Prof. G. Jakubson
Economics 321
Spring 2007
PRELIMINARY EXAM #1  ANSWER
KEY
1.
Class intervals
for infant mortality rate
Countries
cum rel
freq
rel freq
freq
0 to <10
24
24/131 = .18
24/131 = .18
10
to < 20
24
24/131 = .18
48/131 = .36
20 to <30
14
14/131 = .11
62/131 = .47
30 to <50
18
18/131 = .14
80/131 = .61
50 to <75
18
18/131 = .14
98/131 = .75
75 to <100
16
16/131 = .12
114/131 = .87
100 to <140
13
13/131 = .10
127/131 = .97
140 to 180
4
4/131 = .03
131/131 = 1
Total number of countries
131
a.
Median
infant mortality rate is in the
30 to < 50 category.
Since we were asked for the median and
not just the class, we must interpolate: Median
= 30 + [.5  .47)/.14]*(5030) = 34.3.
This assumes a
uniform distribution within the class.
b.
See above.
c.
Use grouped mean and variance expressions:
Mean =
xf
x
n
n
k
k
k
k
k
k
∑∑
=
and
Variance =
n
n
xx
f
k
k
k
−
−
∑
1
2
()
, where x
k
is the class mark and f
k
is relative frequency.
2.
a.
[]
Z=
1
n
∑−
=∑ −∑=−=
XX
s
n
X
sn
X
s
X
s
X
s
ix
i
x
x
/
11
0
or use the linear transformation formulae: Z
i
= a + bX
i
where
x
x
s
1
b
and
s
X

a
=
=
b.
s
z
= (1/s
x
) s
x
= 1.
c.
Since T
i
is just a standardized version of Y
i
, its mean and variance are also 0 and 1,
respectively.
d.
Variance of
ee
.
Here, X=log(food expenditure).
Evaluated at
s
x
≈
2
2
X , this yields
.
Therefore, the std of e
.
.
e
821 2
2
46
2861746
=
X
≈
2861746 =1692.
Note that this
approximates the std of food expenditure.
In order to compare this to the standard deviation of F
from the table, we must adjust the units. Log(food expenditure) (LF) uses food expenditure data in
$1983, while food expenditure (F) uses data in 1,000’s of $1983.
So, we must divide the std we
got by 1000 (=1.692) to put it into $1983.
Compare 1.692 to the std of F from the table, which is
1.85.
The approximation is about 9% lower than the actual.
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 Spring '07
 MOLINARI
 Economics, Econometrics, Linear Regression, Regression Analysis, Applied Econometrics Preliminary, Econometrics Preliminary Exam

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