STAT 854/454
Short Solutions to Assignment 2
1. Let
N
= 7
,
800 be the number of public schools in Canada. We wish to estimate
p
=
M/N
,
where
M
is the number of schools that have Wireless Internet Access (WIA).
(a) Find the required sample size
n
such that the estimate is accurate within 2 percentage
points, 19 times out of 20 under SRSWOR, assuming prior information on
p
is not
available.
Soln:
Use
n
=
{
Z
2
α/
2
p
(1

p
)
}
/
{
e
2
+
Z
2
α/
2
p
(1

p
)
/N
}
with
e
= 0
.
02,
α
= 0
.
05 and
p
= 0
.
5.
(b) Redo (a) assuming
p
= 5% for the purpose of sample size calculation.
Soln:
Omitted.
(c) If we do inverse sampling with
m
= 10 under selection without replacement, what is
the expected number of schools we have to sample? Assume
p
= 5% for the purpose
of planning.
Soln:
The probability function for the random sample size
n
is given by
P
(
n
=
j
) =
(
M
m

1
)(
N

M
j

m
)
(
N
j

1
)
M

m
+ 1
N

j
+ 1
,
j
=
m, m
+ 1
,
· · ·
, N

M
+
m .
Note that
∑
N

M
+
m
j
=
m
P
(
n
=
j
) = 1 for any
N
,
M
and
m < M
, and also note that
1
j
(
N
j

1
) =
1
N
+ 1
(
N
+1
j
)
and (
M
m

1
) =
m
M
+ 1
(
M
+1
m
)
,
we can show that
E
(
n
) =
∑
jP
(
n
=
j
) =
m
(
N
+ 1)
/
(
M
+ 1).
Use
M
=
pN
and
p
= 0
.
05 to get the final answer.
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 Fall '09
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 soln

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