Chapter 12 Simple Linear RegressionSimple Linear Regression ModelSlide 3Simple Linear Regression EquationSlide 5Slide 6Slide 7Estimation ProcessLeast Squares MethodThe Least Squares MethodSlide 11Example: Reed Auto SalesSlide 13Slide 14Slide 15The Coefficient of DeterminationSlide 17Slide 18The Correlation CoefficientSlide 20Model AssumptionsTesting for SignificanceSlide 23Slide 24Testing for Significance: t TestSlide 26Slide 27Slide 28Confidence Interval for 1Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Some Cautions about the Interpretation of Significance TestsSlide 37Chapter 12Chapter 12 Simple Linear Regression Simple Linear RegressionSimple Linear Regression ModelSimple Linear Regression ModelLeast Squares Method Least Squares Method Coefficient of DeterminationCoefficient of DeterminationModel AssumptionsModel AssumptionsTesting for SignificanceTesting for SignificanceUsing the Estimated Regression EquationUsing the Estimated Regression Equation for Estimation and Predictionfor Estimation and PredictionComputer SolutionComputer SolutionResidual Analysis: Validating Model Residual Analysis: Validating Model AssumptionsAssumptionsSimple Linear Regression ModelSimple Linear Regression ModelThe equation that describes how y is related to The equation that describes how y is related to x and an error term is called the x and an error term is called the regression regression modelmodel..The The simple linear regression modelsimple linear regression model is: is:yy = = 00 + + 11xx + +• 00 and and 11 are called are called parameters of the parameters of the modelmodel..•  is a random variable called theis a random variable called the error error termterm..Simple Linear Regression EquationSimple Linear Regression EquationThe The simple linear regression equationsimple linear regression equation is: is:EE((yy) = ) = 00 + + 11xx• Graph of the regression equation is a Graph of the regression equation is a straight line.straight line.• 00 is the is the yy intercept of the regression line. intercept of the regression line.• 11 is the slope of the regression line. is the slope of the regression line.• EE((yy) is the expected value of ) is the expected value of yy for a given for a given xx value.value.Simple Linear Regression EquationSimple Linear Regression EquationPositive Linear RelationshipPositive Linear RelationshipEE((yy))xxSlope Slope 11is positiveis positiveRegression lineRegression lineInterceptIntercept00Simple Linear Regression EquationSimple Linear Regression EquationNegative Linear RelationshipNegative Linear RelationshipEE((yy))xxSlope Slope 11is negativeis negativeRegression lineRegression lineInterceptIntercept00Simple Linear Regression EquationSimple Linear Regression EquationNo RelationshipNo RelationshipEE((yy))xxSlope Slope 11is 0is 0Regression lineRegression lineInterceptIntercept00Estimated Simple Linear Regression Estimated Simple Linear Regression EquationEquationThe The estimated simple linear regression estimated simple linear regression equationequation is: is:• The graph is called the estimated The graph is called the estimated regression line.regression line.• bb00 is the is the yy intercept of the line. intercept of the line.• bb11 is the slope of the line. is the slope of the line.• is the estimated value of is the estimated value of yy for a given for a given xx value.value.0 1ˆy b b x 0 1ˆy b b x ˆyˆyEstimation ProcessEstimation ProcessRegression ModelRegression Modelyy = = 00 + + 11xx + +Regression EquationRegression EquationEE((yy) = ) = 00 + + 11xxUnknown ParametersUnknown Parameters00, , 11Sample Data:Sample Data:x yx yxx11 y y11. .. . . .. . xxnn yynnEstimatedEstimatedRegression EquationRegression Equation Sample StatisticsSample Statisticsbb00, , bb11bb00 and and bb11provide estimates ofprovide estimates of00 and and 110 1ˆy b b x 0 1ˆy b b x Least Squares MethodLeast Squares MethodLeast Squares CriterionLeast Squares Criterionwhere:where:yyii = = observedobserved value of the dependent value of the dependent variablevariable for the for the iith observationth observationyyii = = estimatedestimated value of the dependent value of the dependent variablevariable for the for the iith observationth observationmin (y yi i)2min (y yi i)2^^Slope for the Estimated Regression EquationSlope for the Estimated Regression Equationbx y x y nx x ni i i ii i12 2( )/( ) /bx y x y nx x ni i i ii i12 2( )/( ) /The Least Squares MethodThe Least Squares Methodwhere:where:xxii = value of independent variable for = value of independent variable for iith th observationobservationyyii = value of dependent variable for = value of dependent variable for iith th observationobservation nn = total number of observations = total number of observationsyy-Intercept for the Estimated Regression -Intercept for the Estimated Regression EquationEquationwhere:where: xx = mean value for independent variable = mean value for independent variable yy = mean value for dependent variable = mean value for dependent variable nn = total number of observations = total number of observations____The Least Squares MethodThe Least Squares Method0 1b y b x 0 1b y b x Example: Reed Auto SalesExample: Reed Auto SalesSimple Linear RegressionSimple Linear Regression Reed Auto periodically has a special week-Reed Auto periodically has a special week-long sale. As part of the advertising campaign long sale. As part of the advertising campaign Reed runs one or more television commercials Reed runs one or more television commercials during the weekend preceding the sale. Data during the weekend preceding the sale. Data from a sample of 5 previous sales are shown from a sample of 5 previous sales are shown on the next slide.on the next slide.Example: Reed Auto SalesExample: Reed Auto SalesSimple Linear RegressionSimple