t1solf06 - STA 302 1001 H1F Fall 2006 Test 1 LAST NAME...

STA 302 / 1001 H1F – Fall 2006
Test 1
October 18, 2006
LAST NAME: SOLUTIONS FIRST NAME:
STUDENT NUMBER:
ENROLLED IN: (circle one) STA 302 STA 1001
INSTRUCTIONS:
• Time: 90 minutes
• Aids allowed: calculator.
• A table of values from the t distribution is on the last page (page 7).
• Total points: 50
Some formulae:
b
1
=
∑
( X
i
- X )( Y
i
- Y )
∑
( X
i
- X ) 2
=
∑
X
i
Y
i
- n X Y
∑
X
2
i
- n X
2
b
0
= Y - b
1
X
Var( b
1
) =
σ
2
∑
( X
i
- X ) 2
Var( b
0
) = σ
2 1
n
+
X
2
∑
( X
i
- X ) 2
Cov( b
0
, b
1
) = -
σ
2
X
∑
( X
i
- X ) 2
SSTO =
∑
( Y
i
- Y )
2
SSE =
∑
( Y
i
-
ˆ
Y
i
)
2
SSR = b
2
1
∑
( X
i
- X )
2
=
∑
(
ˆ
Y
i
- Y )
2
σ
2
{
ˆ
Y
h
} = Var(
ˆ
Y
h
) = σ
2 1
n
+
( X
h
- X )
2
∑
( X
i
- X )
2
σ
2
{ pred } = Var( Y
h
-
ˆ
Y
h
) = σ
2
1 +
1
n
+
( X
h
- X )
2
∑
( X
i
- X )
2
r =
∑
( X
i
- X )( Y
i
- Y )
∑
( X
i
- X )
2
∑
( Y
i
- Y )
2
1 2ab 2cdef 2gh 3
1

1. A simple linear regression model
Y
i
= β
0
+ β
1
X
i
+
i
is fit using least squares to n data points. Assume that the Gauss-Markov conditions hold
and that the error terms are normally distributed with mean 0 and variance σ
2
.
(a) (3 marks) What is the probability distribution of b
1
? What is the probability distribu-
tion of β
1
?
b
1
∼ N β
1
,
σ
2
∑
( X
i
- X ) 2
(2 marks)
β
1
is not random (1 mark)
(b) (4 marks) Describe the method of least squares. How is it related to R
2
?
Find the slope and intercept of the line that minimizes the sum of squares of the vertical
deviations of the data points from the line. (2 marks)
The quantity that is being minimized as a proportion of variation in the Y ’s is 1 - R
2
.
(2 marks)
(c) (3 marks) Suppose the regression model is being used to predict blood pressure as a
function of weight. Explain the difference between a confidence interval for the mean
response at a new X and a prediction interval at a new X in this context. (Do not
discuss the details of the formulae for calculating the intervals.)
For a particular weight, there is a probability distribution for the possible values of blood
pressure. The confidence interval for the mean of Y at that weight gives an interval
that captures the mean of this probability distribution ( β
0
+ β
1
X ) 100(1 - α ) % of the
time. The prediction interval gives an interval that captures the actual value of the blood