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. 2

Regression Models Probabilistic Multiple Regression Model Y = 0 + 1X1 + 2X2 + 3X3 + . . . + kXk+  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. 4

Estimated Regression Model Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

Multiple Regression Model with Two Independent Variables (First-Order) Population Model Estimated Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

Response Plane for First-Order Two-Predictor Multiple Regression Model Y  Vertical Intercept Y1 Response Plane X2 X1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Least Squares Equations for k = 2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Real Estate Data Observation Y X1 X2 1 63.0 65.1 1,605 35 13 79.7 2,121 14 2 2,489 45 84.5 2,485 9 3 69.9 7 1,553 20 15 96.0 2,300 19 4 76.8 2,404 32 16 109.5 2,714 5 73.9 1,884 25 17 102.5 2,463 6 77.9 1,558 18 121.0 3,076 74.9 1,748 8 104.9 3,048 78.0 3,105 10 128.0 3,267 79.0 1,682 28 21 129.0 3,069 63.4 2,470 30 22 117.9 4,765 11 79.5 1,820 23 140.0 4,540 12 83.9 2,143 Market Price ($1,000) Square Feet Age (Years) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 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 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Predicting the Price of Home Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

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. 13

Testing the Overall Model for the Real Estate Example ANOVA df SS MS F p Regression 2 8189.723 4094.86 28.63 .000 Residual (Error) 20 2861.017 143.1 Total 22 11050.74 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Significance Test of the Regression Coefficients for the Real Estate Example tCal = 5.63 > 2.086, reject H0. Coefficients Std Dev t Stat p x1 (Sq.Feet) 0.0177 0.003146 5.63 .000 x2 (Age) -0.666 0.2280 -2.92 .008 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

Residuals and Sum of Squares Error for the Real Estate Example SSE Observation Y 1 43.0 42.466 0.534 0.285 13 59.7 65.602 -5.902 34.832 2 45.1 51.465 -6.365 40.517 14 64.5 75.383 -10.883 118.438 3 49.9 51.540 -1.640 2.689 15 76.0 65.442 10.558 111.479 4 56.8 58.622 -1.822 3.319 16 89.5 82.772 6.728 45.265 5 53.9 54.073 -0.173 0.030 17 82.5 77.659 4.841 23.440 6 57.9 55.627 2.273 5.168 18 101.0 87.187 13.813 190.799 7 54.9 62.991 -8.091 65.466 19 84.9 89.356 -4.456 19.858 8 58.0 85.702 -27.702 767.388 20 108.0 91.237 16.763 280.982 9 59.0 48.495 10.505 110.360 21 109.0 85.064 23.936 572.936 10 63.4 61.124 2.276 5.181 22 97.9 114.447 -16.547 273.815 11 59.5 68.265 -8.765 76.823 23 120.0 112.460 7.540 56.854 12 63.9 71.322 -7.422 55.092 2861.017 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

MINITAB Residual Diagnostics for the Real Estate Problem 3 2 1 - 6 5 4 R e s i d u a l F r q n c y H t o g m f O b v N I C h X = 7 . E S L 9 8 P M D Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

SSE and Standard Error of the Estimate ANOVA df SS MS F P Regression 2 8189.7 4094.9 28.63 .000 Residual (Error) 20 2861.0 143.1 Total 22 11050.7 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Coefficient of Multiple Determination (R2) SSE ANOVA df SS MS F p Regression 2 8189.7 4094.89 28.63 .000 Residual (Error) 20 2861.0 143.1 Total 22 11050.7 SSYY SSR Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

Adjusted R2 SSYY SSE n-k-1 n-1 ANOVA df SS MS F p Regression 2 8189.7 4094.9 28.63 .000 Residual (Error) 20 2861.0 143.1 Total 22 11050.7 SSYY SSE n-k-1 n-1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

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

Demonstration Problem 14.1: Excel Output Regression Statistics Multiple R 0.812 R Square 0.659 Adjusted R Square 0.488 Standard Error 44273.86677 Observations 7 ANOVA   df SS MS F Sig. F Regression 2 15148592381 7.57E+09 3.86 0.116 Residual 4 7840701114 1.96E+09 Total 6 22989293495 Coefficients t Stat P-value Intercept -26425.45085 67624.93769 -0.39 0.716 Length 0f Roads 0.101820862 0.043495015 2.34 0.079 Commercial Vehicles 0.04094856 0.017121018 2.39 0.075 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.