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.