(
)
*)
1
(
*
*
)
1
(
,
~
2
p
p
m
z
n
n
p
p
p
N
p
−
=
⇒
−
p
ˆ

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Comparing two independent samples
We often need to compare 2 treatments used on
independent
samples. We
can compute the difference between the 2 sample proportions and compare it
to the corresponding, approximately normal sampling distribution for (
1
–
2
):
p
ˆ
p
ˆ

Large-sample CI for two proportions
For two independent SRSs of sizes
n
1
and
n
2
with sample
proportion of successes
1
and
2
respectively, an
approximate
level
C
confidence interval
for
p
1
– p
2
is
2
2
2
1
1
1
2
1
)
ˆ
1
(
ˆ
)
ˆ
1
(
ˆ
*
*
error
of
margin
the
is
,
)
ˆ
ˆ
(
n
p
p
n
p
p
z
SE
z
m
m
m
p
p
diff
−
+
−
=
=
±
−
Use this method only when the populations are at least 10 times larger than
the samples and the number of successes and the number of failures are each
at least 10 in both samples.
C is the area under the standard normal curve between
−
z* and z*.
p
ˆ
p
ˆ

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