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3
CHAPTER 2
SOLUTIONS TO PROBLEMS
2.2
(i) Let
y
i
=
GPA
i
,
x
i
=
ACT
i
, and
n
= 8.
Then
x
= 25.875,
y
= 3.2125,
1
n
i
=
∑
(
x
i
–
x
)(
y
i
–
y
) =
5.8125, and
1
n
i
=
∑
(
x
i
–
x
)
2
= 56.875.
From equation (2.9), we obtain the slope as
1
ˆ
β
=
5.8125/56.875
≈
.1022, rounded to four places after the decimal.
From (2.17),
0
ˆ
=
y
–
1
ˆ
x
≈
3.2125 – (.1022)25.875
≈
.5681.
So we can write
n
GPA
=
.5681 + .1022
ACT
n
= 8.
The intercept does not have a useful interpretation because
ACT
is not close to zero for the
population of interest.
If
ACT
is 5 points higher,
n
GPA
increases by .1022(5) = .511.
(ii) The fitted values and residuals — rounded to four decimal places — are given along with
the observation number
i
and
GPA
in the following table:
i GPA
n
GPA
ˆ
u
1
2.8
2.7143
.0857
2
3.4
3.0209
.3791
3
3.0
3.2253
–.2253
4
3.5
3.3275
.1725
5
3.6
3.5319
.0681
6
3.0
3.1231
–.1231
7
2.7
3.1231
–.4231
8
3.7
3.6341
.0659
You can verify that the residuals, as reported in the table, sum to
−
.0002, which is pretty close to
zero given the inherent rounding error.
(iii) When
ACT
= 20,
n
GPA
= .5681 + .1022(20)
≈
2.61.
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4
(iv) The sum of squared residuals,
2
1
ˆ
n
i
i
u
=
∑
, is about .4347 (rounded to four decimal places),
and the total sum of squares,
1
n
i
=
∑
(
y
i
–
y
)
2
, is about 1.0288.
So the
R
squared from the
regression is
R
2
=
1 – SSR/SST
≈
1 – (.4347/1.0288)
≈
.577.
Therefore, about 57.7% of the variation in
GPA
is explained by
ACT
in this small sample of
students.
2.3
(i) Income, age, and family background (such as number of siblings) are just a few
possibilities.
It seems that each of these could be correlated with years of education.
(Income
and education are probably positively correlated; age and education may be negatively correlated
because women in more recent cohorts have, on average, more education; and number of siblings
and education are probably negatively correlated.)
(ii) Not if the factors we listed in part (i) are correlated with
educ
.
Because we would like to
hold these factors fixed, they are part of the error term.
But if
u
is correlated with
educ
then
E(
ueduc
)
≠
0, and so SLR.4 fails.
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 Spring '11
 Suzuki
 Econometrics, Regression Analysis, Household income in the United States, prior consent, U.S. Edition

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