QUESTION 1

1. Please look at the following output from the regression. 

   What does the value of R-square tell us about our model? 

   Note: It is not sufficient to just provide some general answers. Use the numbers from the output, and write your answers specific to our regression model.

   Model Summary

   Model

   R

   R Square

   Adjusted R Square

   Std. Error of the Estimate

   1

   .771a

   .594

   .580

   21.741

   a. Predictors: (Constant), DENSITY

        — Font family –Andale MonoArialArial BlackBook AntiquaComic Sans MSCourier NewGeorgiaHelveticaImpactSymbolTahomaTerminalTimes New RomanTrebuchet MSVerdanaWebdingsWingdings — Font size –1 (8pt)2 (10pt)3 (12pt)4 (14pt)5 (18pt)6 (24pt)7 (36pt)                      — Format –HeadingSub Heading 1Sub Heading 2ParagraphFormatted Code — Font family — — Font size —                                                                              

   Path: p

   Words:0

12.5 points   

QUESTION 2

1. Please look at the following output from the regression. 

   What do the value of F-test and its P-value tell us about our model? 

   Note: It is not sufficient to just provide some general answers. Use the numbers from the output, and write your answers specific to our regression model.

   ANOVAa

   Model

   Sum of Squares

   df

   Mean Square

   F

   Sig.

   1

   Regression

   20747.246

   1

   20747.246

   43.895

   .000b

   Residual

   14179.629

   30

   472.654

   Total

   34926.875

   31

   a. Dependent Variable: SALES

   b. Predictors: (Constant), DENSITY

        — Font family –Andale MonoArialArial BlackBook AntiquaComic Sans MSCourier NewGeorgiaHelveticaImpactSymbolTahomaTerminalTimes New RomanTrebuchet MSVerdanaWebdingsWingdings — Font size –1 (8pt)2 (10pt)3 (12pt)4 (14pt)5 (18pt)6 (24pt)7 (36pt)                      — Format –HeadingSub Heading 1Sub Heading 2ParagraphFormatted Code — Font family — — Font size —                                                                              

   Path: p

   Words:0

12.5 points   

QUESTION 3

1. The regression coefficient output is shown below. 

   Does Density matter in terms of explaining sales? Can you provide an explanation of the coefficient estimate for Density? (Note: the unit of Density is number of homes per acre, and the unit of Sales is dollars per thousand homes ). 

   Coefficientsa

   Model

   Unstandardized Coefficients

   Standardized Coefficients

   t

   Sig.

   B

   Std. Error

   Beta

   1

   (Constant)

   141.525

   9.109

   15.538

   .000

   DENSITY

   -12.893

   1.946

   -.771

   -6.625

   .000

   a. Dependent Variable: SALES

   For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac).     — Font family –Andale MonoArialArial BlackBook AntiquaComic Sans MSCourier NewGeorgiaHelveticaImpactSymbolTahomaTerminalTimes New RomanTrebuchet MSVerdanaWebdingsWingdings — Font size –1 (8pt)2 (10pt)3 (12pt)4 (14pt)5 (18pt)6 (24pt)7 (36pt)                      — Format –HeadingSub Heading 1Sub Heading 2ParagraphFormatted Code — Font family — — Font size —                                                                              

   5

   Path: p

   Words:28

12.5 points   

QUESTION 4

1. Managers suspect that the effect of Density on Sales can be nonlinear; in other words, as density increases, there will a decreasing marginal effect on density. To test this idea, they ran an additional regression, with Density and Density_Squared (i.e. Density*Density) as the independent variables (again, Sales as the dependant variable), and the output of the new regression shows below.

   Can you explain what the R_square and F-test tell us about the new model? Is the new model better than the model with only Density as the independent variable?

   Model Summary

   Model

   R

   R Square

   Adjusted R Square

   Std. Error of the Estimate

   1

   .910a

   .829

   .817

   14.354

   a. Predictors: (Constant), Density2, DENSITY

   ANOVAa

   Model

   Sum of Squares

   df

   Mean Square

   F

   Sig.

   1

   Regression

   28951.384

   2

   14475.692

   70.253

   .000b

   Residual

   5975.491

   29

   206.051

   Total

   34926.875

   31

   a. Dependent Variable: SALES

   b. Predictors: (Constant), Density2, DENSITY

   Coefficientsa

   Model

   Unstandardized Coefficients

   Standardized Coefficients

   t

   Sig.

   B

   Std. Error

   Beta

   1

   (Constant)

   212.595

   12.768

   16.650

   .000

   DENSITY

   -47.293

   5.601

   -2.827

   -8.444

   .000

   Density2

   3.419

   .542

   2.113

   6.310

   .000

   a. Dependent Variable: SALES

        — Font family –Andale MonoArialArial BlackBook AntiquaComic Sans MSCourier NewGeorgiaHelveticaImpactSymbolTahomaTerminalTimes New RomanTrebuchet MSVerdanaWebdingsWingdings — Font size –1 (8pt)2 (10pt)3 (12pt)4 (14pt)5 (18pt)6 (24pt)7 (36pt)                      — Format –HeadingSub Heading 1Sub Heading 2ParagraphFormatted Code — Font family — — Font size —                                                                              

   Path: p

   Words:0

12.5 points   

QUESTION 5

1. (This is a Bonus Question)

   Continuing from Question 4, can you explain the meaning of the coefficient estimate of Density_Squared (i.e. Density*Density)? (Hint: the effect of Density on Sales is negative, while the effect of Density_Squared on Sales is positive).