1
NOTES: Confidence intervals from sample proportions KEY
Suppose that we are estimating an unknown population proportion
p
. We do this by first finding a sample
proportion
p
ˆ
and then calculating its confidence interval. In theory, the formula for the confidence
interval would be
)
ˆ
(*
ˆpSDzp ±
or
n
pp
zp )1(
*
ˆ−
±
where
*
z
depends on the Confidence Level.
But this formula involves the parameter (
p
) that we are trying to estimate! We therefore use the
Standard Error for
p
ˆ
(in other words, replace
p
with
p
ˆ
in the standard deviation formula) instead of
the Standard Deviation for
p
ˆ
. The formula for the confidence interval then becomes
)
ˆ
(*
ˆpSEzp ±
OR
n
pp
zp )
ˆ
1(
ˆ
*
ˆ−
±
*
z
can be found with the following, where
C
is the given Confidence Level inputted in decimal form.
=
*
z
InvNorm
+5.0
2
C
Let L denote the left endpoint of the confidence interval and let R denote the right endpoint of the
confidence interval. Therefore, the confidence interval written in interval notation would be (L,R). The
margin of error (m) is the distance that our confidence interval branches out in either direction from
p
ˆ
.
Think of this as
)
ˆ
(RpL
mm
→←
. Thus, the confidence interval formula can be thought of as
mp
±
ˆ
. It follows that the margin of error for these confidence intervals is
n
pp
z)
ˆ
1(
ˆ
*
−
. In other
words
n
pp
zm )
ˆ
1(
ˆ
*
−
=
. Also,
(
)
2
LR
m
−
=
.
INTERPRETATION: “We are ___% confident that
p
is between
L
and
R
.”
Suppose that the confidence level is 95%. Then there is a 95% probability that choosing any
confidence interval will give us
p
in it, because
p
is in 95% of the confidence intervals that we can
create. Once we fix
p
ˆ
and the confidence interval it generates, then
p
is either in that particular
confidence interval or it is not. Once
p
ˆ
and its confidence interval is fixed then we say that “we are 95%
confident” that
p
is in that particular confidence interval. That is, in repeated trials, we expect that 95% of
the confidence intervals in the long run, to contain
p
.
CONDITIONS/REQUIREMENTS: Before we are allowed to use the confidence interval formula certain
conditions must be met:
1. The sample proportion
p
ˆ
must be obtained from a Simple Random Sample (SRS).
2. The number of successes in the sample is at least 5, preferably at least 10, i.e.,
10
ˆ
≥
pn
.
3. The number of failures in the sample is at least 5, preferably at least 10, i.e.,
10)
ˆ
1(
≥
−
pn
.
4. The population size is at least ten times the sample size (
n
).
