1
EC120B
Midterm Solution
Prof. Berman
Winter 2004
(20) 1.
A. Given a random sample of 40 observations of variables Y and X from some population,
provide a formula to estimate
β
1
, the slope of the population linear regression line.
Estimate for
β
1
:
β
ˆ
1
=
S
xy
/S
xx
=
Σ
(X
i

X
)(Y
i

Y
)/
Σ
(X
i

X
)
2
B. Write a formula to estimate a 95% confidence interval for
β
1
.
95% confidence interval for
β
1
:
β
ˆ
1
±
1.96
*
se(
β
ˆ
1
)
C. Now assume that 140 of your classmates sample from that population and estimate a
confidence interval as in part A. Let z be the number of confidence intervals (between 0 and 140) that
do
not
contain
β
1
. Estimate the expected value of z, E(z)?
Using the CLT to approximate, prob(interval does not contain
β
1
)
≈
0.05 for each classmate.
E(z)
≈
140(0.05)=7
(10 Bonus) D. What’s the variance of z, V(z)?
The variance of a binary variable is p(1p), where p is the probability of a “1.”
The variance of the sum of n independent binary variables is just n times the variance of
each.
V(z)
≈
np(1p)=140(0.05)(10.05)=6.65
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2
(40) 2.
For each statement below, indicate if it is true or false and write a sentence justifying your
answer.
A. If X
1
, ... X
N
are a random sample from a population with E(X) =μ
:
then
Σ
(X
i
 μ)=0
True [ ] / False [X]
Explain:
Σ
(X
i

X
)=0
But
X
≠
μ in a finite sample
So
Σ
(X
i
 μ)
≠
0
B. In random sampling from a population with expectation E(Y) = μ
:
,
Σ
(Y
i

Y
)
2
is greater than
Σ
(Y
i

μ
)
2
since the mean of Y is an imprecise estimate of the population mean
:
.
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