Stat 306: Finding Relaonships in Data....stasc (“esmator”) Esmator as a Random Variable β 0 b 0 B 0 β 1 b 1 B 1 σ2 2 s2 S Step 0: From θ, deﬁne es1mator, Step 1: Consider

Stat306:FindingRela1onshipsinData.

Lecture6Sec1on2.6

Recapfromlastlecture2.5(con1nued)

Popula'onparameteror“somethingwewouldliketoes'mate”

Samplesta's'c(“es'mator”)

Es'matorasaRandomVariable

ExpectedValueofthees'mator

Varianceofthees'mator

StandardErrorofes'mator

ConfidenceInterval

β0 b0 B0 E[B0] Var[B0]

se(b0) C.I.forβ0

β1 b1 B1 E[B1] Var[B1] se(b1) C.I.forβ1

σ2

s2 S2 E[S2] Var[S2] se(s2) C.I.forσ2

E Var C.I.for

Step0:Fromθ,definees1mator,

Step1:Considerthesamplesta1s1c,,asarandomvariable

Step2:DetermineE[](toconfirmit’sunbiased)Var[](tocalculatese)

Step3:Definese()=

Step4:Define(1-α)%C.I.=

•  Confusedabouthomogeneityvs.non-consistentwidthofconfidenceintervals?

0 20 40 60 80 100

020

4060

80100

x

y

σ2isthevarianceofY;constantregardlessofthevalueofx.

Thebluedashedlineistheconfidenceintervalforthesubpopula1onmean.Inotherwords,itrepresentsthevariabilityinoures1mateofthemeanofYasxchanges.

Supposewenowwanttomakeapredic'onforanewvalueofx.Example:Supposewewouldliketopredicthowmuchmoney(Y),

someoneaged50yearsold(X=50)willhave.

0 20 40 60 80 100

020

4060

80100

x

y

Predic'onsandpredic'onintervals

Example:Supposewewouldliketopredicthowmuchmoney(Y), someoneagedX=50yearsoldwillhave.

thishypothe1calnewpersonaged50issome1mescalled“anout-of-sampleunitwithvaluex*”,Wherex*=50.Ourbestes1mate,alsoknownasthe“pointpredic1on”,wouldbeequaltob0+b1(50)=45.1



0 20 40 60 80 100

020

4060

80100

x

y


Ourpredic1onThetruth

Thedifferencebetweenourpredic1onandthetruthistheerror




Thedifferencebetweenourpredic1onandthetruthistheerror

Cov()isequalto0,sincethetwotermsareindependent.




is

Note


se(E)=


0 20 40 60 80 100

020

4060

80100

x

y


Agevs.MoneyObjec've: Thepurposeofthisobserva1onalstudywasto

demonstrateif,andtowhatextent,ageis associatedwithmoney.

DesignandMethods: Wecollectedarandomsampleofindividualsandforeach

determinedtheirage(recordedinyears)andtheamount ofmoney(indollars)intheiraccounts.Analysisof thedatawasdoneusinglinearregression.

Results: Weobtainedarandomsampleofn=9subjects. Thereisa

sta1s1callysignificantassocia1onbetweenageandmoney(p-value=0.036). Foreveryaddi1onalyearinage,anindividual’samountofmoneyincreases onaveragebyanes1matedof$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: Theanalysisrestsonthefollowingassump1ons:

- theobserva1onsareindependentlyandiden1callydistributed. - theresponsevariable,money,isnormallydistributed. - Homoscedas1cityofresidualsorequalvariance. - therela1onshipbetweenresponseandpredictorvariablesislinear.

Forparameterβ1:

se(subpopula'onmean)VS.se(predic'onerror)

Subpopula1onmean:

Whereas is:

2.6Explana1onofStudenttquan1lesintheintervales1mates

2.6.1.Historylessonaboutthet-test2.6.2.Threeimportantthingstoknowaboutanormalrandomvariable2.6.3Es1matorsasRandomVariables(onemore1me!)2.6.4Explana1onofStudenttquan1les

2.6.1.Historylessonaboutthet-testStudentisthepublica1onpseudonymfor

WilliamGosset,whodevelopedmethodsforinferenceofmeansforsmallsampleswhileworkingatGuinnessBrewery(Ireland)inearly1900s.htps://en.wikipedia.org/wiki/William_Sealy_Gosset

WilliamSealyGosset(aka“Student”):“Isthisbatchofbeeranydifferentthanthestandard?”“Let’shaveatastetest!…t-testanyone?”


2.6.1.Historylessonaboutthet-test2.6.2.Threeimportantthingstoknowaboutanormalrandomvariable2.6.3Es1matorsasRandomVariables(onemore1me!)2.6.4Explana1onofStudenttquan1les

•  Thing1:–  Linearcombina1onsofindependentnormalrandomvariablesalsohavenormaldistribu1ons!(seeAppendixB)

2.6.2.Threeimportantthingstoknow

aboutanormalrandomvariable

•  Thing1:–  Linearcombina1onsofindependentnormalrandomvariablesalsohavenormaldistribu1ons!(seeAppendixB)

Forexample: Let: W1beanormalrandomvariable andW2beanormalrandomvariable, Then: W3=aW1+bW2isanormalr.v. foranynumbersaandb.



•  Thing2:–  Anormalrandomvariablecanbeconvertedtoastandardnormalrandomvariable.



•  Thing3:–  Ifthevarianceisunknown,wemustusethetdistribu1on.




2.6.1.Historylessonaboutthet-test2.6.2.Threeimportantthingstoknowaboutanormalrandomvariable2.6.3Es'matorsasRandomVariables(onemore'me!)2.6.4Explana1onofStudenttquan1les




ExpectedValueofthees'mator

Varianceofthees'mator

StandardErrorofes'mator

ConfidenceInterval

β0 b0 B0 E[B0] Var[B0]

se(b0) C.I.forβ0

β1 b1 B1 E[B1] Var[B1] se(b1) C.I.forβ1

σ2

s2 S2 E[S2] Var[S2] se(s2) C.I.forσ2

E Var C.I.for



Step2:DetermineE[](toconfirmit’sunbiased)Var[](tocalculatese)

Step3:Definese()=

Step4:Define(1-α)%C.I.=




β0 b0 B0

β1 b1 B1

σ2

s2 S2



SinceB1isalinearcombina1onoftheYis(NormalRVs),then(withThing1):

Recall:

and:

and:

,where:




β0 b0 B0

β1 b1 B1

σ2

s2 S2



Recall:

Also:

Fortheintercept,wecan,again,makeuseofthefactthatB0isalinearcombina1onofnormalrandomvariables(Thing1):




β0 b0 B0

β1 b1 B1

σ2

s2 S2






β0 b0 B0

β1 b1 B1

σ2

s2 S2



Wehavethat:

Andagain,alinearcombina1onofnormalrandomvariablesisanormalrandomvariable(Thing1):




β0 b0 B0

β1 b1 B1

σ2

s2 S2




2.6.1.Historylessonaboutthet-test2.6.2.Threeimportantthingstoknowaboutanormalrandomvariable2.6.3Es1matorsasRandomVariables(onemore1me!)2.6.4Explana'onofStudenttquan'les




β0 b0 B0

β1 b1 B1

σ2

s2 S2



WithThing2,wehave:

Butwedonotknowthevariance.Weonlyhaveanes1mateofthevariance,so(withThing3):

Andtherefore:

Butwedonotknowthevariance,sowithThing3,wehave:

WithThing2,wehave:

Andtherefore:

WithThing1,wehave:

2.6.4Explana'onofStudenttquan'lesEs'matorasaRandomVariable

B0

B1

S2


WithThing2,wehave:

WithThing1,wehave:

2.6.4Explana'onofStudenttquan'les

Andtherefore:

withα=0.05:


B0

B1

S2


WithThing2,wehave:

Andtherefore:

WithThing1,wehave:

95%C.I.forβ1=

where: ,

2.6.4Explana'onofStudenttquan'lesEs'matorasaRandomVariable

B0

B1

S2


WithThing2,wehave:

WithThing1,wehave:


Andtherefore:

withα=0.05:


B0

B1

S2


WithThing2,wehave:

WithThing1,wehave:


Andtherefore:


B0

B1

S2

Butwedonotknowthevariance,sowithThing3,wehave…

WithThing2,wehave…

WithThing1,wehave:


Andtherefore:

where:


B0

B1

S2

HypothesisTestH0:β1=0H1:β1≠0

Wehave:

“Null” hy)othesis

“Alter1ative” hy)othesis



Wehave:



=0



Wehave:



=0

Therefore,“underthenull”,wehave:


Evenifwedecidetorecord“Age”(x)inmonthsand“Money”(Y)inpennies,“underthenull”,wes1llhave:


Evenifwedecidetorecord“Age”(x)inmonthsand“Money”(Y)inpennies,“underthenull”,wes1llhave:

Therefore…Ifβ1(theslope)wasactuallyequalto0,itwouldbeveryunlikelythattheabsolutet-statwouldbeverylarge.

and:


Explana1onofThing3:


Agevs.Money

Popula'on

dollars($)Inbankaccount

Popula1onparameters

HypothesisTest

Sample,n=9Samplesta1s1cs

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:

linearregression

Documents

Stat 306: Finding Relaonships in Data....stasc (“esmator”) Esmator as a Random Variable β 0 b 0 B 0 β 1 b 1 B 1 σ2 2 s2 S Step 0: From θ, deﬁne es1mator, Step 1: Consider