Kata Bognar
[email protected]
Economics 41
Statistics for Economists
UCLA
Fall 2010
Homework Assignment 5.
 suggested solutions by Yujing Xu 
NOTE: Please show your calculations for Questions 12, 613 and 1619. The exercises are from the
textbook (Tanis and Hogg: A Brief Course in Mathematical Statistics).
Distribution of the sample mean.
1. The table below provides the wealth of the world’s six richest people in 2003. Consider these six
people a population of interest.
Person
Wealth ($ billions)
William H. Gates III (G)
41
Warren E. Buﬀett (B)
31
26
Paul G. Allen (P)
20
Prince Alwaleed Bin Talal Alsaud (T)
18
Lawrence J. Ellison (E)
17
(a) Calculate the mean wealth of the six people,
μ
.
Answer:
μ
= (41 + 31 + 26 + 20 + 18 + 17)
/
6 = 153
/
6 = 25
.
5
(b) List all the possible samples of size 2 from this population. Calculate the mean wealth for
all possible samples. Denote
¯
X
2
the sample mean wealth for samples of size 2. Derive the
distribution of
¯
X
2
. What is the expected value of
¯
X
2
?
Answer:
The possible samples of size 2 are
GB,GA,GP,GT,GE,BA,BP,BT,BE,AP,AT,AE,PT,PE,TE
Please refer to the following table for the sample mean wealth for these samples.
mean wealth
G
B
A
P
T
E
G
36
33.5
30.5
29.5
29
B
28.5
25.5
24.5
24
A
23
22
21.5
P
19
18.5
T
17.5
The sample mean for samples of size 2 assumes the values above with equal probabilities
of
P
(¯
x
j
) = 1
/
15
.
The expected value of the sample mean for samples of size 2 is
E
(
¯
X
2
) =
∑
j
¯
x
j
/
15 = 25
.
5
.
Notice that
E
¯
X
2
=
μ.
(c) List all the possible samples of size 4 from this population. Calculate the mean wealth for
all possible samples. Denote
¯
X
4
the sample mean wealth for samples of size 4. Derive the
distribution of
¯
X
4
. What is the expected value of
¯
X
4
?
Answer:
The possible samples of size 4 are
APTE,BPTE,BATE,BAPE,BGAP,GPTE,GATE,GAPE,GAPT,GBTE,GBPE,GBPT,
BGAE,BGAT,GBAT
.
The sample mean wealth for all possible samples are shown in the following table. In the
table, entry (X,Y) means the mean wealth of the sample
{
G,B,A,P,T,E
}
/
{
X,Y
}
, where
X,Y
can be
G,B,A,P,T,E
. For example, entry (G,B) means the mean wealth of sample
{
A,P,T,E
}
mean wealth
G
B
A
P
T
E
G
20.25
21.5
23
23.5
23.75
B
24
25.5
26
26.25
A
26.75
27.25
27.5
P
28.75
29
T
29.5
1
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View Full DocumentThe sample mean for samples of size 4 assumes the values above with equal probabilities
of
P
(¯
x
j
) = 1
/
15
.
The expected value of the sample mean for samples of size 4 is
E
(
¯
X
4
) =
∑
j
¯
x
j
/
15 = 25
.
5
.
Notice that
E
¯
X
4
=
μ.
(d) Compare the distributions in part (b) and (c).
Answer:
Both distributions are centered around the population mean, i.e.
E
(
¯
X
2
) =
E
(
¯
X
4
) =
μ.
However the distribution of
¯
X
4
is more concentrated around that value. You
can see this if you plot the values and/or looking at the range of the values for both random
variables. Alternatively you can calculate the standard deviation for both random variables
to reach the same conclusion.
2. According to the U.S. Census Bureau, the mean price of new mobile homes is $61,300. The
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 Fall '07
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 Economics, Normal Distribution, Standard Deviation, Variance

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