SSTB031 TUTORIAL 5
2022
QUESTION 1
A foam beverage insulator (beer hugger) manufacturer produces their product for firms
that want their logo on beer huggers for marketing purposes. The firm’s cost analyst
wants to estimate their cost function. She interprets as the fixed cost of a production
run, and x1 as the unit variable cost (or marginal cost). Based on five production runs
she observes the following pairs ( X i ; Yi ) where X i is the number of beer huggers
produced in the i th production run (in 1000s), and Yi was the total cost of the run (in R
1000).
(i)
Xi
Yi
3.0
4
7.0
6.5
5.0
5
4.0
7.5
6.0
7.0
Obtain the following matrices and vectors: X, Y, X' X, (X' X) 1 , b, e and
Variance-covariance matrix.
(ii)
Interpret the intercept and the slope of the model in (i) based on the given
statement.
(iii)
Test for the overall model.
(iv)
Based on your findings from (iii) is it necessary to test for the H O : 1  0
using the t-test?
(v)
Suppose we failed to reject the null hypothesis from (iii), what would be the
matrix of X .
(vi)
Find the hat-matrix ( H ) using the formula in Chapter 2 from the study guide.
SSTB031 TUTORIAL 5
2022
QUESTION 2
A company’s motor pool collected the following information concerning cost of
maintenance and repairs of the vehicles during the last year. NB: The age 0 means
the car is one year old.
Vehicle no. Age Mileage
Cost (‘000)
(thousands)
A
3
12
1.2
B
3
36
1.5
C
2
10
0.9
D
4
14
1.5
E
3
16
1.0
F
5
22
4.0
G
2
14
0.8
H
2
10
0.8
I
2
24
1.0
J
5
14
2.0
K
5
18
2.0
L
5
21
3.0
M
5
28
3.2
N
0*
27
0.5
O
5
14
1.9
a) Obtain the following matrices and vectors: X, Y, X' X, (X' X) 1 , b, e Interpret the
meaning of the estimates in part a)
b) Suppose another car was one-year old and was driven 15,000 mile. Estimate
the maintenance and repair costs and interpret the meaning.
c) Test for the overall model.
d) Regardless of the results from c), test for the significance of the individual
predictors using the t-test.
e) Based on your findings from d) what would be your final model?
f) Construct and interpret a 95% confidence intervals for the slopes.