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ISYE 6414 - ALL UNITS COMPLETELY SOLVED GRADED A 2024
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ISYE 6414 - ALL UNITS COMPLETELY SOLVED GRADED A 2024
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ISYE 6414 - MIDTERM 1 PREPQUESTIONS AND VERIFIED ANSWERS
2024
Ifvλ=1v-vans✔✔wevdovnotvtransform
non-deterministicv-
vans✔✔Regressionvanalysisvisvonevofvthevsimplestvwaysvwevhavevinvstatisticsvtovinvestiga
tevthevrelationshipvbetweenvtwovorvmorevvariablesvinvav___vway
randomv-
vans✔✔Thevresponsevvariablevisvav___vvariable,vbecausevitvvariesvwithvchangesvinvthevpr
edictingvvariable,vorvwithvothervchangesvinvthevenvironment
fixedv-
vans✔✔Thevpredictingvvariablevisvav___vvariable.vItvisvsetvfixed,vbeforevthevresponsevisvme
asured.
simplevlinearvregressionv-
vans✔✔regressionvanalysisvinvolvingvonevindependentvvariablevandvonevdependentvvaria
blevinvwhichvthevrelationshipvbetweenvthevvariablesvisvapproximatedvbyvavstraightvline
MultiplevLinearvRegressionv-
vans✔✔Avstatisticalvmethodvusedvtovmodelvthevrelationshipvbetweenvonevdependentv(orvr
esponse)vvariablevandvtwovorvmorevindependentv(orvexplanatory)vvariablesvbyvfittingvavlin
earvequationvtovobservedvdata
,polynomialvregressionv-
vans✔✔avregressionvmodelvwhichvdoesvnotvassumevavlinearvrelationship;vavcurvilinearvcor
relationvcoefficientvisvcomputedv(wevcanvthinkvofvXvandvX-
squaredvasvtwovdifferentvpredictingvvariables)
threevobjectivesvinvregressionv-vans✔✔1)vPrediction
2)vModeling
3)vTestingvhypothesis
Predictionv-
vans✔✔Wevwantvtovseevhowvthevresponsevvariablevbehavesvinvdifferentvsettings.vForvexa
mple,vforvavdifferentvlocation,vifvwevthinkvaboutvavgeographicvprediction,vorvinvtime,vifvwevthi
nkvaboutvtemporalvprediction
Modelingv-
vans✔✔modelingvthevrelationshipvbetweenvthevresponsevvariablevandvthevexplanatoryvvar
iables,vorvpredictingvvariables
Testingvhypothesesv-vans✔✔ofvassociationvrelationships
usefulvrepresentationvofvrealityv-
vans✔✔Wevdovnotvbelievevthatvthevlinearvmodelvrepresentsvavtruevrepresentationvofvrealit
y.vRather,vwevthinkvthat,vperhaps,vitvprovidesvav___
β0v-vans✔✔interceptvparameterv(thevvaluevatvwhichvthevlinevintersectsvthevy-axis)
β1v-vans✔✔slopevparameterv(slopevofvthevlinevwevarevtryingvtovfit)
epsilonv(ε)v-vans✔✔isvthevdeviancevofvthevdatavfromvthevlinearvmodel
tovfindvβ0vandvβ1v-
vans✔✔tovfindvthevlinevthatvdescribesvavlinearvrelationship,vsuchvthatvwevfitvthisvmodel.
simplevlinearvregressionvdatavstructurev-
vans✔✔pairsvofvdatavconsistingvofvavvaluevforvthevresponsevvariable,andvavvaluevforvthevpr
edictingvvariable.vAndvwevhavevnvsuchvpairs
modelingvframeworkvforvthevsimplevlinearvregression:v-
vans✔✔1)videntifyingvdatavstructure
2)vclearlyvstatingvthevmodelvassumptions
linearvregressionvassumptionsv-vans✔✔1)vlinearity
2)vconstantvvariancevassumption
3)vindependencevassumption
, linearityvassumptionv-
vans✔✔meanvzerovassumption,vmeansvthatvthevexpectedvvaluevofvtheverrorsvisvzero.
Avviolationvofvthisvassumptionvwillvleadvtovdifficultiesvinvestimatingvβ0,vandvmeansvthatvyou
rvmodelvdoesvnotvincludevavnecessaryvsystematicvcomponent.
constantvvariancevassumptionv-
vans✔✔whichvmeansvthatvthevvariancev(σ^2)vofvtheverrorvtermsvorvdeviancesvisvconstantvf
orvthevgivenvpopulation.vAvviolationvofvthisvassumptionvmeansvthatvthevestimatesvarevnotva
svefficientvasvtheyvcouldvbevinvestimatingvthevtruevparameters
IndependencevAssumptionv-
vans✔✔whichvmeansvthatvthevdeviancesvarevindependentvrandomvvariables.
Violationvofvthisvassumptionvcanvleadvtovmisleadingvassessmentsvofvthevstrengthvofvthevre
gression.
normalityvassumptionv-
vans✔✔errorsv(ε)varevnormallyvdistributed.vThisvisvneededvforvstatisticalvinference,vforvexa
mple,vconfidencevorvpredictionvintervals,vandvhypothesisvtesting.vIfvthisvassumptionvisvviol
ated,vhypothesisvtestsvandvconfidencevandvpredictionvintervalsvcanvbevmisleading.v
thirdvparameterv-vans✔✔thevvariancevofvtheverrorvtermsv(σ^2)
Onevapproachvisvtovminimizevthevsumvofvsquaredvresidualsvorverrorsvwithvrespectvtovβ0van
dvβ1.vThisvtranslatedvintovfindingvthevlinevsuchvthatvthevtotalvsquaredvdeviancesvfromvthevli
nevisvminimum.v-
vans✔✔Howvcanvwevgetvestimatesvofvthevregressionvcoefficientsvorvparametersvinvlinear
regressionvanalysis?
fittedvvaluesv-vans✔✔tovbevthevregressionvlinevwherevthevparametersvarevreplaced
byvthevestimatedvvaluesvofvthevparameters.
Residualsv-vans✔✔arevsimplyvthevdifference
betweenvobservedvresponsevandvfittedvvalues,vandvtheyvarevproxiesvofvtheverrorvtermsvin
thevregressionvmodel
MSEv-vans✔✔Thevestimatorvforvsigmavsquarevisvsigmavsquarevhat,vandvisvthe
sumvofvthevsquaredvresiduals,vdividedvbyvnv-v2.
σ^2v(samplevdistributionvofvthevvariancevestimator)v-vans✔✔isvchi-
squaredvdistributionvwithvnv-v2vdegreesvofvfreedomv(We
losevtwovdegreesvofvfreedomvbecausevwevreplacedvthevtwovparametersvß0vandvß1vwith
theirvestimatorsvtovobtainvthevresiduals.)
epsilonvivhatv-vans✔✔proxiesvforvthevdeviancesvorvtheverrorvterms
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