2.
A company representative claims that 45 percent of Burger
King sales are made at the drivethrough window.
3.
A survey of homes in the Chicago area indicated that 85
percent of the new construction had central air conditioning.
4.
A recent survey of married men between the ages of 35 and
50 found that 63 percent felt that both partners should earn a
living.
305
Using the Normal Distribution to
Approximate the Binomial Distribution
To develop a confidence interval for a proportion, we need to meet the
following assumptions.
1. The binomial conditions, discussed in Chapter 6, have been met. Briefly,
these conditions are:
a. The sample data is the result of counts.
b. There are only two possible outcomes.
c. The probability of a success remains the same from one trial
to
the next.
d. The trials are independent. This means the outcome on one
trial does not affect the outcome on another.
2. The values
n
ʌ
and
n
(1
ʌ
) should both be greater than or equal to 5.
This condition allows us to invoke the central limit theorem and employ
the standard normal distribution, that is,
z
, to complete a confidence
interval.
306
Assumptions when using a Normal
approximation for confidence intervals
with a proportion (
short version
)
1.
It meets the conditions of a
BINOMIAL
distribution
2.
Both
n*
ʌ
and
n*(1
ʌ
)
are at least
5
306
Confidence Interval for a Population
Proportion  Formula
306
Confidence Interval for a Population
Proportion

Example
The union representing the Bottle
Blowers of America (BBA) is
considering a proposal to
merge with the Teamsters
Union. According to BBA union
bylaws, at least threefourths of
the union membership must
approve any merger. A random
sample of
2,000
current BBA
members reveals
1,600
plan to
vote for the merger proposal.
What is the estimate of the
population proportion?
Develop a
95 percent
confidence
interval for the population
proportion. Basing your
decision on this sample
information, can you conclude
that the necessary proportion of
BBA members favor the
merger? Why?
.
membership
union
the
of
percent
75
than
greater
values
includes
estimate
interval
the
because
pass
likely
will
proposal
merger
The
:
Conclude
)
818
.
0
,
782
.
0
(
018
.
80
.
2,000
)
80
.
1
(
80
.
96
.
1
80
.
0
)
1
(
C.I.
C.I.
95%
the
Compute
80
.
0
2000
1,600
:
proportion
sample
the
compute
First,
2
/
!
r
!
0
r
!
0
r
!
!
!
!
n
p
p
z
p
n
x
p
D
307
FinitePopulation Correction Factor
z
A population that has a fixed upper bound is said to be finite.
z
For a finite population, where the total number of objects is
N
and the
size of the sample is
n
, the following adjustment is made to the
standard errors of the sample means and the proportion:
z
However, if
n
/
N
< .05
, the finitepopulation correction factor may be
ignored.
1
0
0
!
N
n
N
n
x
V
V
1
)
1
(
0
0
0
!
N
n
N
n
p
p
p
V
Standard Error of the Mean
Standard Error of the Proportion
309
Effects on FPC when
n/N
Changes
Observe that FPC approaches 1 when
n/N
becomes smaller
309
Confidence Interval Formulas for Estimating Means
and Proportions with Finite Population Correction
1
0
0
r
N
n
N
n
z
X
V
C.I. for the Mean (
P
)
1
)
1
(
0
0
0
r
N
n
N
n
p
p
z
p
C.I. for the Proportion (
S
)
1
0
0
r
N
n
N
n
s
t
X
C.I. for the Mean (
P
)
309
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 Spring '11
 Leany
 Normal Distribution, Standard Deviation, pilot study, American Management Association