STA 6208 – Spring 2004 – Exam 1
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UFID:
All questions are based on the following two regression models, where SIMPLE REGRESSION
refers to the case where
p
=1
, and X is of full column rank (no linear dependencies among the
predictor variables)
Model 1:
Y
i
=
β
0
+
β
1
X
i
1
+
···
+
β
p
X
ip
+
ε
i
i
,...,n ε
i
∼
NID
(0
,σ
2
)
Model 2: Y
=
X
β
+
ε
X
≡
n
×
p
±
β
≡
p
±
×
1
ε
∼
N
(
0
2
I
)
Cochran’s Theorem
Suppose
Y
is distributed as follows with nonsingular matrix
V
:
Y
∼
N
(
μ,
V
σ
2
)
r
(V) =
n
then:
1.
Y
±
(
1
σ
2
A
)
Y
is distributed noncentral
χ
2
with:
(a) Degrees of freedom =
r
(
A
)
(b) Noncentrality parameter = Ω =
1
2
σ
2
μ
±
A
μ
if
AV
is idempotent
2.
Y
±
AY
and
Y
±
BY
are independent if
AVB
=
0
3.
Y
±
and linear function
BY
are independent if
BVA
=
0
1) Based on
Model 1
,
derive
the normal equations for the simple linear regression model.
2) Show that
∑
n
i
=1
(
ˆ
Y
i

Y
) = 0. You may do this based on either
Model 1
or
Model 2
.
3) A simple linear regression is Ft, relating Frst weekend revenues (
Y
) to advertising expenditures (
X
) for
n
= 5 randomly selected horror Flms:
±ilm
i
Sales
Ad Exp
Scarier Movie
1
25.0
8.0
I Know What You Did Last Winter
2
15.0
6.0
Rural Legend
3
12.0
4.0
Shout
4
30.0
10.0
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This note was uploaded on 07/28/2011 for the course STA 6208 taught by Professor Park during the Fall '08 term at University of Florida.
 Fall '08
 Park
 Regression Analysis

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