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EC 420, Spring 2008
Midterm
Show all necessary work.
You will have the entire class period to complete this exam.
This exam is “open book” but “closed note”; you may use the course text (Wooldridge),
but notes, old homework questions and answers, etc., are strictly forbidden.
Part One (short answer and proof)
:
1. What is a sampling distribution?
What does the Central Limit Theorem state, and
why is it so awesome (important for us statisticians)? (3 points)
A sampling distribution describes how a sample statistic (such as the sample mean,
the sample regression slope, the sample regression intercept, etc) varies from one
sample to the next.
The Central Limit Theorem is indeed awesome – it says that the
sampling distribution of the sample mean and sample regression “betas” are all
approximately normally distributed, as long as the sample size is “large enough”
(n>20 is a typical rule of thumb).
It is important because it means that we can
compute the probability of observing a particular sample statistics if we knew the
population parameter – this is fundamental for computing pvalues and doing
hypothesis tests.
2. What are the similarities between the slope of a simple regression line, the correlation
between two variables, and the rsquared of a simple regression model?
What are the
differences? (3 points)
If the rsquared equals zero, so does the correlation and the slope coefficient.
The
correlation and the slope coefficient always are the same sign, and the rsquared just
equals the square of the correlation between
x
and
y
.
The differences are that, while the slope tells us how steep a regression line is, it does
not really tell us how well the line fits the data…the correlation and the rsquared do
that.
3. Consider the following data for (
x
,
y
): (1,2), (4,5), (1,4), (3,5) and (4,7). I run a
regression of
y
on
x
, and I find that
1
ˆ
β
= 1 and
0
ˆ
= 2. (3 points)
a. Graph the underlying data.
b. Graph the predicted values for each observation.
c. Graph the regression line.
d. Indicate the residuals for each observation.
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Provide proofs for the following. Provide an explicit description of each of the steps
you take. (1 point each)
a. Show that
0
)
(
1
=

∑
=
X
X
n
i
i
.
0
)
(
1
1
1
=

=

=

∑
∑
∑
=
=
=
X
n
X
n
X
X
X
X
n
i
n
i
i
n
i
i
Where the third equals sign comes from the definition of
X
and the fact
that the sum of a constant (like
X
) is just n times that constant.
b. Using the result from part a, show that
0
)
(
1
=

∑
=
X
X
X
n
i
i
OK, this one is really easy:
0
0
)
(
)
(
1
1
=
=

=

∑
∑
=
=
X
X
X
X
X
X
X
n
i
i
n
i
i
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View Full DocumentPart Two
:
Many people think that the quality of college that one attends will affect one’s yearly
salary later in life.
You wish to investigate this idea.
You have access to a data set that
includes information from 1000 college graduates on their annual salary at age 40
(“
salary
”) and a measure of the quality of the college they attended (“
quality
”).
This
measure is the college’s score from the U.S. News “America’s Best Colleges” survey,
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This note was uploaded on 03/30/2008 for the course ECON 420 taught by Professor Toddelder during the Spring '08 term at Michigan State University.
 Spring '08
 ToddElder

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