STAT 420
Spring 2010
Homework #9
(due Friday, April 9, by 4:00 p.m.)
1.
Recall
Problem
5.6
:
In order to determine the profitability of hiring an additional salesman, the sales manager
decided to determine the relationship between the number of salesmen and the average
sales in thousands of dollars.
After looking at a scatter diagram of the data
(
obtained
over several years
),
X
(
number of salesmen
)
2
3
4
5
6
Y
(
sales in $1000’s
)
20
27
33
38
43
he postulates the model
Y
i
=
α
+
β
x
i
+
ε
i
,
i
= 1, 2, … , 5,
where the
ε
i
’s are uncorrelated normal variables with mean zero and variance
σ
2
.
After examining the residuals vs. fitted values plot
(
Homework 5
) it was concluded that
linear model does NOT seem to be appropriate here.
Consider the second-order model
Y
i
=
β
0
+
β
1
x
i
+
β
2
x
i
2
+
ε
i
,
i
= 1, 2, … , 5,
where the
ε
i
’s are uncorrelated normal variables with mean zero and variance
σ
2
.
a)
Obtain the least squares estimates
0
β
ˆ
,
1
β
ˆ
, and
2
β
ˆ
.
> x = c( 2, 3, 4, 5, 6)
> y = c(20,27,33,38,43)
> fit2 = lm(y ~ x + I(x^2))
> summary(fit2)
Call:
lm(formula = y ~ x + I(x^2))
Residuals:
1
2
3
4
5
-0.08571
0.14286
0.08571 -0.25714
0.11429