This preview shows pages 1–3. Sign up to view the full content.
STAT 420
Examples #3
Spring 2008
Y
i
=
β
0
+
β
1
x
i
1
+ … +
β
p
x
i
p
–1
+
e
i
,
i
= 1, 2, … ,
n
,
E
(
e
i
)
= 0, Var
(
e
i
)
=
σ
2
, Cov
(
e
i
,
e
j
)
= 0,
i
≠
j
.
Y
=
X
β
+
e
,
E
(
e
)
=
0
, Var
(
e
)
=
((
Cov
(
e
i
,
e
j
)
))
i
j
=
σ
2
I
n
.
Y
=
±
²
³
³
³
³
³
´
µ
n
Y
...
Y
Y
2
1
,
X
=
±
²
³
³
³
³
³
´
µ



1
1
1
...
1
...
...
...
...
...
...
1
...
1
2
1
2
2
2
1
2
1
2
1
1
1
p
n
n
n
p
p
x
x
x
x
x
x
x
x
x
,
β
=
±
²
³
³
³
³
³
´
µ

1
...
1
0
p
,
e
=
±
²
³
³
³
³
³
´
µ
n
e
e
e
...
2
1
.
ˆ
=
(
X
T
X
)
–
1
X
T
Y
, E
(
ˆ
)
=
β
, Var
(
ˆ
)
=
σ
2
(
X
T
X
)
–
1
.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
The (normal) simple linear regression model:
y
i
=
β
0
+
β
1
x
i
+
ε
i
,
where
ε
i
’s are independent Normal
(
0
,
σ
2
) (
iid Normal
(
0
,
σ
2
)
).
β
0
,
β
1
, and
σ
2
are unknown model parameters.
Suppose
x
i
’s are fixed (not random).
¶
Y
i
’s are independent Normal
(
β
0
+
β
1
x
i
,
σ
2
) random variables.
1
ˆ
=
( )
( )
·
·


2
Y
x
x
x
x
i
i
i
~ N
( )
¸
¸
¸
¹
º
»
»
»
¼
½

·
2
2
1
±
,
x
x
i
0
ˆ
=
x
ˆ
Y
1

~ N
( )
¸
¸
¸
¹
º
»
»
»
¼
½

·
·
2
2
2
0
±
,
x
x
n
x
i
i
= N
( )
¸
¸
¸
¹
º
»
»
»
¼
½
¸
¸
¸
¹
º
»
»
»
¼
½

+
·
1
,
2
2
2
0
±
x
x
x
n
i
( )
2
1
0
2
ˆ
ˆ
Y
2
1
S
i
i
e
x
n
·



=
( )
2
2
±
S
2
e
n

~
χ
2
(
n
– 2
)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 1.
The owner of
Momma Leona’s Pizza
restaurant chain believes that if a restaurant
is located near a college campus, then there is a linear relationship between sales
and the size of the student population. Suppose data were collected from a
sample of 10
Momma Leona’s Pizza
restaurants located near college campuses.
For the
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 02/10/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 STEPANOV
 Statistics

Click to edit the document details