Stat 332
R.J. MacKay, University of Waterloo, 2005
Chapter 7 Solutions 1
Chapter 7 Exercise Solutions
1.
Consider the sampling protocols defined in Example 1.
a)
Show that the inclusion probability for each unit in the frame is 1/100 for
every protocol.
For each protocol, the model is uniform. That is, the chance of any possible sample is
equal. To find the inclusion probability, we need only count the number of samples that
contain a particular unit. Once the unit is in the sample, we count the ways of selecting
the remaining 99 units.
SRS: there are
99
9999
ways to select the other units so
100
1
100
10000
99
9999
=
=
i
p
Stratified sampling: there are
9
10
1000
9
999
ways to select the other units so
10
1
10
1000
10
1000
9
999
10
9
=
=
i
p
Cluster sampling: there are
9
999
ways to choose the remaining clusters so
100
1
10
1000
9
999
=
=
i
p
Systematic sampling: there is only one way to select the sample so
p
i
=
1 100
/
Two stage sampling: there are
1
9
ways to select the second primary unit. Then the other
99 secondary units can be selected in
50
1000
49
999
ways so
10
1
50
1000
2
10
50
1000
49
999
1
9
2
=
=
i
p
b)
On a final examination, a student once defined simple random sampling as follows:
“simple random sampling is a method of selecting units from a population so that
every unit has the same chance of selection”. Is this a correct answer?
No because there are many sampling protocols that satisfy this definition as shown in a)
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Stat 332
R.J. MacKay, University of Waterloo, 2005
Chapter 7 Solutions 2
c)
Show that the estimator corresponding to the sample average
$
μ
ε
=
∑
y
n
i
i s
is unbiased
for
μ
for each of the protocols.
Let
I
i
i
N
i
=
R
S
T
=
1
1
if unit is in thesample
0 otherwise
,...,
so that
E I
p
i
i
(
)
=
. Then we can write
~
μ
ε
=
∑
y I
n
i
i
i
U
and
E
y E I
n
y p
n
i
i
i
U
i
i
i
U
(
~
)
(
)
μ
μ
ε
ε
=
=
=
∑
∑
since
p
n
N
i
=
/
for all five protocols.
2.
Consider the estimate
$
(
)
σ
ε
=


∑
y
y
n
i
i s
2
1
and the corresponding estimator
~
σ
.
a)
For SRS, show that
~
σ
2
is an unbiased estimator for
σ
2
. [Hint: Use the fact that
(
)
y
y
y
ny
i
i s
i
i s

=

∑
∑
2
2
2
ε
ε
].
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 Spring '09
 Xu(Sunny)Wang
 Probability

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