Kata Bognar
kbognar@ucla.edu
Economics 41
Statistics for Economists
UCLA
Fall 2010
Homework Assignment 5.
- suggested solutions by Yujing Xu -
NOTE: Please show your calculations for Questions 1-2, 6-13 and 16-19. 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