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Dec 30, 2018

15 - 1 2017 Cengage Learning. All Rights Reserved.

May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 15 Multiple Regression Learning Objectives 1. Understand how multiple regression analysis can be used to develop relationships involving one

dependent variable and several independent variables. 2. Be able to interpret the coefficients in a multiple regression analysis. 3. Know the assumptions necessary to conduct statistical tests involving the hypothesized regression

model. 4. Understand the role of computer packages in performing multiple regression analysis. 5. Be able to interpret and use computer output to develop the estimated regression equation. 6. Be able to determine how good a fit is provided by the estimated regression equation. 7. Be able to test for the significance of the regression equation. 8. Understand how multicollinearity affects multiple regression analysis. 9. Know how residual analysis can be used to make a judgement as to the appropriateness of the model,

identify outliers, and determine which observations are influential. 10. Understand how logistic regression is used for regression analyses involving a binary dependent

variable.

Chapter 15

15 - 2 2017 Cengage Learning. All Rights Reserved.

May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Solutions: 1. a. b1 = .5906 is an estimate of the change in y corresponding to a 1 unit change in x1 when x2 is held

constant. b2 = .4980 is an estimate of the change in y corresponding to a 1 unit change in x2 when x1 is held

constant. b. y = 29.1270 + .5906(180) + .4980(310) = 289.82 2. a. Partial Minitab output follows:

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 10021.2 10021.2 15.53 0.004 X1 1 10021.2 10021.2 15.53 0.004 Error 8 5161.7 645.2 Lack-of-Fit 7 5157.2 736.7 163.72 0.060 Pure Error 1 4.5 4.5 Total 9 15182.9 Model Summary S R-sq R-sq(adj) R-sq(pred) 25.4009 66.00% 61.75% 49.59% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 45.1 25.4 1.77 0.114 X1 1.944 0.493 3.94 0.004 1.00 Regression Equation Y = 45.1 + 1.944 X1

The estimated regression equation is y = 45.1 + 1.944x1 An estimate of y when x1 = 45 is y = 45.1 + 1.944(45) = 132.58 b. Partial Minitab output follows:

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 3363.4 3363.4 2.28 0.170 X2 1 3363.4 3363.4 2.28 0.170 Error 8 11819.5 1477.4 Lack-of-Fit 6 11010.5 1835.1 4.54 0.192 Pure Error 2 809.0 404.5 Total 9 15182.9

Multiple Regression

15 - 3 2017 Cengage Learning. All Rights Reserved.

May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Model Summary S R-sq R-sq(adj) R-sq(pred) 38.4374 22.15% 12.42% 0.00% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 85.2 38.4 2.22 0.057 X2 4.32 2.86 1.51 0.170 1.00 Regression Equation Y = 85.2 + 4.32 X2

The estimated regression equation is y = 85.2 + 4.32x2 An estimate of y when x2 = 15 is y = 85.2 + 4.32(15) = 150 c. Partial Minitab output is shown below:

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 2 14052 7026.1 43.50 0.000 X1 1 10689 10688.7 66.17 0.000 X2 1 4031 4030.9 24.95 0.002 Error 7 1131 161.5 Total 9 15183 Model Summary S R-sq R-sq(adj) R-sq(pred) 12.7096 92.55% 90.42% 87.95% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant -18.4 18.0 -1.02 0.341 X1 2.010 0.247 8.13 0.000 1.00 X2 4.738 0.948 5.00 0.002 1.00 Regression Equation Y = -18.4 + 2.010 X1 + 4.738 X2

The estimated regression equation is y = -18.4 + 2.01x1 + 4.738x2

Chapter 15

15 - 4 2017 Cengage Learning. All Rights Reserved.

An estimate of y when x1 = 45 and x2 = 15 is y = -18.4 + 2.01(45) + 4.738(15) = 143.12 3. a. b1 = 3.8 is an estimate of the change in y corresponding to a 1 unit change in x1 when x2, x3, and x4

are held constant. b2 = -2.3 is an estimate of the change in y corresponding to a 1 unit change in x2 when x1, x3, and x4

are held constant. b3 = 7.6 is an estimate of the change in y corresponding to a 1 unit change in x3 when x1, x2, and x4

are held constant. b4 = 2.7 is an estimate of the change in y corresponding to a 1 unit change in x4 when x1, x2, and x3

are held constant. b. y = 17.6 + 3.8(10) 2.3(5) + 7.6(1) + 2.7(2) = 57.1 4. a. y = 25 + 10(15) + 8(10) = 255; sales estimate: $255,000 b. Sales can be expected to increase by $10 for every dollar increase in inventory investment when

advertising expenditure is held constant. Sales can be expected to increase by $8 for every dollar increase in advertising expenditure when inventory investment is held constant.

5. a. Partial Minitab output follows:

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 16.640 16.640 11.27 0.015 Televison_Advertising_($1000s) 1 16.640 16.640 11.27 0.015 Error 6 8.860 1.477 Lack-of-Fit 4 6.360 1.590 1.27 0.485 Pure Error 2 2.500 1.250 Total 7 25.500 Model Summary S R-sq R-sq(adj) R-sq(pred) 1.21518 65.26% 59.46% 28.39% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 88.64 1.58 56.02 0.000 Televison_Advertising_($1000s) 1.604 0.478 3.36 0.015 1.00 Regression Equation Weekly Gross_Revenue_($1000s) = 88.64 + 1.604 Televison_Advertising_($1000s)

Multiple Regression

15 - 5 2017 Cengage Learning. All Rights Reserved.

b. Partial Minitab output follows:

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 2 23.435 11.7177 28.38 0.002 Televison_Advertising_($1000s) 1 23.425 23.4247 56.73 0.001 Newspaper_Advertising_($1000s) 1 6.795 6.7953 16.46 0.010 Error 5 2.065 0.4129 Total 7 25.500 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.642587 91.90% 88.66% 68.19% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 83.23 1.57 52.88 0.000 Televison_Advertising_($1000s) 2.290 0.304 7.53 0.001 1.45 Newspaper_Advertising_($1000s) 1.301 0.321 4.06 0.010 1.45 Regression Equation Weekly Gross_Revenue_($1000s) = 83.23 + 2.290 Televison_Advertising_($1000s) + 1.301 Newspaper_Advertising_($1000s)

c. No, it is 1.60 in part (a) and 2.29 above. In part (b) it represents the marginal change in revenue due

to an increase in television advertising with newspaper advertising held constant. d. Revenue = 83.2 + 2.290(3.5) + 1.301(1.8) = $93.5568 or $93,566.80 6. a. Partial Minitab output follows:

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 4814.3 4814.3 19.11 0.001 Yds/Att 1 4814.3 4814.3 19.11 0.001 Error 14 3527.4 252.0 Lack-of-Fit 13 3037.6 233.7 0.48 0.829 Pure Error 1 489.8 489.8 Total 15 8341.7 Model Summary S R-sq R-sq(adj) R-sq(pred) 15.8732 57.71% 54.69% 44.88% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant -58.8 26.2 -2.25 0.041 Yds/Att 16.39 3.75 4.37 0.001 1.00

Chapter 15

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Regression Equation Win% = -58.8 + 16.39 Yds/Att Fits and Diagnostics for Unusual Observations Std Obs Win% Fit Resid Resid 14 81.30 47.77 33.53 2.19 R R Large residual

b. Partial Minitab output follows:

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 3653 3652.8 10.91 0.005 Int/Att 1 3653 3652.8 10.91 0.005 Error 14 4689 334.9 Lack-of-Fit 11 3536 321.4 0.84 0.644 Pure Error 3 1153 384.4 Total 15 8342 Model Summary S R-sq R-sq(adj) R-sq(pred) 18.3008 43.79% 39.77% 26.48% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 97.5 13.9 7.04 0.000 Int/Att -1600 485 -3.30 0.005 1.00 Regression Equation Win% = 97.5 - 1600 Int/Att Fits and Diagnostics for Unusual Observations Obs Win% Fit Resid Std Resid 8 12.50 55.93 -43.43 -2.45 R R Large residual

c. Partial Minitab out

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