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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鈥檚 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 伪 = .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) 尾2, 尾3, and 尾4 represent the differences among the mean sale prices for homes in the four regions. 尾2 represents difference in mean sale price of NE region and west region, 尾3 represents difference in mean sale price of NW region and west region, and 尾4represents 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

尾2, 尾3, and 尾4 represent the differences among the mean sale prices for homes in the four regions. 螔2 represents difference in mean sale price of NE region and west region, 尾3 represents difference in mean sale price of NW region and west region, and 尾4represents 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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Most popular questions from this chapter

Question: Reality TV and cosmetic surgery. Refer to the Body Image: An International Journal of Research (March 2010) study of the impact of reality TV shows on a college student鈥檚 decision to undergo cosmetic surgery, Exercise 12.43 (p. 739). The data saved in the file were used to fit the interaction model, E(Y)=尾0+尾1x1+尾2x4+尾3x1x4, where y = desire to have cosmetic surgery (25-point scale),x1= {1 if male, 0 if female}, and x4= impression of reality TV (7-point scale). From the SPSS printout (p. 739), the estimated equation is:y^=11.78-1.97x1+0.58x4-0.55x1x4

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b. Repeat part a for male students.

Question: Failure times of silicon wafer microchips. Researchers at National Semiconductor experimented with tin-lead solder bumps used to manufacture silicon wafer integrated circuit chips (International Wafer-Level Packaging Conference, November 3鈥�4, 2005). The failure times of the microchips (in hours) were determined at different solder temperatures (degrees Celsius). The data for one experiment are given in the table. The researchers want to predict failure time (y) based on solder temperature (x).

  1. Construct a scatterplot for the data. What type of relationship, linear or curvilinear, appears to exist between failure time and solder temperature?
  2. Fit the model,E(y)=尾0+尾1x+尾2x2, to the data. Give the least-squares prediction equation.
  3. Conduct a test to determine if there is upward curvature in the relationship between failure time and solder temperature. (use.伪=0.05)

Write a model relating E(y) to one qualitative independent variable that is at four levels. Define all the terms in your model.

State casket sales restrictions. Some states permit only licensed firms to sell funeral goods (e.g., caskets, urns) to the consumer, while other states have no restrictions. States with casket sales restrictions are being challenged in court to lift these monopolistic restrictions. A paper in the Journal of Law and Economics (February 2008) used multiple regression to investigate the impact of lifting casket sales restrictions on the cost of a funeral. Data collected for a sample of 1,437 funerals were used to fit the model. A simpler version of the model estimated by the researchers is E(y)=尾0+尾1x1+尾2x2+尾3x1x2, where y is the price (in dollars) of a direct burial, x1 = {1 if funeral home is in a restricted state, 0 if not}, and x2 = {1 if price includes a basic wooden casket, 0 if no casket}. The estimated equation (with standard errors in parentheses) is:

y^=1432 + 793x1- 252x2+ 261x1x2, R2= 0.78

(70) (134) (109)

  1. Calculate the predicted price of a direct burial with a basic wooden casket at a funeral home in a restricted state.

  2. The data include a direct burial funeral with a basic wooden casket at a funeral home in a restricted state that costs \(2,200. Assuming the standard deviation of the model is \)50, is this data value an outlier?

  3. The data also include a direct burial funeral with a basic wooden casket at a funeral home in a restricted state that costs \(2,500. Again, assume that the standard deviation of the model is \)50. Is this data value an outlier?

Question: Failure times of silicon wafer microchips. Refer to the National Semiconductor study of manufactured silicon wafer integrated circuit chips, Exercise 12.63 (p. 749). Recall that the failure times of the microchips (in hours) was determined at different solder temperatures (degrees Celsius). The data are repeated in the table below.

  1. Fit the straight-line modelEy=尾0+尾1xto the data, where y = failure time and x = solder temperature.

  2. Compute the residual for a microchip manufactured at a temperature of 149掳C.

  3. Plot the residuals against solder temperature (x). Do you detect a trend?

  4. In Exercise 12.63c, you determined that failure time (y) and solder temperature (x) were curvilinearly related. Does the residual plot, part c, support this conclusion?
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