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34
CHAPTER 7
SOLUTIONS TO PROBLEMS
7.1
(i) The coefficient on
male
is 87.75, so a man is estimated to sleep almost one and one-half
hours more per week than a comparable woman.
Further,
t
male
= 87.75/34.33
≈
2.56, which is
close to the 1% critical value against a two-sided alternative (about 2.58).
Thus, the evidence for
a gender differential is fairly strong.
(ii) The
t
statistic on
totwrk
is
−
.163/.018
≈
−
9.06, which is very statistically significant.
The
coefficient implies that one more hour of work (60 minutes) is associated with .163(60)
≈
9.8
minutes less sleep.
(iii) To obtain
2
r
R
, the
R
-squared from the restricted regression, we need to estimate the
model without
age
and
age
2
.
When
age
and
age
2
are both in the model,
age
has no effect only if
the parameters on both terms are zero.
7.3
(i) The
t
statistic on
hsize
2
is over four in absolute value, so there is very strong evidence that
it belongs in the equation. We obtain this by finding the turnaround point;
this is the value of
hsize
that maximizes
ˆ
sat
(other things fixed):
19.3/(2
⋅
2.19)
≈
4.41.
Because
hsize
is measured
in hundreds, the optimal size of graduating class is about 441.
(ii) This is given by the coefficient on
female
(since
black
= 0):
nonblack females have SAT
scores about 45 points lower than nonblack males.
The
t
statistic is about –10.51, so the
difference is very statistically significant.
(The very large sample size certainly contributes to
the statistical significance.)
(iii) Because
female
= 0, the coefficient on
black
implies that a black male has an estimated
SAT score almost 170 points less than a comparable nonblack male.
The
t
statistic is over 13 in
absolute value, so we easily reject the hypothesis that there is no ceteris paribus difference.
(iv) We plug in
black
= 1,
female
= 1 for black females and
black
= 0 and
female
= 1 for
nonblack females.
The difference is therefore –169.81 + 62.31 =
−
107.50.
Because the estimate
depends on two coefficients, we cannot construct a
t
statistic from the information given.
The
easiest approach is to define dummy variables for three of the four race/gender categories and
choose nonblack females as the base group.
We can then obtain the
t
statistic we want as the
coefficient on the black female dummy variable.
7.5
(i) Following the hint,
n
colGPA
=
0
ˆ
β
+
0
ˆ
δ
(1 –
noPC
) +
1
ˆ
β
hsGPA
+
2
ˆ
β
ACT
= (
0
ˆ
β
+
0
ˆ
δ
)
−
0
ˆ
δ
noPC
+
1
ˆ
β
hsGPA
+
2
ˆ
β
ACT
.
For the specific estimates in equation (7.6),
0
ˆ
β
= 1.26 and
0
ˆ
δ
=
.157, so the new intercept is 1.26 + .157 = 1.417.
The coefficient on
noPC
is –.157.

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