisforIndependCheckingforNormalitynExaminetheStem-and-LeafDisplayoftheResidualsnExaminetheBoxplotoftheResidualsnExaminetheHistogramoftheResidualsnConstructaNormalProbabilityPlotoftheResiduals43BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.CheckingforNormalityExamineResidualAnalysisforNormalityPercentResidualWhenusinganormalprobabilityplot,normalerrorswillapproximatelydisplayinastraightline-3-2-10123010044BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ResidualAnalysisforNormalitResidualAnalysisforEqualVarianceNon-constantvarianceConstantvariancexxYxxYresidualsresiduals45BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ResidualAnalysisforEqualVSimpleLinearRegressionExample:ExcelResidualOutputRESIDUALOUTPUTPredicted House Price Residuals1251.92316-6.9231622273.8767138.123293284.85348-5.8534844304.062843.9371625218.99284-19.992846268.38832-49.388327356.2025148.797498367.17929-43.179299254.667464.3326410284.85348-29.85348Doesnotappeartoviolateanyregressionassumptions46BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SimpleLinearRegressionExampInferencesAbouttheSlopenThestandarderroroftheregressionslopecoefficient(b1)isestimatedbywhere:=Estimateofthestandarderroroftheslope=Standarderroroftheestimate47BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.InferencesAbouttheSlopeTheInferencesAbouttheSlope:tTestnttestforapopulationslopenIstherealinearrelationshipbetweenXandY?nNullandalternativehypothesesnH0:1=0(nolinearrelationship)nH1:10(linearrelationshipdoesexist)nTeststatisticwhere:b1=regression slope coefficient 1=hypothesized slope Sb1=standard error of the slope48BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.InferencesAbouttheSlope:tInferencesAbouttheSlope:tTestExampleHousePricein$1000s(y)SquareFeet(x)2451400312160027917003081875199110021915504052350324245031914252551700Estimated Regression Equation:The slope of this model is 0.1098 Is there a relationship between the square footage of the house and its sales price?49BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.InferencesAbouttheSlope:tInferencesAbouttheSlope:tTestExampleH0:1=0H1:10FromExceloutput:CoefficientsStandardErrortStatP-valueIntercept98.2483358.033481.692960.12892SquareFeet0.109770.032973.329380.01039b1PredictorCoefSECoefTPConstant98.2558.031.690.129SquareFeet0.109770.032973.330.010FromMinitaboutput:b150BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.InferencesAbouttheSlope:tInferencesAbouttheSlope:tTestExampleTest Statistic:tSTAT=3.329There is sufficient evidence that square footage affects house priceDecision:Reject H0RejectH0RejectH0/2=.025-t/2DonotrejectH00t/2/2=.025-2.30602.30603.329d.f.=10-2=8H0:1=0H1:1051BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.InferencesAbouttheSlope:tInferencesAbouttheSlope:tTestExampleH0:1=0H1:10FromExceloutput:CoefficientsStandardErrortStatP-valueIntercept98.2483358.033481.692960.12892SquareFeet0.109770.032973.329380.01039p-valueThere is sufficient evidence that square footage affects house price.Decision:Reject H0,since p-value PredictorCoefSECoefTPConstant98.2558.031.690.129SquareFeet0.109770.032973.330.010FromMinitaboutput:52BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.InferencesAbouttheSlope:tFTestforSignificancenFTeststatistic:wherewhereFSTATfollowsanFdistributionwith1numeratorand(n2)denominatordegreesoffreedom53BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.FTestforSignificanceFTestF-TestforSignificanceExcelOutputRegressionStatisticsMultipleR0.76211RSquare0.58082AdjustedRSquare0.52842StandardError41.33032Observations10ANOVAdfSSMSFSignificanceFRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000With1and8degreesoffreedomp-valuefortheF-Test54BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.F-TestforSignificanceExcelF-TestforSignificanceMinitabOutputAnalysisofVarianceSourceDFSSMSFPRegression1189351893511.080.010ResidualError8136661708Total932600With1and8degreesoffreedomp-valuefortheF-Test55BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.F-TestforSignificanceMinitaH0:1=0H1:10=.05df1=1df2=8TestStatistic:Decision:Conclusion:RejectH0at=0.05Thereissufficientevidencethathousesizeaffectssellingprice0=.05F.05=5.32RejectH0DonotrejectH0CriticalValue:F=5.32FTestforSignificance(continued)F56BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.H0:1=0TestStatistic:RejeConfidenceIntervalEstimatefortheSlopeConfidenceIntervalEstimateoftheSlope: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-257BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ConfidenceIntervalEstimateSincetheunitsofthehousepricevariableis$1000s,weare95%confidentthattheaverageimpactonsalespriceisbetween$33.74and$185.80persquarefootofhousesizeCoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept98.2483358.033481.692960.12892-35.57720232.07386SquareFeet0.109770.032973.329380.010390.033740.18580This95%confidenceintervaldoesnotinclude0.Conclusion:Thereisasignificantrelationshipbetweenhousepriceandsquarefeetatthe.05levelofsignificanceConfidenceIntervalEstimatefortheSlope(continued)58BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.SincetheunitsofthehouseptTestforaCorrelationCoefficientnHypothesesH0:=0(nocorrelationbetweenXandY)H1:0(correlationexists)nTeststatistic(withn2degreesoffreedom)59BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.tTestforaCorrelationCoefft-testForACorrelationCoefficientIsthereevidenceofalinearrelationshipbetweensquarefeetandhousepriceatthe.05levelofsignificance?H0:=0(Nocorrelation)H1:0(correlationexists)=.05,df=10-2=8(continued)60BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.t-testForACorrelationCoefft-testForACorrelationCoefficientConclusion:Thereisevidenceofalinearassociationatthe5%levelofsignificanceDecision:RejectH0RejectH0RejectH0/2=.025-t/2DonotrejectH00t/2/2=.025-2.30602.30603.329d.f.=10-2=8(continued)61BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.t-testForACorrelationCoeffEstimatingMeanValuesandPredictingIndividualValuesYXXiY=b0+b1Xi ConfidenceIntervalforthemeanofY,givenXiPredictionIntervalforanindividualY,givenXiGoal:FormintervalsaroundYtoexpressuncertaintyaboutthevalueofYforagivenXiY 62BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.EstimatingMeanValuesandPreConfidenceIntervalfortheAverageY,GivenXConfidenceintervalestimateforthemeanvalueofYgivenaparticularXiSizeofintervalvariesaccordingtodistanceawayfrommean,X63BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.ConfidenceIntervalfortheAPredictionIntervalforanIndividualY,GivenXPredictionintervalestimateforanIndividualvalueofYgivenaparticularXiThisextratermaddstotheintervalwidthtoreflecttheaddeduncertaintyforanindividualcase64BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.PredictionIntervalforanInEstimationofMeanValues:ExampleFindthe95%confidenceintervalforthemeanpriceof2,000square-foothousesPredictedPriceYi=317.85($1,000s)ConfidenceIntervalEstimateforY|X=XTheconfidenceintervalendpointsare280.66and354.90,orfrom$280,660to$354,900i65BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.EstimationofMeanValues:ExaEstimationofIndividualValues:ExampleFindthe95%predictionintervalforanindividualhousewith2,000squarefeetPredictedPriceYi=317.85($1,000s)PredictionIntervalEstimateforYX=XThepredictionintervalendpointsare215.50and420.07,orfrom$215,500to$420,070i66BusinessStatistics:AFirstCourse,5e2009Prentice-Hall,Inc.EstimationofIndividualValueFindingConfidenceandPredictionInter
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