ISYE 6414 MIDTERM, SUMMER 2021
Thedpredictiondintervaldofdonedmemberdofdthedpopulationdwilldalwaysdbedlargerdthandthedc
onfidencedintervaldofdthedmeandresponsedfordalldmembersdofdthedpopulationdwhendusingdt
hedsamedpredictingdvalues.d-dans✔✔true
Seed1.7dRegressiondLine:dEstimationd&dPredictiondExamples
"Justdtodwrapdupdthedcomparison,dthedconfidencedintervalsdunderdestimationdarednarrower
dthandthedpredictiondintervalsdbecausedthedpredictiondintervalsdhavedadditionaldvariancedfr
omdthedvariationdofdadnewdmeasurement."
IndANOVA,dthedlinearitydassumptiondisdassesseddusingdadplotdofdthedresponsedagainstdthe
dpredictingdvariable.d-dans✔✔false
Seed2.2.dEstimationdMethod
LinearitydisdnotdandassumptiondofdANOVA.
Ifdthedmodeldassumptionsdhold,dthendthedestimatordfordthedvariance,dσd^d2,disdadrandomdva
riable.d-dans✔✔true
Seed1.8dStatisticaldInference
Wedassumedthatdthederrordtermsdaredindependentdrandomdvariables.dTherefore,dthedresid
ualsdaredindependentdrandomdvariables.dSincedσd^d2disdadcombinationdofdthedresiduals,ditdi
sdalsodadrandomdvariable.
ThedmeandsumdofdsquaredderrorsdindANOVAdmeasuresdvariabilitydwithindgroups.d-
dans✔✔true
Seed2.4dTestdfordEqualdMeans
MSEd=dwithin-groupdvariability
Thedsimpledlineardregressiondcoefficient,dβd^d0,disduseddtodmeasuredthedlineardrelationship
dbetweendthedpredictingdanddresponsedvariables.d-dans✔✔false
Seed1.2dEstimationdMethod
βd^d0disdthedinterceptdandddoesdnotdtelldusdaboutdthedrelationshipdbetweendthedpredictingda
nddresponsedvariables.
,Thedsamplingddistributiondfordthedvariancedestimatordindsimpledlineardregressiondisdχd2d(ch
i-squared)dregardlessdofdthedassumptionsdofdtheddata.d-dans✔✔false
Seed1.2dEstimationdMethod
"Thedsamplingddistributiondofdthedestimatordofdthedvariancedisdchi-squared,dwithdnd-
d2ddegreesdofdfreedomd(moredondthisdindadmoment).dThisdisdunderdthedassumptiondofdnor
malitydofdthederrordterms."
βd^d1disdandunbiaseddestimatordfordβd0.d-dans✔✔False
Seed1.4dStatisticaldInference
"Whatdthatdmeansdisdthatdβd^d1disdandunbiaseddestimatordfordβd1."dItdisdnotdandunbiaseddes
timatordfordβd0.
IfdthedpairwisedcomparisondintervaldbetweendgroupsdindandANOVAdmodeldincludesdzero,d
wedconcludedthatdthedtwodmeansdaredplausiblydequal.d-dans✔✔true
Seed2.8dDatadExample
Ifdthedcomparisondintervaldincludesdzero,dthendthedtwodmeansdarednotdstatisticallydsignific
antlyddifferent,danddaredthus,dplausiblydequal.
Underdthednormalitydassumption,dthedestimatordfordβd1disdadlineardcombinationdofdnormall
yddistributeddrandomdvariables.d-dans✔✔true
Seed1.4dStatisticaldInference
"Underdthednormalitydassumption,dβd1disdthusdadlineardcombinationdofdnormallyddistributed
drandomdvariables...dβd^d0disdalsodlineardcombinationdofdrandomdvariables"
AndANOVAdmodeldwithdadsingledqualitativedpredictingdvariabledcontainingdkdgroupsdwilldha
vedkd+d1dparametersdtodestimate.d-dans✔✔true
Seed2.2dEstimationdMethod
Wedhavedtodestimatedthedmeansdofdthedkdgroupsdanddthedpooleddvariancedestimator,dsdpd
ododldeddd2.
Indsimpledlineardregressiondmodels,dwedlosedthreeddegreesdofdfreedomdwhendestimatingdt
hedvariancedbecausedofdthedestimationdofdthedthreedmodeldparametersdβd0d,dβd1d,dσd2.d-
dans✔✔false
Seed1.2dEstimationdMethod
"Thedestimatordfordσd2disdσd^d2,danddisdthedsumdofdthedsquareddresiduals,ddivideddbydnd-
d2."
, Thedpooleddvariancedestimator,dsdpdododldeddd2,dindANOVAdisdsynonymousdwithdthedvarian
cedestimator,dσd^d2,dindsimpledlineardregressiondbecausedtheydbothdusedmeandsquaredderr
ord(MSE)dfordtheirdcalculations.d-dans✔✔true
Seed1.2dEstimationdMethoddfordsimpledlineardregression
Seed2.2dEstimationdMethoddfordANOVA
Thedpooleddvariancedestimatordis,dindfact,dthedvariancedestimator.
Thednormalitydassumptiondstatesdthatdthedresponsedvariabledisdnormallyddistributed.d-
dans✔✔false
Seed1.8dDiagnostics
"Normalitydassumption:dthederrordtermsdarednormallyddistributed."
Thedresponsedmaydordmaydnotdbednormallyddistributed,dbutdthederrordtermsdaredassumedd
todbednormallyddistributed.
IfdthedconstantdvariancedassumptiondindANOVAddoesdnotdhold,dthedinferencedondthedequa
litydofdthedmeansdwilldnotdbedreliable.d-dans✔✔true
Seed2.8dDatadExample
"Thisdisdimportantdsincedwithoutdadgooddfit,dwedcannotdrelydondthedstatisticaldinference."
Onlydwhendthedmodeldisdadgooddfit,di.e.dalldmodeldassumptionsdhold,dcandwedrelydondthedst
atisticaldinference.
Adnegativedvaluedofdβd1disdconsistentdwithdandinversedrelationshipdbetweendthedpredictord
variabledanddthedresponsedvariable.d-dans✔✔true
Seed1.2dEstimationdMethod
"Adnegativedvaluedofdβd1disdconsistentdwithdandinversedrelationship"
Thedp-valuedisdadmeasuredofdthedprobabilitydofdrejectingdthednulldhypothesis.d-
dans✔✔false
Seed1.5dStatisticaldInferencedDatadExample
"p-
valuedisdadmeasuredofdhowdrejectabledthednulldhypothesisdis...dIt'sdnotdthedprobabilitydofdrej
ectingdthednulldhypothesis,dnordisditdthedprobabilitydthatdthednulldhypothesisdisdtrue."
Wedassessdthedconstantdvariancedassumptiondbydplottingdthederrordterms,dϵdi,dagainstdfitte
ddvalues.d-dans✔✔false
Seed1.2dEstimationdMethod
"Wedusedϵd^didasdproxiesdfordtheddeviancesdordthederrordterms.dWeddon'tdhavedtheddevianc
esdbecausedweddon'tdhavedβd0danddβd1.
Thedpredictiondintervaldofdonedmemberdofdthedpopulationdwilldalwaysdbedlargerdthandthedc
onfidencedintervaldofdthedmeandresponsedfordalldmembersdofdthedpopulationdwhendusingdt
hedsamedpredictingdvalues.d-dans✔✔true
Seed1.7dRegressiondLine:dEstimationd&dPredictiondExamples
"Justdtodwrapdupdthedcomparison,dthedconfidencedintervalsdunderdestimationdarednarrower
dthandthedpredictiondintervalsdbecausedthedpredictiondintervalsdhavedadditionaldvariancedfr
omdthedvariationdofdadnewdmeasurement."
IndANOVA,dthedlinearitydassumptiondisdassesseddusingdadplotdofdthedresponsedagainstdthe
dpredictingdvariable.d-dans✔✔false
Seed2.2.dEstimationdMethod
LinearitydisdnotdandassumptiondofdANOVA.
Ifdthedmodeldassumptionsdhold,dthendthedestimatordfordthedvariance,dσd^d2,disdadrandomdva
riable.d-dans✔✔true
Seed1.8dStatisticaldInference
Wedassumedthatdthederrordtermsdaredindependentdrandomdvariables.dTherefore,dthedresid
ualsdaredindependentdrandomdvariables.dSincedσd^d2disdadcombinationdofdthedresiduals,ditdi
sdalsodadrandomdvariable.
ThedmeandsumdofdsquaredderrorsdindANOVAdmeasuresdvariabilitydwithindgroups.d-
dans✔✔true
Seed2.4dTestdfordEqualdMeans
MSEd=dwithin-groupdvariability
Thedsimpledlineardregressiondcoefficient,dβd^d0,disduseddtodmeasuredthedlineardrelationship
dbetweendthedpredictingdanddresponsedvariables.d-dans✔✔false
Seed1.2dEstimationdMethod
βd^d0disdthedinterceptdandddoesdnotdtelldusdaboutdthedrelationshipdbetweendthedpredictingda
nddresponsedvariables.
,Thedsamplingddistributiondfordthedvariancedestimatordindsimpledlineardregressiondisdχd2d(ch
i-squared)dregardlessdofdthedassumptionsdofdtheddata.d-dans✔✔false
Seed1.2dEstimationdMethod
"Thedsamplingddistributiondofdthedestimatordofdthedvariancedisdchi-squared,dwithdnd-
d2ddegreesdofdfreedomd(moredondthisdindadmoment).dThisdisdunderdthedassumptiondofdnor
malitydofdthederrordterms."
βd^d1disdandunbiaseddestimatordfordβd0.d-dans✔✔False
Seed1.4dStatisticaldInference
"Whatdthatdmeansdisdthatdβd^d1disdandunbiaseddestimatordfordβd1."dItdisdnotdandunbiaseddes
timatordfordβd0.
IfdthedpairwisedcomparisondintervaldbetweendgroupsdindandANOVAdmodeldincludesdzero,d
wedconcludedthatdthedtwodmeansdaredplausiblydequal.d-dans✔✔true
Seed2.8dDatadExample
Ifdthedcomparisondintervaldincludesdzero,dthendthedtwodmeansdarednotdstatisticallydsignific
antlyddifferent,danddaredthus,dplausiblydequal.
Underdthednormalitydassumption,dthedestimatordfordβd1disdadlineardcombinationdofdnormall
yddistributeddrandomdvariables.d-dans✔✔true
Seed1.4dStatisticaldInference
"Underdthednormalitydassumption,dβd1disdthusdadlineardcombinationdofdnormallyddistributed
drandomdvariables...dβd^d0disdalsodlineardcombinationdofdrandomdvariables"
AndANOVAdmodeldwithdadsingledqualitativedpredictingdvariabledcontainingdkdgroupsdwilldha
vedkd+d1dparametersdtodestimate.d-dans✔✔true
Seed2.2dEstimationdMethod
Wedhavedtodestimatedthedmeansdofdthedkdgroupsdanddthedpooleddvariancedestimator,dsdpd
ododldeddd2.
Indsimpledlineardregressiondmodels,dwedlosedthreeddegreesdofdfreedomdwhendestimatingdt
hedvariancedbecausedofdthedestimationdofdthedthreedmodeldparametersdβd0d,dβd1d,dσd2.d-
dans✔✔false
Seed1.2dEstimationdMethod
"Thedestimatordfordσd2disdσd^d2,danddisdthedsumdofdthedsquareddresiduals,ddivideddbydnd-
d2."
, Thedpooleddvariancedestimator,dsdpdododldeddd2,dindANOVAdisdsynonymousdwithdthedvarian
cedestimator,dσd^d2,dindsimpledlineardregressiondbecausedtheydbothdusedmeandsquaredderr
ord(MSE)dfordtheirdcalculations.d-dans✔✔true
Seed1.2dEstimationdMethoddfordsimpledlineardregression
Seed2.2dEstimationdMethoddfordANOVA
Thedpooleddvariancedestimatordis,dindfact,dthedvariancedestimator.
Thednormalitydassumptiondstatesdthatdthedresponsedvariabledisdnormallyddistributed.d-
dans✔✔false
Seed1.8dDiagnostics
"Normalitydassumption:dthederrordtermsdarednormallyddistributed."
Thedresponsedmaydordmaydnotdbednormallyddistributed,dbutdthederrordtermsdaredassumedd
todbednormallyddistributed.
IfdthedconstantdvariancedassumptiondindANOVAddoesdnotdhold,dthedinferencedondthedequa
litydofdthedmeansdwilldnotdbedreliable.d-dans✔✔true
Seed2.8dDatadExample
"Thisdisdimportantdsincedwithoutdadgooddfit,dwedcannotdrelydondthedstatisticaldinference."
Onlydwhendthedmodeldisdadgooddfit,di.e.dalldmodeldassumptionsdhold,dcandwedrelydondthedst
atisticaldinference.
Adnegativedvaluedofdβd1disdconsistentdwithdandinversedrelationshipdbetweendthedpredictord
variabledanddthedresponsedvariable.d-dans✔✔true
Seed1.2dEstimationdMethod
"Adnegativedvaluedofdβd1disdconsistentdwithdandinversedrelationship"
Thedp-valuedisdadmeasuredofdthedprobabilitydofdrejectingdthednulldhypothesis.d-
dans✔✔false
Seed1.5dStatisticaldInferencedDatadExample
"p-
valuedisdadmeasuredofdhowdrejectabledthednulldhypothesisdis...dIt'sdnotdthedprobabilitydofdrej
ectingdthednulldhypothesis,dnordisditdthedprobabilitydthatdthednulldhypothesisdisdtrue."
Wedassessdthedconstantdvariancedassumptiondbydplottingdthederrordterms,dϵdi,dagainstdfitte
ddvalues.d-dans✔✔false
Seed1.2dEstimationdMethod
"Wedusedϵd^didasdproxiesdfordtheddeviancesdordthederrordterms.dWeddon'tdhavedtheddevianc
esdbecausedweddon'tdhavedβd0danddβd1.
