Multiple-Regression-and-Correlation.pptx

Chapter 16Multiple Regression and Correlationto accompanyIntroduction to Business Statisticsfourth edition,by Ronald M.WeiersPresentation by Priscilla Chaffe-Stengel Donald N.Stengel 2002 The Wadsworth GroupChapter 16 Learning ObjectivesObtain and interpret the multiple regression equationMake estimates using the regression model:Point value of the dependent variable,yIntervals:Confidence interval for the conditional mean of yPrediction interval for an individual y observationConduct and interpret hypothesis tests on theCoefficient of multiple determinationPartial regression coefficients 2002 The Wadsworth GroupChapter 16-Key TermsPartial regression coefficientsMultiple standard error of the estimateConditional mean of yIndividual y observationCoefficient of multiple determinationCoefficient of partial determinationGlobal F-testStandard deviation of bi 2002 The Wadsworth GroupThe Multiple Regression ModelProbabilistic Modelyi=b0+b1x1i+b2x2i+.+bkxki+ei whereyi=a value of the dependent variable,yb0=the y-interceptx1i,x2i,.,xki=individual values of the independent variables,x1,x2,.,xkb1,b2,.,bk=the partial regression coefficients for the independent variables,x1,x2,.,xkei=random error,the residual 2002 The Wadsworth GroupThe Multiple Regression ModelSample Regression Equation =b0+b1x1i+b2x2i+.+bkxki where =the predicted value of the dependent variable,y,given the values of x1,x2,.,xkb0=the y-interceptx1i,x2i,.,xki=individual values of the independent variables,x1,x2,.,xkb1,b2,.,bk=the partial regression coefficients for the independent variables,x1,x2,.,xk 2002 The Wadsworth GroupThe Amount of Scatter in the DataThe multiple standard error of the estimate where yi=each observed value of y in the data set =the value of y that would have been estimated from the regression equationn=the number of data values in the setk=the number of independent(x)variablesmeasures the dispersion of the data points around the regression hyperplane.se=(yi y i)2nk1 2002 The Wadsworth GroupApproximating a Confidence Interval for a Mean of yA reasonable estimate for interval bounds on the conditional mean of y given various x values is generated by:where =the estimated value of y based on the set of x values providedt=critical t value,(1a)%confidence,df=n k 1se=the multiple standard error of the estimatenesty 2002 The Wadsworth GroupApproximating a Prediction Interval for an Individual y ValueA reasonable estimate for interval bounds on an individual y value given various x values is generated by:where =the estimated value of y based on the set of x values providedt=critical t value,(1a)%confidence,df=n k 1se=the multiple standard error of the estimate y tse 2002 The Wadsworth GroupCoefficient of Multiple DeterminationThe proportion of variance in y that is explained by the multiple regression equation is given by:R2=1S(yi y i)2S(yiy)2=1 SSESST=SSRSST 2002 The Wadsworth GroupCoefficients of Partial DeterminationFor each independent variable,the coefficient of partial determination denotes the proportion of total variation in y that is explained by that one independent variable alone,holding the values of all other independent variables constant.The coefficients are reported on computer printouts.2002 The Wadsworth GroupTesting the Overall Significance of the Multiple Regression ModelIs using the regression equation to predict y better than using the mean of y?The Global F-TestI.H0:b1=b2=.=bk=0The mean of y is doing as good a job at predicting the actual values of y as the regression equation.H1:At least one bi does not equal 0.The regression model is doing a better job of predicting actual values of y than using the mean of y.2002 The Wadsworth GroupTesting Model SignificanceII.Rejection RegionGiven a and numerator df=k,denominator df=n k 1Decision Rule:If F critical value,reject H0.2002 The Wadsworth GroupTesting Model SignificanceIII.Test StatisticwhereSSR=SST SSESST=SSE=If H0 is rejected:At least one bi differs from zero.The regression equation does a better job of predicting the actual values of y than using the mean of y.S(yi y)2S(yi y)2 2002 The Wadsworth GroupTesting the Significance of a Single Regression CoefficientIs the independent variable xi useful in predicting the actual values of y?The Individual t-TestI.H0:bi=0The dependent variable(y)does not depend on values of the independent variable xi.(This can,with reason,be structured as a one-tail test instead.)H1:bi 0The dependent variable(y)does change with the values of the independent variable xi.2002 The Wadsworth GroupTesting the Impact on y of a Single Independent VariableII.Rejection RegionGiven a and df=n k 1Decision Rule:If t critical valueor t critical value,reject H0.2002 The Wadsworth GroupTesting the Impact on y of a Single Independent VariableIII.Test Statisticwherebi=estimate for bi for the multiple regression equation =the standard deviation of biIf H0 is rejected:The dependent variable(y)does change with the independent variable(xi).t=bi 0sbi 2002 The 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