Question

11.29) A builder wants to predict the relationship between house size (as measured by number of rooms). and selling price. Two difference neighbourhoods are compare. A random sample of 215 houses is selected, with the results as follows.

House	Number of rooms	Selling Price ($000)	Neighbourhood
1	8	345	0
2	8	360	1
3	6	325	0
4	8	400	1
5	7	350	1
6	9	360	0
7	13	405	0
8	6	299	0
9	8	405	1
10	9	365	0
11	10	520	1
12	8	330	0
13	14	600	1
14	9	370	0
15	7	395	1

(a) Is there a relationship between selling price and two independent variables at the 0.05 level of significance?
(b) Does the independent variable make a contribution to the regression model?
(c) Add an interaction term to the model and at the 0.05 level of significance determine whether it makes a significant contribution to the model.0

Answer 1

Solution:

(a) Is there a relationship between selling price and two independent variables at the 0.05 level of significance?

Accessing relationship between Selling price and Number of rooms :

r = correlation coefficient = 0.7679632

critical correlation coefficient at the 0.05 level of significance (n = 15) = 0.514

Since r > 0.514, there is significant linear relation ship between Selling price and Number of rooms

Accessing relationship between Selling price and neighbourhood :

r = correlation coefficient = 0.55288

critical correlation coefficient at the 0.05 level of significance (n = 15) = 0.514

Since r > 0.514, there is significant linear relation ship between Selling price and neighbourhood

b ) Does the independent variable make a contribution to the regression model?

Fro the regression model

selling_price ~ number_of_rooms + neighbourhood

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 137.504     36.754 3.741 0.00282 **

number_of_rooms 24.985      4.081 6.122 5.16e-05 ***

neighbourhood 74.059 17.839 4.152 0.00134 **

The p value of the coefficients of the independent variables in the regression model is < 0.05

Therefore independent variable make a significant contribution to the regression model.

(c) Add an interaction term to the model and at the 0.05 level of significance determine whether it makes a significant contribution to the model.

After adding the interaction term in the model , the regression equation becomes

selling_price ~ number_of_rooms + neighbourhood + neighbourhood * number_of_rooms

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 231.750     35.398   6.547 4.15e-05 ***

number_of_rooms 13.897      4.047   3.434 0.00558 **

neighbourhood -110.781     50.178 -2.208 0.04941 *

number_of_rooms:neighbourhood 21.316      5.611   3.799 0.00295 **

The p value of the coefficient of the interaction variable (number_of_rooms*neighbourhood ) is 0.00295 << 0.05 significance level, Therefore this interaction term makes a significant contribution to the model