CHAPTER 8
LESSON 8
Date __________________
Confidence Interval for a Proportion
Objectives:
 To investigate what is meant by “margin of error” and “confidence interval”.
 To calculate a confidence interval for an unknown population proportion.
 To determine the sample size needed to obtain an estimate with a prescribed margin of
error and confidence level.
Investigate
On the web and in the press we are bombarded with survey results like the ones captured in the
headlines below.
Intense National Concern
Almost Half of Canadians Don’t Trust Their Water
— Especially in Towns and Cities
The national representative sample of 500 Canadian adults was interviewed May 9-10, 2001.
Samples of this size are deemed accurate to within 4.5 percentage points 19 times out of 20.
A National Post/COMPAS Survey, May 11, 2001.
Canadian majority supports endangered-species law
A new survey indicates 94 per cent support for measures to protect wildlife
Kate Jaimet Vancouver Sun
OTTAWA -- More than 90 per cent of Canadians would support a law to protect endangered
species, a new survey by Pollara pollsters indicates. The survey of 1,024 adults was
commissioned by the Sierra Legal Defence Fund, an environmental group lobbying for
strong endangered species legislation. The polling firm is independent of the environmental
group, and results are considered to reflect Canadian public opinion with an accuracy within
3.1 percentage points, 19 times out of 20.
In this section we will discover how these polls are able to claim an accuracy within a few
percentage points “19 times out of 20” based on only a small sample of Canadians.
Estimation of a Population Proportion
For each of the examples on the previous page a study was done with the goal of obtaining
estimates of an unknown population proportion.
For example what proportion of Canadians
 trust their water supply?
 support endangered species law?
A properly designed random sample allows you
to estimate these unknown population
proportions with remarkably good accuracy.
Population
Unknown
proportion
of interest
Symbolism:
p=
Sample
Estimate of
the unknown
proportion
p =
Note:
p =
X
n
where n is the
,
and X is the
.
Before the sample is actually taken, X and p are variables.
After the sample is taken we will have specific values for each of the variables X and p .
Example 1:
Population:
B.C. high school students
Proportion of interest:
p = proportion with Internet access at home
To estimate this proportion a researcher plans to randomly sample 800 B.C. high school students.
a) In words, what do the variables X and p represent in this particular application?
b) After the random sample was taken, 576 of the 800 students had Internet access at home.
What are the values of X and p for this sample?
What is a Confidence Interval?
In a properly designed survey, it is possible to not only give an estimate of an unknown
population quantity, but also to attach a margin of error to it.
In general, a confidence interval is just an interval estimate of an unknown population quantity
with a specified confidence level (like 95% or 99%) attached to the interval.
The general form of a confidence interval is:
Estimate ± (Margin of Error)
For the specific case of an unknown population proportion,
a confidence interval for p is of the form:
p ± (Margin of Error)
Example 2:
Suppose that the sample proportion is 0.74 and the margin of error for a 95% confidence level is
calculated to be 0.03, what would be the 95% confidence interval for the population proportion?
Example 3:
The relevant sentences specifying the margin of error and the confidence level for each of the
two surveys presented on the first page of this section are repeated below.
For each one identify the margin of error and the confidence level.
a) “The national representative sample of 500 Canadian adults was interviewed May 9-10,
2001. Samples of this size are deemed accurate to within 4.5 percentage points 19 times out
of 20.”
Margin of error =
Confidence level =
b) “The survey of 1,024 adults was commissioned by the Sierra Legal Defence Fund, an
environmental group lobbying for strong endangered species legislation. The polling firm is
independent of the environmental group, and results are considered to reflect Canadian
public opinion with an accuracy within 3.1 percentage points, 19 times out of 20.”
Margin of error =
Confidence level =
How is the Margin of Error and Confidence Interval for a Proportion Calculated?
X
where X is the number of “successes” in the sample.
n
Here X is a variable that could take on any integer value between 0 and n inclusive.
The sample proportion p =
It follows that X has a
with mean =
So p =
mean =
distribution,
and sd =
.
X
has
n
and
sd =
The sd of p =
X
is called the
n
.
Furthermore, in Section 8.7 we showed that for a large sample we can approximate the binomial
distribution with a normal distribution.
95% Margin of Error and Confidence Interval for a Proportion
For large samples, the sample proportion p has approximately a
normal distribution with mean p and standard deviation
From the 68-95-99 Rule it follows that about 95%
of the time p will be within 2 sd’s of its mean.
More precisely, 95% of the time p will be within
sd’s of its mean.
So the 95% margin of error is:
and the 95% confidence interval for p is:
pq
.
n
-4
-3
-2
-1
0
1
2
3
4
Example 4:
In a random sample of 600 Canadians, 492 supported increased health funding.
a) Calculate the sample proportion supporting increased health funding.
b) Use the sample results to obtain a 95% margin of error and a 95% confidence interval for the
proportion of all Canadians that support increased health funding.
What about confidence levels like 90% or 99%?
It is easy to construct confidence intervals for confidence levels other than 95%, but first we
need to introduce a new notation.
Notice that for a 95% confidence interval we used
1.96 as a z-value cutoff to capture the central 0.95
area of a standard normal distribution.
0.95
-4
-3
-2
-1.96
-1
0
1
1.96
2
3
4
1
2
3
4
1
2
3
4
What z-value would cut-off a central 0.90 area?
0.90
-4
-3
-2
-1
0
What z-value would cut-off a central 0.99 area?
0.99
In general, the notation z/2 is used to represent the
z-value cut-off for the central (1– ) area of a
standard normal distribution.
-4
-3
-2
-1
0
/2
/2
1– 
-4
-3
-2
-1
0
1
2
3
4
Margin of Error and Confidence Interval for a Proportion
For large samples, the sample proportion p has approximately a normal distribution with
mean p and standard deviation
pq
.
n
1– 
-4
-3
--2z/2-1
0
1
2
z/2
3
4
So the (1– ) margin of error is:
the (1–) confidence interval for p is:
the approximate (1–) confidence interval for p is:
Example 5:
In a random sample of 300 B.C. high school students, 96 owned cell phones.
a) Find a 90% confidence interval for the population proportion of B.C. high school students
that own cell phones. What is the 90% margin of error?
b) Find a 99% confidence interval for the population proportion of B.C. high school students
that own cell phones. What is the 99% margin of error?
Sample Size Needed to Guarantee a Desired Margin of Error and Confidence Level
Case A: Some Prior Information Is Available About the Unknown Population Proportion
Based on the results of the last B.C. election the Green Party appears to have the support of at
most 15% of the electorate. A polling company is hired to conduct a survey to obtain an estimate
of the current level of support for the Greens.
How large a random sample is needed to obtain an estimate that is within 3% of the true
percentage support for the Greens at a 95% confidence level?
Case B: No Prior Information Is Available About the Population Proportion
On the first page of this section there is reference to a survey done by the COMPAS polling
company to determine the percentage of Canadians who don’t trust their water supply. Prior to
conducting the survey, the COMPAS polling company would not know anything about this
unknown population percentage, but it would like to take a random sample large enough so that
its estimate is within 4.5 percentage points 19 times out of 20. How large a sample should it
take?
Compare your answer to the actual sample size quoted on the first page.