A1.pdf - Stat 331 Applied Linear Models – Assignment 1 Due...

Stat 331 Applied Linear Models – Assignment 1
Due on May 29th (Friday) 11 pm to Crowdmark. If a question involves R, use R Markdown to
write your solution and show your R codes and outputs.
1. Under the simple linear regression model, let ˆ y
i
=
ˆ
β
0
+
ˆ
β
1
x
i
denote the fitted value, and
r
i
= y
i
- ˆ y
i
denote the corresponding residual. We want to show that
ˆ σ
2
=
∑
n
i =1
r
2
i
n - 2
is an unbiased estimator of σ
2
. Prove the following
(a) Show that
∑
n
i =1
r
2
i
= s
yy
-
s
2
xy
s
xx
where s
yy
=
∑
n
i =1
( y
i
- ¯ y )
2
and s
xy
=
∑
n
i =1
( x
i
- ¯ x )( y
i
- ¯ y ) .
(b) Show that E ( s
yy
) = ( n - 1) σ
2
+ β
2
1
s
xx
and E ( s
2
xy
) = β
2
1
s
2
xx
+ σ
2
s
xx
.
(c) Show that E
(∑
n
i =1
r
2
i
)
= ( n - 2) σ
2
.
2. We look at weekly reports on the box office ticket sales for plays on Broadway in New York.
We shall consider the data for a particular week (referred to below as the current week). The
data are in the form of the gross box office results for the current week and the gross box
office results for the previous week. The data are available on the Learn website in the file
a1q2.csv.