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Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-1 Business Statistics, 4e by Ken Black Chapter 14 Multiple Regression Analysis.

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Presentation on theme: "Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-1 Business Statistics, 4e by Ken Black Chapter 14 Multiple Regression Analysis."— Presentation transcript:

1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-1 Business Statistics, 4e by Ken Black Chapter 14 Multiple Regression Analysis

2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-2 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.

3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-3 Regression Models Probabilistic Multiple Regression Model Y =  0 +  1 X 1 +  2 X 2 +  3 X 3 +... +  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

4 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-4 Estimated Regression Model

5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-5 Multiple Regression Model with Two Independent Variables (First-Order) Population Model Estimated Model

6 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-6 Response Plane for First-Order Two- Predictor Multiple Regression Model X1X1 X2X2 Response Plane Y1Y1 Vertical Intercept Y 

7 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-7 Least Squares Equations for k = 2

8 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-8 Real Estate Data ObservationYX1X1 X2X2 YX1X1 X2X2 163.0 65.1 1,6053513 79.72,12114 22,4894514 84.52,4859 369.9 7 1,5532015 96.02,30019 476.82,4043216109.52,7144 573.91,8842517102.52,4635 677.91,5581418 121.03,0767 774.91,748819104.93,0483 878.03,1051020 128.03,2676 979.01,6822821 129.03,06910 63.42,4703022117.94,76511 79.51,820223 140.04,5408 1283.92,1436 Market Price ($1,000) Square Feet Age (Years) Market Price ($1,000) Square Feet Age (Years)

9 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-9 MINITAB Output for the Real Estate Example The regression equation is Price = 57.4 + 0.0177 Sq.Feet - 0.666 Age Predictor Coef StDev T P Constant 57.35 10.01 5.73 0.000 Sq.Feet 0.017718 0.003146 5.63 0.000 Age -0.6663 0.2280 -2.92 0.008 S = 11.96 R-Sq = 74.1% R-Sq(adj) = 71.5% Analysis of Variance Source DF SS MS F P Regression 2 8189.7 4094.9 28.63 0.000 Residual Error 20 2861.0 143.1 Total 22 11050.7

10 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-10 Predicting the Price of Home

11 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-11 Evaluating the Multiple Regression Model Significance Tests for Individual Regression Coefficients Testing the Overall Model

12 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-12 Testing the Overall Model for the Real Estate Example ANOVA df SSMSF p Regression28189.7234094.8628.63.000 Residual (Error)202861.017 143.1 Total2211050.74

13 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-13 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)0.01770.003146 5.63.000 x 2 (Age)-0.6660.2280-2.92.008 t.025,20 = 2.086

14 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-14 Residuals and Sum of Squares Error for the Real Estate Example SSE ObservationY Y 143.042.4660.5340.2851359.765.602-5.90234.832 245.151.465-6.36540.5171464.575.383-10.883118.438 349.951.540-1.6402.6891576.065.44210.558111.479 456.858.622-1.8223.3191689.582.7726.72845.265 553.954.073-0.1730.0301782.577.6594.84123.440 657.955.6272.2735.16818101.087.18713.813190.799 754.962.991-8.09165.4661984.989.356-4.45619.858 858.085.702-27.702767.38820108.091.23716.763280.982 959.048.49510.505110.36021109.085.06423.936572.936 1063.461.1242.2765.1812297.9114.447-16.547273.815 1159.568.265-8.76576.82323120.0112.4607.54056.854 1263.971.322-7.42255.0922861.017

15 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-15 MINITAB Residual Diagnostics for the Real Estate Problem 3020100-10-20-30 6 5 4 3 2 1 0 Residual F r e q u e n c y Histogram of Residuals 20100 40 30 20 10 0 -10 -20 -30 -40 Observation Number R e s i d u a l I Chart of Residuals X=-7.2E-14 3.0SL=31.26 -3.0SL=-31.26 14013012011010090807060 20 10 0 -10 -20 -30 Fit R e s i d u a l Residuals vs. Fits 210-1-2 20 10 0 -10 -20 -30 Normal Plot of Residuals Normal Score R e s i d u a l Residual Model Diagnostics

16 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-16 SSE and Standard Error of the Estimate SSE ANOVA df SSMSF P Regression28189.74094.928.63.000 Residual (Error)202861.0143.1 Total2211050.7

17 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-17 Coefficient of Multiple Determination (R 2 ) SSE ANOVA df SSMSF p Regression28189.74094.8928.63.000 Residual (Error)202861.0143.1 Total2211050.7 SS YY SSR

18 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-18 Adjusted R 2 ANOVA df SSMSF p Regression28189.74094.928.63.000 Residual (Error)202861.0143.1 Total2211050.7 SS YY SSE n-k-1 n-1

19 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-19 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 278,806 673,239 5,010,000 Brazil 178,359 1,031,693 1,371,127 India 144,000 1,342,000 1,980,000 Germany 138,975 395,367 2,923,000 Italy 125,171 188,597 2,745,500 Spain 105,824 206,271 2,859,438 Mexico 96,049 157,036 3,758,034

20 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14-20 Demonstration Problem 14.1: Excel Output Regression Statistics Multiple R0.812 R Square0.659 Adjusted R Square0.488 Standard Error44273.86677 Observations7 ANOVA dfSSMSFSig. F Regression2151485923817.57E+093.860.116 Residual478407011141.96E+09 Total622989293495 CoefficientsStandard Errort StatP-value Intercept-26425.4508567624.93769-0.390.716 Length 0f Roads0.1018208620.0434950152.340.079 Commercial Vehicles0.040948560.0171210182.390.075

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