1.
Consider the following data set:
x
0
2
4
6
8
y
110
123
119
86
62
a)
Construct a scatter plot. Does the plot suggest that a linear relationship is
appropriate?
Consider the model
Y
i
=
β
0
+
β
1
x
i
+
e
i
,
i
= 1, 2, 3, 4, 5,
where
e
i
’s are i.i.d.
N
(
0
,
σ
2
)
.
Σ
x
= 20,
Σ
y
= 500,
Σ
x
2
= 120,
Σ
y
2
= 52,630,
Σ
x
y
= 1,734,
Σ
(
x
–
x
)
2
= 40,
Σ
(
y
–
y
)
2
= 2,630,
Σ
(
x
–
x
)
(
y
–
y
)
=
Σ
(
x
–
x
)
y
= –
266.
OR
X
T
X
=
°
±
²
³
´
µ
120
20
20
5
(
X
T
X
)
–
1
=
°
±
²
³
´
µ


025
.
0
1
.
0
1
.
0
6
.
0
X
T
Y
=
°
±
²
³
´
µ
1,734
500
b)
Find the equation of the leastsquares regression line. Add the regression line to
the scatter plot.
Σ
(
y
–
y
ˆ
)
2
= 861.1.
c)
What proportion of the observed variation in
y
values is explained by a straightline
relationship with
x
?
d)
Is regression significant at a 5% level of significance?
e)
Test
H
0
:
β
1
= 0 vs.
H
1
:
β
1
< 0 at a 5% level of significance.
f)
Test
H
0
:
β
0
= 100 vs.
H
1
:
β
0
> 100 at a 5% and at a 10% level of significance.
g)
Construct a 90% prediction interval for the future value of
y
corresponding to
x
= 10.
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Consider the model
Y
i
=
β
0
+
β
1
x
i
+
β
2
x
i
2
+
e
i
,
i
= 1, 2, 3, 4, 5,
where
e
i
’s are i.i.d.
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 Spring '07
 STEPANOV
 Linear Regression, Regression Analysis, 5%, Yi, Scatter plot

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