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27
CHAPTER
6
SOLUTIONS TO PROBLEMS
6.1
This would make little sense.
Performances on math and science exams are measures of
outputs of the educational process, and we would like to know how various educational inputs
and school characteristics affect math and science scores.
For example, if the staff-to-pupil ratio
has an effect on both exam scores, why would we want to hold performance on the science test
fixed while studying the effects of
staff
on the math pass rate?
This would be an example of
controlling for too many factors in a regression equation.
The variable
scill
could be a dependent
variable in an identical regression equation.
6.2
(i) Because
ˆ
exp( 1.96 )
1
σ
−<
and
2
ˆ
exp(
/ 2)
1
>
, the point prediction is always above the
lower bound. The only issue is whether the point prediction is below the upper bound. This is the
case when
2
ˆˆ
exp(
/ 2)
exp(1.96 )
≤
or, taking logs,
2
/ 2 1.96
≤
, or
ˆ
2(1.96)
3.92
≤
=
.
Therefore, the point prediction is in the approximate 95% prediction interval for ˆ
3.92
≤
.
Because ˆ
is the estimated standard deviation in the regression with log(
y
) as the dependent
variable, 3.92 is a very large value for the estimated standard deviation of the error, which is on
the order of 400 percent. Most of the time, the estimated SER is well below that.
(ii) In the CEO salary regression, ˆ
.505
=
, which is well below 3.92.
6.5
(i) The turnaround point is given by
1
ˆ
β
/(2|
2
ˆ
|), or .0003/(.000000014)
≈
21,428.57;
remember, this is sales in millions of dollars.
(ii) Probably.
Its
t
statistic is about –1.89, which is significant against the one-sided
alternative H
0
:
1
< 0 at the 5% level (
cv
≈
–1.70 with
df
= 29).
In fact, the
p
-value is about
.036.
(iii) Because
sales
gets divided by 1,000 to obtain
salesbil
, the corresponding coefficient gets
multiplied by 1,000:
(1,000)(.00030) = .30.
The standard error gets multiplied by the same
factor.
As stated in the hint,
salesbil
2
=
sales
/1,000,000, and so the coefficient on the quadratic
gets multiplied by one million:
(1,000,000)(.0000000070) = .0070; its standard error also gets
multiplied by one million.
Nothing happens to the intercept (because
rdintens
has not been
rescaled) or to the
R
2
:
n
rdintens
=
2.613
+ .30
salesbil
– .0070
salesbil
2
(0.429)
(.14)
(.0037)
n
= 32,
R
2
= .1484.