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Chapter 12: Q95E (page 767)

Recently sold, single-family homes. The National Association of Realtors maintains a database consisting of sales information on homes sold in the United States. The next table lists the sale prices for a sample of 28 recently sold, single-family homes. The table also identifies the region of the country in which the home is located and the total number of homes sold in the region during the month the home sold.

a) Propose a complete second-order model for the sale price of a single-family home as a function of region and sales volume.

b) Give the equation of the curve relating sale price to sales volume for homes sold in the West.

c) Repeat part b for homes sold in the Northwest.

d) Which b’s in the model, part a, allow for differences among the mean sale prices for homes in the four regions?

e) Fit the model, part a, to the data using an available statistical software package. Is the model statistically useful for predicting sale price? Test using α = .01.

Short Answer

Expert verified

a) A complete second-order model for the sale price of a single-family home as a function of region and sales volume can be written as

b) The equation of the curve relating sale price to sales volume for homes sold in the West can be written when X2= 0, X3 = 0 and X4= 0.The equation becomes.

c) The equation of the curve relating sale price to sales volume for homes sold in the NorthWest can be written when X2= 0, X3 = 1 and X4= 0.The equation becomes.

d) β2, β3, and β4 represent the differences among the mean sale prices for homes in the four regions. β2 represents difference in mean sale price of NE region and west region, β3 represents difference in mean sale price of NW region and west region, and β4represents difference in mean sale price of south region and west region.

e) It can be concluded with 99% confidence interval that the model is not statistically useful for predicting sale price.

Step by step solution

01

Second order model

A complete second-order model for the sale price of a single-family home as a function of region and sales volume can be written as

Where, x1= sales volume

X2= 1 if region is NE; 0 otherwise

X3= 1 if region is NW; 0 otherwise

X4= 1 if region is S; 0 otherwise

02

Subsequent order model equation

The equation of the curve relating sale price to sales volume for homes sold in the

West can be written when X2= 0, X3 = 0 and X4= 0

03

Next order model equation 

The equation of the curve relating sale price to sales volume for homes sold in the

Northwest can be written when X2= 0, X3 = 1 and X4= 0

04

Interpretation of β

β2, β3, and β4 represent the differences among the mean sale prices for homes in the four regions. Β2 represents difference in mean sale price of NE region and west region, β3 represents difference in mean sale price of NW region and west region, and β4represents difference in mean sale price of south region and west region.

05

Model fitted to “part a” equation

The excel output is presented below

SUMMARY OUTPUT

















Regression Statistics








Multiple R

0.921704








R Square

0.849538








Adjusted R Square

0.746095








Standard Error

24365.83








Observations

28

















ANOVA









df

SS

MS

F

Significance F




Regression

11

5.36E+10

4.88E+09

8.212628

0.000112




Residual

16

9.5E+09

5.94E+08






Total

27

6.31E+10













Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

5568060

4015347

1.386695

0.184553

-2944095

14080214

-2944095

14080214

Sales Volume(x1)

-107.648

73.57124

-1.46319

0.162784

-263.613

48.31561

-263.613

48.31561

x2

-3663319

4478880

-0.81791

0.425422

-1.3E+07

5831482

-1.3E+07

5831482

x3

-3503659

4058018

-0.86339

0.40068

-1.2E+07

5098955

-1.2E+07

5098955

x4

1628588

5945303

0.273929

0.787644

-1.1E+07

14232068

-1.1E+07

14232068

x1^2

0.00054

0.000337

1.604055

0.128257

-0.00017

0.001254

-0.00017

0.001254

x1*x2

37.21225

103.005

0.361266

0.722626

-181.149

255.5732

-181.149

255.5732

x1*x3

59.45824

75.18596

0.790816

0.440617

-99.9289

218.8454

-99.9289

218.8454

x1*x4

13.35528

93.25644

0.14321

0.887912

-184.34

211.0501

-184.34

211.0501

x1^2*x2

0.000181

0.000733

0.246847

0.808166

-0.00137

0.001736

-0.00137

0.001736

x1^2*x3

-0.00024

0.000351

-0.68403

0.503745

-0.00098

0.000504

-0.00098

0.000504

x1^2*x4

-0.00022

0.000385

-0.58005

0.56996

-0.00104

0.000593

-0.00104

0.000593

To test the significance of the model F-test is conducted. The null and alternate hypothesis would be

Therefore, the model is not statistically useful for predicting sale price.

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