This problem extends the home prices example used previously to 76 homes (Sec tion 6.1 contains a complete case study of these data). We wish to model the relationship between the price of a single-family home (Y, in $ thousands) and the following predictors:
X1 = floor size (thousands of square feet)
X 2 = lot size category (from l to I I-see page 78)
x 3 = number of bathrooms (with half-bathrooms counting as “0.1”)
Xi = number of bedrooms (between 2 and 6)
Xs = age (standardized: (year built – 1970)/ l 0)
X6 = garage size (0, l ,2, or 3 cars)
D 1 = indicator for “active listing” (reference: pending or sold)
D 8 = indicator for Edison Elementary (reference: Edgewood Elementary)
D 9 = indicator for Harris Elementary (reference: Edgewood Elementary)
Consider the following model, which includes an interaction between X3 and Xi:
E(Y ) = bo+b1X1 + biX2 +b3X3 +b4XJ +bsX3XJ
+ b s +h? X f +b s +b9D1 + b1oDs +b 1 1 D9 .
The regression results for this model are:
Predictor variable
Parameter estimate
1vo tail p-value
X1
56.72
0.05
X2
9.92
O.Ql
X3
-98.16
0.02
Xi
-78.91
0.01
X3XJ
30.39
0.01
Xs
3.30
0.30 x2
5
l.64
0.03
X6
13.12
0.12
D1
27.42
0.02
Ds
67.06
o.oo
D9
47.27
0.00
Test whether the relationship between home price (Y) and number of bathrooms (X3) depends on number of bedrooms (Xi), all else equal (significance level 5% ).
(b) Does the association between number of bathrooms and home price vary with number of bedrooms? We can investigate this by isolating the parts of the model involving just X3: the “X3 effect” is given by b3X3 + b5X3XJ = (bJ + bsXi )X3. For example, when Xi =2, this effect is estimated to be (-98.16+30.39(2))X3 =
-37.38X3. Thus, for two-bedroom homes, there is a negative relationship between number of bathrooms and home price (for each additional bathroom, price drops by $37,380, all else being equal-perhaps adding extra bathrooms to two-bedroom homes is considered a waste of space and so has a negative impact on price). Use a similar calculation to show the relationship between number of bathrooms and home price for three-bedroom homes, and also for four-bedroom homes.
Hint: Understanding the example on page 142 will help you solve this part.
