Question

Problem 3. A heating and plumbing contractor estimates material and labor costs for new homes (1) based on the total area of the house (x) and whether central air conditioning will be installed (x2). a) Write a main effects model relating the mean cost of material and labor (E(y)) to the area and central air conditioning variables. We will refer to this model as the reduced model. b) Write a complete second-order model for E(y) as a function of the same variables. c) What are the null and the alternative hypotheses that you would test in order to determine whether the second-order terms are useful for predicting the mean cost? d) Refer to part c. The contractor samples 100 recent jobs and fits both the complete second-order model (part b) and the reduced main effects model (part a), so that a test can be conducted to determine whether the additional complexity of the second-order model is necessary. The resulting SSE and R values are shown in the following table. Is there sufficient evidence to conclude that the second-order terms are important for predicting the mean cost? Use 0.05. SSE Main effects 8.548 0.950 Second-order 6.133 0.964

Answer 1

Given,

y=material and labor costs for new homes

x1= total area of the house (x1) and

x2= whether central air conditioning will be installed (binary 1-0 variable)

a) equation for main effect only:

E(y)= b0+b1*x1 +b2*x2

b) second-order model for E(y):

E(y) =b0+b1*x1 +b2*x2 + b3*x1*x3 (the last term stands for intercation effect)

c) We use F-test to test for significance difference between SSR (sum of sq due to regression) of both the models.

H0: there is no significant difference

H: there is significant difference

d) Given, n=25

We form the following F-test:

	SSE	R2	SSR	F-stat	P-value
Main effect	8.548	0.95	162.412	1.011182	0.489021
Second-order	6.133	0.964	164.2281

Since p-value>0.05, we cant reject H0. Thus, we conclude that there is no significance increase in predictive power of the model. Thus dont use interaction effect.

Excel formula:

	SSE	R2	SSR	F-stat	P-value
Main effect	8.548	0.95	=B6*C6/(1-C6)	=D7/D6	=1-F.DIST(E6,25,25,TRUE)
Second-order	6.133	0.964	=B7*C7/(1-C7)