2.5.2Deriva+onsRecall:
Recall:
0
Agevs.Money
Popula'on
Cashinpocketdollars($)
Popula+onparameters
HypothesisTest
Sample,n=9Samplesta+s+cs
β0, σ2β1,
H0:β1=0H1:β1≠0
82
22
4571
29
129
1824
x y
71
54
43452111304510
AgeinYears
PREDICTOR variable
x
RESPONSE variable
Y
b0=17.7b1=0.55s=15.5R2=0.49
Forparameterβ1:
simplelinearregression
Agevs.MoneyObjec've: Thepurposeofthisobserva+onalstudywasto
demonstrateif,andtowhatextent,ageis associatedwithmoney.
DesignandMethods: Wecollectedarandomsampleofindividualsandforeach
determinedtheirage(recordedinyears)andtheamount ofmoney(indollars)intheiraccounts.Analysisof thedatawasdoneusinglinearregression.
Results: Weobtainedarandomsampleofn=9subjects. Thereisa
sta+s+callysignificantassocia+onbetweenageandmoney(p-value=0.036). Foreveryaddi'onalyearinage,anindividual’samountofmoneyincreases onaveragebyanes'matedof$0.55(95%C.I.=[$0.05,$1.05]).
Conclusions: Wefoundthat,ashypothesized,ageisassociatedwithmoney. Inoursampleageaccountedforabouthalfofthevariability observedinmoney(R2=0.49).Wepredictthata50yearoldwill have$45.1(95%P.I.=[$5.6,$84.5]),whereasa40year oldwillhave$39.6(95%P.I.=[$0.8,$78.4]).
SmallPrint: Theanalysisrestsonthefollowingassump+ons:
- theobserva+onsareindependentlyandiden+callydistributed. - theresponsevariable,money,isnormallydistributed. - Homoscedas+cityofresidualsorequalvariance. - therela+onshipbetweenresponseandpredictorvariablesislinear.
“Ourresearch(usinglinearregression)indicatesthatolderpeopleholdandusemorecash.”
Stat306:FindingRela+onshipsinData.
Lecture7Sec+on3.1Leastsquareswithtwoormore
explanatoryvariables
Agevs.Money
Popula'on
Cashinpocketdollars($)
Popula+onparameters
HypothesisTest
Sample,n=9Samplesta+s+cs
β0, σ2β1,
H0:β1=0H1:β1≠0
82
22
4571
29
129
1824
x y
71
54
43452111304510
AgeinYears
PREDICTOR variable
x
RESPONSE variable
Y
b0=17.7b1=0.55s=15.5R2=0.49
Forparameterβ1:
simplelinearregression
Agevs.Money
Popula'on Popula+onparameters
HypothesisTest
Sample,n=9Samplesta+s+cs
β0, σ2β1,β2,
H0:β1=0H1:β1≠0
82
22
4571
29
129
1824
x1 x2 y
71
54
43
4521113045
10
AgeinYearsIncomeinthousandsof$.
PREDICTOR variables
x1
x2
RESPONSE variable
Y
b0=23.26b1=0.68b2=-0.28s=13.9R2=0.65Forparameterβ1:
mul'plelinearregression
26
37
4976
40
20
1092
[0.18,1.18]
0.016
Cashinpocketdollars($)
Chapter3
3.1Leastsquareswithtwoormoreexplanatoryvariables3.4Sta+s+calsogwareoutputformul+pleregression
-R2andadjR2and3.4.1Proper+esofR2andσ2-Sumofsquaresdecomposi+on
3.5Importantexplanatoryvariables3.6Intervales+matesandstandarderrors3.7DenominatoroftheresidualSD3.8Residualplots3.9Categoricalexplanatoryvariables3.10Par+alcorrela+on
3.1Leastsquareswithtwoormoreexplanatoryvariables
0 20 40 60 80 100
0
20
40
60
80
100
Age (years)
Mon
ey ($
)
predic'onequa'on:y=b0+b1x
3.1Leastsquareswithtwoormoreexplanatoryvariables
“hyperplaneequa'on”
3.1Leastsquareswithtwoormoreexplanatoryvariables
“hyperplaneequa'on”
3.1Leastsquareswithtwoormoreexplanatoryvariables
Onceagainwecanminimizetheleastsquareswithsimplecalculus:
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
3.1Leastsquareswithtwoormoreexplanatoryvariables
designmatrixordatamatrix
Thesystemofnormalequa5ons
3.1Leastsquareswithtwoormoreexplanatoryvariables
Asbefore,YisarandomvectorandXisfixed.
3.1Leastsquareswithtwoormoreexplanatoryvariables
Asbefore,YisarandomvectorandXisfixed.
Agevs.Money
Popula'on Popula+onparameters
HypothesisTest
Sample,n=9Samplesta+s+cs
β0, σ2β1,β2,
H0:β1=0H1:β1≠0
82
22
4571
29
129
1824
x1 x2 y
71
54
43
4521113045
10
AgeinYearsIncomeinthousandsof$.
PREDICTOR variables
x1
x2
RESPONSE variable
Y
b0=23.26b1=0.68b2=-0.28s=13.9R2=0.65Forparameterβ1:
mul'plelinearregression
26
37
4976
40
20
1092
[0.18,1.18]
0.016
Cashinpocketdollars($)
3.1Leastsquareswithtwoormoreexplanatoryvariables
Chapter3
3.1Leastsquareswithtwoormoreexplanatoryvariables3.4Sta's'calsoUwareoutputformul'pleregression
-R2andadjR2and3.4.1Proper'esofR2andσ2-Sumofsquaresdecomposi'on
3.5Importantexplanatoryvariables3.6Intervales+matesandstandarderrors3.7DenominatoroftheresidualSD3.8Residualplots3.9Categoricalexplanatoryvariables3.10Par+alcorrela+on
3.4Sta+s+calsogwareoutputfor
mul+pleregression
3.4Sta+s+calsogwareoutputfor
mul+pleregression
3.4Sta+s+calsogwareoutputfor
mul+pleregression
3.4Sta+s+calsogwareoutputfor
mul+pleregression
Agevs.Money
Popula'on Popula+onparameters
HypothesisTest
Sample,n=9Samplesta+s+cs
β0, σ2β1,β2,
H0:β1=0H1:β1≠0
82
22
4571
29
129
1824
x1 x2 y
71
54
43
4521113045
10
AgeinYearsIncomeinthousandsof$.
PREDICTOR variables
x1
x2
RESPONSE variable
Y
b0=23.26b1=0.68b2=-0.28s=13.9R2=0.65Forparameterβ1:
mul'plelinearregression
26
37
4976
40
20
1092
[0.18,1.18]
0.016
Cashinpocketdollars($)
3.1Leastsquareswithtwoormoreexplanatoryvariables
“hyperplaneequa'on”
hips://commons.wikimedia.org/wiki/File:2d_mul+ple_linear_regression.gif
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