# Regression Models

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1. The owner of an apartment building in Eurelia believes that her property tax bill is too high because of an overassessment of the property value by the city tax assessor. The owner has hired an independent real-estate appraiser to investigate the appropriateness of the city assessment. The appraiser used regression analysis to explore the relationship between the sale prices of apartment buildings sold in Eurelia and various characteristics of the properties. Twenty-five apartment buildings were selected at random from all the apartment buildings that were sold in a recent year. The variables used were:

price = the sales price in Eurelian dollars

apts = the number of apartments in the building

age = the age of the structure in years

lotsize = the number of square feet in the lot

park = the number of on-site parking places

area = the gross building area in square feet

(See data in the attached file)

QUESTIONS TO BE ANSWERED. Wherever possible answers should be justified using numbers from the printouts.

a) The owner has an 82-year-old building with 5 apartments but no parking spots. The lot is 7500 square feet and the gross building area is 9542 square feet. What sales price is predicted by MODEL I ?

b) State why you know that all three models are significant for predicting the sales price. Since this is the case, why would a statistician prefer MODEL III to MODEL II ?

c) Fill the missing numbers in the analysis-of-variance table for MODEL III.

d) From MODEL II to MODEL III the proportion of explained variation in sales price reduces from 98% to 97.7%. Give precisely the reason you know that this is not a significant reduction in R2.

e) Use MODEL III to answer this part. On the average, what does an extra apartment add to the sales price? Give a 95% confidence interval for the marginal contribution of the 'apts' variable.

Data:

MODEL I

The regression equation is

price = 93074 + 4152 apts - 855 age + 0.92 lotsize + 2692 park + 15.5 area

Predictor Coef StDev T P

Constant 93074 28721 3.24 0.004

apts 4152 1492 2.78 0.012

age -854.9 298.4 -2.86 0.010

lotsize 0.924 2.877 0.32 0.752

park 2692 1577 1.71 0.104

area 15.543 1.463 10.62 0.000

S = 33226 R-Sq = 98.0% R-Sq(adj) = 97.5%

Analysis of Variance

Source DF SS MS F P

Regression 5 1052890000000 210579000000 190.75 0.000

Residual Error 19 20975246806 1103960358

Total 24 1073870000000

MODEL II

The regression equation is

price = 99967 + 4444 apts - 886 age + 2606 park + 15.5 area

Predictor Coef StDev T P

Constant 99967 18661 5.36 0.000

apts 4444 1156 3.84 0.001

age -886.5 275.4 -3.22 0.004

park 2606 1519 1.72 0.102

area 15.487 1.420 10.91 0.000

S = 32472 R-Sq = 98.0% R-Sq(adj) = 97.6%

Analysis of Variance

Source DF SS MS F P

Regression 4 1052780000000 263195000000 249.60 0.000

Residual Error 20 21089200172 1054460009

Total 24 1073870000000

MODEL III

The regression equation is

price = 114369 + 5036 apts - 1057 age + 15.0 area

Predictor Coef StDev T P

Constant 114369 17421 6.56 0.000

apts 5036 1153 4.37 0.000

age -1057.0 268.5 -3.94 0.001

area 14.961 1.449 10.33 0.000

S = 33942 R-Sq = 97.7% R-Sq(adj) = 97.4%

Analysis of Variance

Source DF SS MS F P

Regression * ***** ***** ***** 0.000

Residual Error ** 24193116945 *****

Total ** *****

DATA

price apts age lotsize park area

90300 4 82 4635 0 4266

384000 20 13 17798 0 14391

157500 5 66 5913 0 6615

676200 26 64 7750 6 34144

165000 5 55 5150 0 6120

300000 10 65 12506 0 14552

108750 4 82 7160 0 3040

276538 11 23 5120 0 7881

420000 20 18 11745 20 12600

950000 62 71 21000 3 39448

560000 26 74 11221 0 30000

268000 13 56 7818 13 8088

290000 9 76 4900 0 11315

173200 6 21 5424 6 4461

323650 11 24 11834 8 9000

162500 5 19 5246 5 3828

353500 20 62 11223 2 13680

134400 4 70 5834 0 4680

187000 8 19 9075 0 7392

155700 4 57 5280 0 6030

93600 4 82 6864 0 3840

110000 4 50 4510 0 3092

573200 14 10 11192 0 23704

79300 4 82 7425 0 3876

272000 5 82 7500 0 9542

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