Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Business Statistics, 4e by Ken Black Chapter 14 Multiple Regression Analysis

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Learning Objectives Develop a multiple regression model. Understand and apply significance tests of the regression model and its coefficients. Compute and interpret residuals, the standard error of the estimate, and the coefficient of determination. Interpret multiple regression computer output.

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Regression Models Probabilistic Multiple Regression Model Y =  0 +  1 X 1 +  2 X 2 +  3 X  k X k +  Y = the value of the dependent (response) variable  0 = the regression constant  1 = the partial regression coefficient of independent variable 1  2 = the partial regression coefficient of independent variable 2  k = the partial regression coefficient of independent variable k k = the number of independent variables  = the error of prediction

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Estimated Regression Model

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Multiple Regression Model with Two Independent Variables (First-Order) Population Model Estimated Model

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Response Plane for First-Order Two- Predictor Multiple Regression Model X1X1 X2X2 Response Plane Y1Y1 Vertical Intercept Y 

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Least Squares Equations for k = 2

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Real Estate Data ObservationYX1X1 X2X2 YX1X1 X2X , , , , , , , , , , , , , , , , , , , , , , ,1436 Market Price ($1,000) Square Feet Age (Years) Market Price ($1,000) Square Feet Age (Years)

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons MINITAB Output for the Real Estate Example The regression equation is Price = Sq.Feet Age Predictor Coef StDev T P Constant Sq.Feet Age S = R-Sq = 74.1% R-Sq(adj) = 71.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Predicting the Price of Home

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Evaluating the Multiple Regression Model Significance Tests for Individual Regression Coefficients Testing the Overall Model

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Testing the Overall Model for the Real Estate Example ANOVA df SSMSF p Regression Residual (Error) Total

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Significance Test of the Regression Coefficients for the Real Estate Example t Cal = 5.63 > 2.086, reject H 0. CoefficientsStd Devt Stat p x 1 (Sq.Feet) x 2 (Age) t.025,20 = 2.086

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Residuals and Sum of Squares Error for the Real Estate Example SSE ObservationY Y

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons MINITAB Residual Diagnostics for the Real Estate Problem Residual F r e q u e n c y Histogram of Residuals Observation Number R e s i d u a l I Chart of Residuals X=-7.2E SL= SL= Fit R e s i d u a l Residuals vs. Fits Normal Plot of Residuals Normal Score R e s i d u a l Residual Model Diagnostics

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons SSE and Standard Error of the Estimate SSE ANOVA df SSMSF P Regression Residual (Error) Total

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Coefficient of Multiple Determination (R 2 ) SSE ANOVA df SSMSF p Regression Residual (Error) Total SS YY SSR

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Adjusted R 2 ANOVA df SSMSF p Regression Residual (Error) Total SS YY SSE n-k-1 n-1

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 14.1: Freight Data Country Freight Cargo Shipped by Road (Million Short-Ton Miles) Length 0f Roads (Miles) Number of Commercial Vehicles China 278, ,239 5,010,000 Brazil 178,359 1,031,693 1,371,127 India 144,000 1,342,000 1,980,000 Germany 138, ,367 2,923,000 Italy 125, ,597 2,745,500 Spain 105, ,271 2,859,438 Mexico 96, ,036 3,758,034

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons Demonstration Problem 14.1: Excel Output Regression Statistics Multiple R0.812 R Square0.659 Adjusted R Square0.488 Standard Error Observations7 ANOVA dfSSMSFSig. F Regression E Residual E+09 Total CoefficientsStandard Errort StatP-value Intercept Length 0f Roads Commercial Vehicles