Introduction to Regression Analysis Chapter 2 (2)

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1、InferenceaboutE(Y)andpredictingY'sANOVAandF-testsR2andrChapter21InferenceaboutE(Y)andpredictingY's2ANOVAandF-tests3R2andr1/28InferenceaboutE(Y)andpredictingY'sANOVAandF-testsR2andrInferenceaboutE(Y)andpredictingY'sRevisedslides(forchap2part.pdf,slide29-36havesometypos)tobetterexplaindistinctionbet

2、weencondenceintervalsandpredictionintervalsw.r.t.theY'sCondenceinterval:Forparameterestimation,e.g.1orE(Yh)=0+1XhPredictioninterval:Forpredictingtheoutcomeofarandomvariable,e.g.newobservationYh(new)2/28InferenceaboutE(Y)andpredictingY'sANOVAandF-testsR2andrPredictingthemeanoutcomeE(Yh)atX=XhWhat

3、'stheexpected/meanoutcomeofastudent'sGPA,ifhisACTtestscoreisgivenbyXh(withinscopeofobservedACToutcomes)?EstimatorforE(Yh)givenby:Y^h=b0+b1Xh,withsamplingdistribution:Y^N+X;2fY^g;h01hh2with2fY^g=21+P(XhX)hn(XiX)2Aswiththesamplingdistributionofthebi's,2isunknownandestimatedbys2,whichthe

4、ngivesat-distributionforY^h:Y^hE(Yh)221(XhX)2tn2;withsfY^hg=s+PsfY^hgn(XiX)2UsethisdistributiontoconstructcondenceintervalsforE(Yh):Y^ht(1=2;n2)sfY^hg3/28InferenceaboutE(Y)andpredictingY'sANOVAandF-testsR2andrConndenceintervalforE(Y)Extra(takenotes!)Theboundsofthecondenceintervalsf

5、orE(Yh):Y^ht(1=2;n2)sfY^hg;arerandom,weexpect95/100CIstocontainthetrueE(Y)Therefore;weare(1)%condentthatthe(realized)CIwillcontainthetrueE(Y)4/28InferenceaboutE(Y)andpredictingY'sANOVAandF-testsR2andrFreshmen'sGPA,andtheircollegeentrancetest●●●4.0●●●●●●●●●●●●●●●●95%condence●●●●●●●●●●●●●●●●●

6、interval(CI)for3.5●●●●●●●●●●●●●●●●●●●●●E(Y)atX=20is●●●hh●●●●●●●●●●●3.0●●●●●●●●●●●●givenby[2.7,3.1]●●●●●●●●●●●●2.5●●●Interpretation:We●●●●●●GPA●●●2.0●are95%condent●●●thatthemeanGPA●1.5●forstudentswithtest1.0score20issomewhere95%CIforEYatX=20LSregressionlineb0+b1*Xbetween2.7and3.10.5●points15202530

7、35ACT5/28InferenceaboutE(Y)andpredictingY'sANOVAandF-testsR2andrPredictionofanewobservationWhatoutcomesdoweexpectforonestudent'sGPA,ifhisACTtestscoreisgivenbyXh(withinscopeofobservedACToutcomes)?Howisthisdierent