Chapter 1 Matrix approach to Simple Regression Models

# Chapter 1 Matrix approach to Simple Regression Models -...

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CYM 1.1 Chapter 1 MatrixApproachto SimpleRegressionModel ST3131 Regression Analysis

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CYM 1.2 Overview Leastsquaresestimation Sumofsquares:SST,SSR,andSSE ANOVAtable, F ‐test Varianceof Confidenceintervalsfor β 0 and β 1 Confidenceintervalfor ݕ | ݔ Predictionintervalfor ST3131 Regression Analysis
CYM 1.3 1.1 Simple Regression Model Considerthesimpleregressionmodel where ݅ istheresponse(ordependent)variable, ݅ isthepredictor(orindependent)variable, and areunknownparameters, sareindependentnormalrandomvariables, with and forall i . ST3131 Regression Analysis Deterministic but unknown Random

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CYM 1.4 Simple Regression Model ST3131 Regression Analysis Random Error of Y for this X i value Y X Observed Value of Y for X i Expected Value of Y for X i 0 1 Y β β X X i Slope = β 1 Intercept = β 0 ε i i 0 1 i i Y β β X ε
Simple Regression Model Recall TheLeastSquaresEstimatesfor and inModel(1)are obtainedby ,ఉ TheLSEfor and aregivenby ௜ୀଵ ௜ୀଵ CYM 1.5 ST3131 Regression Analysis

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Simple Regression Model Recall WewanttotestH 0 : againstH 1 : Teststatistic ି଴ ௏௔௥ RejectH 0 if where with CYM 1.6 ST3131 Regression Analysis
Simple Regression Model Model(1)canbeexpressedinamatrixformasfollows: where , , ,and with and . Hence and . CYM 1.7 ST3131 Regression Analysis

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1.2 Least Squares Estimation Leastsquaresestimatorfor β canbeobtainedby minimizing (i.e. ௜ୀଵ ). Since ,therefore ST3131 Regression Analysis CYM 1.8
Least Squares Estimation (Continued) Rewritetheaboveexpressionas Byexpandingtheaboveexpression,wehave Hence and ST3131 Regression Analysis CYM 1.9

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Least Squares Estimation (Continued) Assumingthat isnon‐singular(i.e. ିଵ exists), then ିଵ and ST3131 Regression Analysis CYM 1.10