Assignment 2 - STAT 331/361/SYDE 334 Assignment 2 Due...

STAT 331/361/SYDE 334 Assignment 2
Due: Thursday, February 14, 2008 in class
Note: Question 5 requires using R. Computing outputs need to be integrated into your answers at
appropriate places. Stacking your computing stuff all together and putting them at the back is
NOT acceptable!
1. If y and , then the sample correlation coefficient between and ) ,..., (
1 n
y y = ) ,..., (
1 n
x x x = y x is
defined as ) /(
xx yy xy
s s ) , ( x y r s = . Let ) ,..., (
1 n
e e e = be the residuals from fitting the simple
linear regression model
i i
x
i
y ε β β + + =
1 0
using the least square method. Show that 0 ) , ( = x e r .
2. Suppose that the simple linear regression model with the assumptions given in Section 2.1.1 holds.
Show that
( )
2
1
2
) 2 ( σ − =
∑
=
n e E
n
i
i
. ( Hint : First show
xx xy yy
n
i
i
s s s e /
2
1
2
− =
∑
=
xx
s
, then show
and ).
xx
s n
2
1
2
) 1 β σ + − E
yy
s E ( ) ( =
xx xy
s s
2 2 2
1
2
) ( σ β + =
3. Consider the simple linear regression model , ,..., 1 ,
1 0
n i x y
i i i
= + + = ε β β where
and
2
) ( , 0 ) ( σ ε ε = =
i i
V E
n
ε ε ε , ... , ,
2 1
are independent.