+ 4 and the count of successes is
X
+
2.
4
ns
observatio
all
of
count
2
successes
of
counts
~
+
+
=
p
)
4
(
)
~
1
(
~
*
*
with
,
~
:
+
−
=
=
±
n
p
p
z
SE
z
m
m
p
CI
The “plus four” estimate of
p
is:
And an approximate level
C
confidence interval is:
Use this method when
C
is at least 90% and sample size is at least 10.

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Significance test for
p
The sampling distribution for
is approximately normal for large sample sizes
and its shape depends on
p
and
n
. Thus, we can easily test the null hypothesis:
H
0
:
p = p
0
(a given value we are testing).
z
=
ˆ
p
−
p
0
p
0
(1
−
p
0
)
n
If
H
0
is true, the sampling distribution is known
The likelihood of our sample proportion given the null
hypothesis depends on how far from
p
0
our
is in units of
standard deviation.
This is valid when both expected counts—expected successes
np
0
and expected
failures
n
(1
−
p
0
)—are each 10 or larger.
p
0
(1
−
p
0
)
n
p
0
ˆ
p
p
ˆ
ˆ
p

P
-values and one- or two-sided hypotheses
—
reminder
If the
P
-value is as small or smaller than the chosen significance level
a
, then the
difference is statistically significant and we reject
H
0
.

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