7.3 Large Sample
(CI’s) for
π
and
2
1
μ
μ

•
Confidence interval for proportion
•
A bound on the error of estimation
•
A confidence interval for µ
1
µ
2
Let
π
=
proportion of population having a particular
characteristic. A sample of size n is taken and a natural estimate
for
π
is
n
successes
of
number
=
p
, where
Success
=
possessing the characteristic.
Properties of sampling distribution of the proportion p
/ n
π
π
σ
π
μ
p
p
)
1
(
=
=
If both np and n (1p) greater than 10, then sampling distribution
for proportion is approximately normal.
When n is large, z
=
(1
) /
p
n
π
π
π


~
N ( 0, 1).
Using the earlier approach, 100(1
α
)% CI is
(we need to solve a
Quadratic equation here )
1
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n
z
n
z
n
p
p
c
n
z
c
c
c
z
p
2
2
2
2
)
(
4
)
(
)
1
(
2
)
(
1
+
+
±
+

where
).
2
1
(
α

=
c
Assumptions
1.
SRS sample
2.
Observations independent on each other
3.
If sampling without replacement, the sample size n should be
no more than 10% of the population.
4.
Large “sample size”
10)
)

(1
and
10
(
π
n
nπ
Now
)

(1
100
α
% confidence interval (aaproximate) for a
population proportion
=
2

1
c
α
is
n
p
p
p
)
1
(
z
c

±
,
where
c
z
is a central area critical value for standard normal.
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 Summer '08
 Palaniappan
 Normal Distribution, 10%, Body mass index, The Associated Press

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