SOLUTIONS TO PROBLEMS
The generality is not necessary.
is only about
.30, which shows that
is very statistically insignificant.
Plus, having the squared term has only a minor effect on
the slope even for large values of
(The approximate slope is .0215
, and even
= 25 – about one standard deviation above the average
in the sample – the slope is
.211, as compared with .215 at
(i) The turnaround point is given by
|), or .0003/(.000000014)
remember, this is sales in millions of dollars.
statistic is about –1.89, which is significant against the one-sided
< 0 at the 5% level (
In fact, the
-value is about
gets divided by 1,000 to obtain
, the corresponding coefficient gets
multiplied by 1,000:
(1,000)(.00030) = .30.
The standard error gets multiplied by the same
As stated in the hint,
/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
has not been
rescaled) or to the
(iv) The equation in part (iii) is easier to read because it contains fewer zeros to the right of
Of course the interpretation of the two equations is identical once the different
scales are accounted for.
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
on the math pass rate?
This would be an example of
controlling for too many factors in a regression equation.
could be a dependent
variable in an identical regression equation.
The second equation is clearly preferred, as its adjusted
-squared is notably larger than that
in the other two equations.
The second equation contains the same number of estimated