Chapter 3 Answers
1.
The central limit theorem suggests that when the sample size (
n
) is large, the distribution of the
sample average (
Y
) is approximately
2
,
Y
Y
N
μ
σ
�
�
�
�
�
�
with
2
2
.
Y
n
Y
σ
σ
=
Given a population
100,
Y
μ
=
2
43 0,
Y
σ
=
.
we have
(a)
100,
n
=
2
2
43
100
0 43,
Y
n
Y
σ
σ
=
=
=
.
and
100
101
100
Pr(
101)
Pr
(1.525)
0 9364
0 43
0 43
Y
Y


<
=
<
≈ Φ
=
.
.
.
.
(b)
64,
n
=
2
2
43
64
64
0 6719,
Y
Y
σ
σ
=
=
=
.
and
101
100
100
103
100
Pr(101
103)
Pr
0 6719
0 6719
0 6719
(3 6599)
(1 2200)
0 9999
0 8888
0 1111
Y
Y



<
<
=
<
<
.
.
.
≈ Φ
.
 Φ
.
=
.

.
=
.
.
(c)
165,
n
=
2
2
43
165
0 2606,
Y
n
Y
σ
σ
=
=
=
.
and
100
98
100
Pr(
98)
1
Pr(
98)
1
Pr
0 2606
0 2606
1
( 3 9178)
(3 9178)
1 0000 (rounded to four decimal places)
Y
Y
Y


=

≤
=

≤
.
.
≈
 Φ  .
= Φ
.
= .
.
2.
Each random draw
i
Y
from the Bernoulli distribution takes a value of either zero or one with
probability Pr
(
1)
i
Y
p
=
=
and Pr
(
0)
1
.
i
Y
p
=
=

The random variable
i
Y
has mean
(
)
0
Pr(
0)
1
Pr(
1)
,
i
E Y
Y
Y
p
=
×
=
+ ×
=
=
and variance
2
2
2
2
2
var(
)
[(
) ]
(0
)
Pr(
0)
(1
)
Pr(
1)
(1
)
(1
)
(1
)
i
i
Y
Y
E Y
p
Y
p
Y
i
i
p
p
p
p
p
p
μ
=

=

ä
=
+

=
=

+

=

.
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(a)The fraction of successes is
1
(
1)
(success)
n
i
i
i
Y
# Y
#
p
Y
n
n
n
=
=
=
=
=
=
.
(b)
1
1
1
1
1
(
)
(
)
n
n
n
i
i
i
i
i
Y
E p
E
E Y
p
p
n
n
n
=
=
=
�
�
°
=
=
=
=
.
�
�
�
�
�
�
�
�
(c)
1
2
2
1
1
1
1
(1
)
var(
)
var
var(
)
(1
)
n
n
n
i
i
i
i
i
Y
p
p
p
Y
p
p
n
n
n
n
=
=
=
�
�
°

=
=
=

=
.
�
�
�
�
�
�
�
�
The second equality uses the fact that
1
Y
,
…
,
Y
n
are i.i.d. draws and
cov(
,
)
0,
i
j
Y Y
=
for
.
i
j
3.
Denote each voter’s preference by
.
Y
1
Y
=
if the voter prefers the incumbent and
0
Y
=
if the voter
prefers the challenger.
Y
is a Bernoulli random variable with probability Pr
(
1)
Y
p
=
=
and Pr
(
0)
1
.
Y
p
=
=

From the solution to Exercise 3.2,
Y
has mean
p
and variance
(1
).
p
p

(a)
215
400
0 5375.
p
=
=
.
(b)
ˆ
ˆ
(1
)
0.5375
(1
0.5375)
4
400
ˆ
var(
)
6 2148
10 .
p
p
n
p



=
=
=
.
The standard error is SE
1
2
(
)
(var(
))
0 0249.
p
p
=
=
.
(c)The computed
t
statistic is
0
0 5375
0 5
1 506
SE(
)
0 0249
p
act
p
t
p
μ
,

.

.
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 Spring '10
 Grant
 Statistics, Econometrics, Normal Distribution, Standard Deviation, Statistical hypothesis testing

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