ISYE 6414 - MIDTERM 1 PREP
QUESTIONS AND VERIFIED ANSWERS
2024
Ifdλ=1d-dans✔✔weddodnotdtransform
non-deterministicd-
dans✔✔Regressiondanalysisdisdonedofdthedsimplestdwaysdwedhavedindstatisticsdtodinvestig
atedthedrelationshipdbetweendtwodordmoredvariablesdindad___dway
randomd-
dans✔✔Thedresponsedvariabledisdad___dvariable,dbecauseditdvariesdwithdchangesdindthedp
redictingdvariable,dordwithdotherdchangesdindthedenvironment
fixedd-
dans✔✔Thedpredictingdvariabledisdad___dvariable.dItdisdsetdfixed,dbeforedthedresponsedisdm
easured.
simpledlineardregressiond-
dans✔✔regressiondanalysisdinvolvingdonedindependent dvariabledanddoneddependentdvari
abledindwhichdthedrelationshipdbetweendthedvariablesdisdapproximateddbydadstraightdline
MultipledLineardRegressiond-
dans✔✔Adstatisticaldmethodduseddtodmodeldthedrelationshipdbetweendoneddependentd(ord
response)dvariabledanddtwodordmoredindependentd(ordexplanatory)dvariablesdbydfittingdadli
neardequationdtodobservedddata
polynomialdregressiond-
dans✔✔adregressiondmodeldwhichddoesdnotdassumedadlineardrelationship;dadcurvilineardc
,orrelationdcoefficientdisdcomputedd(wedcandthinkdofdXdanddX-
squareddasdtwoddifferentdpredictingdvariables)
threedobjectivesdindregressiond-dans✔✔1)dPrediction
2)dModeling
3)dTestingdhypothesis
Predictiond-
dans✔✔Wedwantdtodseedhowdthedresponsedvariabledbehavesdinddifferentdsettings.dFordex
ample,dfordaddifferentdlocation,difdwedthinkdaboutdadgeographicdprediction,dordindtime,difdwed
thinkdaboutdtemporaldprediction
Modelingd-
dans✔✔modelingdthedrelationshipdbetweendthedresponsedvariabledanddthedexplanatorydv
ariables,dordpredictingdvariables
Testingdhypothesesd-dans✔✔ofdassociationdrelationships
usefuldrepresentationdofdrealityd-
dans✔✔Weddodnotdbelievedthatdthedlineardmodeldrepresentsdadtruedrepresentationdofdreali
ty.dRather,dwedthinkdthat,dperhaps,ditdprovidesdad___
β0d-dans✔✔interceptdparameterd(thedvaluedatdwhichdthedlinedintersectsdthedy-axis)
β1d-dans✔✔slopedparameterd(slopedofdthedlinedwedaredtryingdtodfit)
epsilond(ε)d-dans✔✔isdtheddeviancedofdtheddatadfromdthedlineardmodel
todfinddβ0danddβ1d-
dans✔✔todfinddthedlinedthatddescribesdadlineardrelationship,dsuchdthatdwedfitdthisdmodel.
simpledlineardregressionddatadstructured-
dans✔✔pairsdofddatadconsistingdofdadvaluedfordthedresponsedvariable,anddadvaluedfordthed
predictingdvariable.dAnddwedhavedndsuchdpairs
modelingdframeworkdfordthedsimpledlineardregression:d-
dans✔✔1)didentifyingddatadstructure
2)dclearlydstatingdthedmodeldassumptions
lineardregressiondassumptionsd-dans✔✔1)dlinearity
2)dconstantdvariancedassumption
3)dindependencedassumption
linearitydassumptiond-
dans✔✔meandzerodassumption,dmeansdthatdthedexpecteddvaluedofdthederrorsdisdzero.
, Adviolationdofdthisdassumptiondwilldleaddtoddifficultiesdindestimatingdβ0,danddmeansdthatdyo
urdmodelddoesdnotdincludedadnecessarydsystematicdcomponent.
constantdvariancedassumptiond-
dans✔✔whichdmeansdthatdthedvarianced(σ^2)dofdthederrordtermsdorddeviancesdisdconstantd
fordthedgivendpopulation.dAdviolationdofdthisdassumptiondmeansdthatdthedestimatesdarednot
dasdefficientdasdtheydcoulddbedindestimatingdthedtruedparameters
IndependencedAssumptiond-
dans✔✔whichdmeansdthatdtheddeviancesdaredindependent drandomdvariables.
Violationdofdthisdassumptiondcandleaddtodmisleadingdassessmentsdofdthedstrengthdofdthedr
egression.
normalitydassumptiond-
dans✔✔errorsd(ε)darednormallyddistributed.dThisdisdneededdfordstatisticaldinference,dfordex
ample,dconfidencedordpredictiondintervals,danddhypothesisdtesting.dIfdthisdassumptiondisdvi
olated,dhypothesisdtestsdanddconfidencedanddpredictiondintervalsdcandbedmisleading.v
thirddparameterd-dans✔✔thedvariancedofdthederrordtermsd(σ^2)
Onedapproachdisdtodminimizedthedsumdofdsquareddresidualsdorderrorsdwithdrespectdtodβ0da
nddβ1.dThisdtranslateddintodfindingdthedlinedsuchdthatdthedtotaldsquaredddeviancesdfromdth
edlinedisdminimum.d-
dans✔✔Howdcandwedgetdestimatesdofdthedregressiondcoefficientsdordparametersdindlinear
regressiondanalysis?
fitteddvaluesd-dans✔✔todbedthedregressiondlinedwheredthedparametersdaredreplaced
bydthedestimateddvaluesdofdthedparameters.
Residualsd-dans✔✔aredsimplydtheddifference
betweendobserveddresponsedanddfitteddvalues,danddtheydaredproxiesdofdthederrordtermsdin
thedregressiondmodel
MSEd-dans✔✔Thedestimatordfordsigmadsquaredisdsigmadsquaredhat,danddisdthe
sumdofdthedsquareddresiduals,ddivideddbydnd-d2.
σ^2d(sampleddistributiondofdthedvariancedestimator)d-dans✔✔isdchi-
squaredddistributiondwithdnd-d2ddegreesdofdfreedomd(We
losedtwoddegreesdofdfreedomdbecausedwedreplaceddthedtwodparametersdß0danddß1dwith
theirdestimatorsdtodobtaindthedresiduals.)
epsilondidhatd-dans✔✔proxiesdfordtheddeviancesdordthederrordterms
sampledvariancedestimatord(s^2)d-
dans✔✔thedestimatordofdthedvariancedofdthederrordtermsd(isdchi-squaredwithdnd-
d1ddegreesdofdfreedom)
QUESTIONS AND VERIFIED ANSWERS
2024
Ifdλ=1d-dans✔✔weddodnotdtransform
non-deterministicd-
dans✔✔Regressiondanalysisdisdonedofdthedsimplestdwaysdwedhavedindstatisticsdtodinvestig
atedthedrelationshipdbetweendtwodordmoredvariablesdindad___dway
randomd-
dans✔✔Thedresponsedvariabledisdad___dvariable,dbecauseditdvariesdwithdchangesdindthedp
redictingdvariable,dordwithdotherdchangesdindthedenvironment
fixedd-
dans✔✔Thedpredictingdvariabledisdad___dvariable.dItdisdsetdfixed,dbeforedthedresponsedisdm
easured.
simpledlineardregressiond-
dans✔✔regressiondanalysisdinvolvingdonedindependent dvariabledanddoneddependentdvari
abledindwhichdthedrelationshipdbetweendthedvariablesdisdapproximateddbydadstraightdline
MultipledLineardRegressiond-
dans✔✔Adstatisticaldmethodduseddtodmodeldthedrelationshipdbetweendoneddependentd(ord
response)dvariabledanddtwodordmoredindependentd(ordexplanatory)dvariablesdbydfittingdadli
neardequationdtodobservedddata
polynomialdregressiond-
dans✔✔adregressiondmodeldwhichddoesdnotdassumedadlineardrelationship;dadcurvilineardc
,orrelationdcoefficientdisdcomputedd(wedcandthinkdofdXdanddX-
squareddasdtwoddifferentdpredictingdvariables)
threedobjectivesdindregressiond-dans✔✔1)dPrediction
2)dModeling
3)dTestingdhypothesis
Predictiond-
dans✔✔Wedwantdtodseedhowdthedresponsedvariabledbehavesdinddifferentdsettings.dFordex
ample,dfordaddifferentdlocation,difdwedthinkdaboutdadgeographicdprediction,dordindtime,difdwed
thinkdaboutdtemporaldprediction
Modelingd-
dans✔✔modelingdthedrelationshipdbetweendthedresponsedvariabledanddthedexplanatorydv
ariables,dordpredictingdvariables
Testingdhypothesesd-dans✔✔ofdassociationdrelationships
usefuldrepresentationdofdrealityd-
dans✔✔Weddodnotdbelievedthatdthedlineardmodeldrepresentsdadtruedrepresentationdofdreali
ty.dRather,dwedthinkdthat,dperhaps,ditdprovidesdad___
β0d-dans✔✔interceptdparameterd(thedvaluedatdwhichdthedlinedintersectsdthedy-axis)
β1d-dans✔✔slopedparameterd(slopedofdthedlinedwedaredtryingdtodfit)
epsilond(ε)d-dans✔✔isdtheddeviancedofdtheddatadfromdthedlineardmodel
todfinddβ0danddβ1d-
dans✔✔todfinddthedlinedthatddescribesdadlineardrelationship,dsuchdthatdwedfitdthisdmodel.
simpledlineardregressionddatadstructured-
dans✔✔pairsdofddatadconsistingdofdadvaluedfordthedresponsedvariable,anddadvaluedfordthed
predictingdvariable.dAnddwedhavedndsuchdpairs
modelingdframeworkdfordthedsimpledlineardregression:d-
dans✔✔1)didentifyingddatadstructure
2)dclearlydstatingdthedmodeldassumptions
lineardregressiondassumptionsd-dans✔✔1)dlinearity
2)dconstantdvariancedassumption
3)dindependencedassumption
linearitydassumptiond-
dans✔✔meandzerodassumption,dmeansdthatdthedexpecteddvaluedofdthederrorsdisdzero.
, Adviolationdofdthisdassumptiondwilldleaddtoddifficultiesdindestimatingdβ0,danddmeansdthatdyo
urdmodelddoesdnotdincludedadnecessarydsystematicdcomponent.
constantdvariancedassumptiond-
dans✔✔whichdmeansdthatdthedvarianced(σ^2)dofdthederrordtermsdorddeviancesdisdconstantd
fordthedgivendpopulation.dAdviolationdofdthisdassumptiondmeansdthatdthedestimatesdarednot
dasdefficientdasdtheydcoulddbedindestimatingdthedtruedparameters
IndependencedAssumptiond-
dans✔✔whichdmeansdthatdtheddeviancesdaredindependent drandomdvariables.
Violationdofdthisdassumptiondcandleaddtodmisleadingdassessmentsdofdthedstrengthdofdthedr
egression.
normalitydassumptiond-
dans✔✔errorsd(ε)darednormallyddistributed.dThisdisdneededdfordstatisticaldinference,dfordex
ample,dconfidencedordpredictiondintervals,danddhypothesisdtesting.dIfdthisdassumptiondisdvi
olated,dhypothesisdtestsdanddconfidencedanddpredictiondintervalsdcandbedmisleading.v
thirddparameterd-dans✔✔thedvariancedofdthederrordtermsd(σ^2)
Onedapproachdisdtodminimizedthedsumdofdsquareddresidualsdorderrorsdwithdrespectdtodβ0da
nddβ1.dThisdtranslateddintodfindingdthedlinedsuchdthatdthedtotaldsquaredddeviancesdfromdth
edlinedisdminimum.d-
dans✔✔Howdcandwedgetdestimatesdofdthedregressiondcoefficientsdordparametersdindlinear
regressiondanalysis?
fitteddvaluesd-dans✔✔todbedthedregressiondlinedwheredthedparametersdaredreplaced
bydthedestimateddvaluesdofdthedparameters.
Residualsd-dans✔✔aredsimplydtheddifference
betweendobserveddresponsedanddfitteddvalues,danddtheydaredproxiesdofdthederrordtermsdin
thedregressiondmodel
MSEd-dans✔✔Thedestimatordfordsigmadsquaredisdsigmadsquaredhat,danddisdthe
sumdofdthedsquareddresiduals,ddivideddbydnd-d2.
σ^2d(sampleddistributiondofdthedvariancedestimator)d-dans✔✔isdchi-
squaredddistributiondwithdnd-d2ddegreesdofdfreedomd(We
losedtwoddegreesdofdfreedomdbecausedwedreplaceddthedtwodparametersdß0danddß1dwith
theirdestimatorsdtodobtaindthedresiduals.)
epsilondidhatd-dans✔✔proxiesdfordtheddeviancesdordthederrordterms
sampledvariancedestimatord(s^2)d-
dans✔✔thedestimatordofdthedvariancedofdthederrordtermsd(isdchi-squaredwithdnd-
d1ddegreesdofdfreedom)
