8-3 Testing a Claim
about a Proportion
This section presents complete procedures for
testing a hypothesis (or claim) made about a
population proportion.
This section uses the components introduced in
the previous section for the P-value method, the
traditional method or the use of confidence
intervals.
Key Concept
Two common methods for testing a claim about a
population proportion are (1) to use a normal
distribution as an approximation to the binomial
distribution, and (2) to use an exact method
based on the binomial probability distribution.
Part 1 of this section uses the approximate
method with the normal distribution, and Part 2 of
this section briefly describes the exact method.
Part 1:
Basic Methods of Testing Claims
about a Population Proportion p
Notation
n
pˆ
= sample size or number of trials
x

n
p
= population proportion
q
=1–p
Requirements for Testing Claims
About a Population Proportion p
1) The sample observations are a simple random sample.
2) The conditions for a binomial distribution are satisfied.
3) The conditions np ≥ 5 and nq ≥ 5 are both satisfied, so
the binomial distribution of sample proportions can be
approximated by a normal distribution with μ = np and
  npq .
Note: p is the assumed proportion not the sample
proportion.
Test Statistic for Testing
a Claim About a Proportion
z 
pˆ  p
pq
n
P-values:
Use the standard normal distribution (Table A-2)
and refer to Figure 8-1.
Critical Values:
Use the standard normal distribution (Table A-2).
Caution
Don’t confuse a P-value with a proportion p.
P-value = probability of getting a test
statistic at least as extreme as
the one representing sample
data
p = population proportion
P-Value Method
Computer programs and calculators usually provide a Pvalue, so the P-value method is used.
If technology is not available, see Figure 8-1 in the text.
Critical Value Method
Use the same method as described in Figure 8-2 in
Section 8-2.
Confidence Interval Method
In general, for two-tailed hypothesis tests, construct a
confidence interval with a confidence level corresponding
to the significance level, as in Table 8-1 in the text.
Caution
When testing claims about a population proportion, the
traditional method and the P-value method are equivalent
and will yield the same result since they use the same
standard deviation based on the claimed proportion p.
However, the confidence interval uses an estimated
standard deviation based upon the sample proportion pˆ .
Consequently, it is possible that the traditional and P-value
methods may yield a different conclusion than the
confidence interval method.
A good strategy is to use a confidence interval to estimate a
population proportion, but use the P-value or traditional
method for testing a claim about the proportion.
Example
Based on information from the National Cyber Security
Alliance, 93% of computer owners believe they have
antivirus programs installed on their computers.
In a random sample of 400 scanned computers, it is found
that 380 of them (or 95%) actually have antivirus software
programs.
Use the sample data from the scanned computers to test
the claim that 93% of computers have antivirus software.
Example - Continued
Requirement check:
1. The 400 computers are randomly selected.
2. There is a fixed number of independent trials with two
categories (computer has an antivirus program or
does not).
3. The requirements np ≥ 5 and nq ≥ 5 are both satisfied
with n = 400
np   400   0.93   372
nq   400   0.07   28
Example - Continued
P-Value Method:
1. The original claim that 93% of computers have
antivirus software can be expressed as p = 0.93.
2. The opposite of the original claim is p ≠ 0.93.
3. The hypotheses are written as:
H 0 : p  0.93
H 1 : p  0.93
Example - Continued
P-Value Method:
4. For the significance level, we select α = 0.05.
5. Because we are testing a claim about a population
proportion, the sample statistic relevant to this test is:
pˆ , approxim ated by a norm al distribution
Example - Continued
P-Value Method:
6. The test statistic is calculated as:
380
z
pˆ  p

 0.93
400
pq
 0.93   0.07 
n
400
 1.57
Example - Continued
P-Value Method:
6. Because the hypothesis test is two-tailed with a test statistic of
z = 1.57, the P-value is twice the area to the right of z = 1.57.
The P-value is twice 0.0582, or 0.1164.
Example - Continued
P-Value Method:
7. Because the P-value of 0.1164 is greater than the significance
level of α = 0.05, we fail to reject the null hypothesis.
8. We fail to reject the claim that 93% computers have antivirus
software. We conclude that there is not sufficient sample
evidence to warrant rejection of the claim that 93% of computers
have antivirus programs.
Example - Continued
Critical Value Method: Steps 1 – 5 are the same as for
the P-value method.
6. The test statistic is computed to be z = 1.57. We now find the
critical values, with the critical region having an area of α = 0.05,
split equally in both tails.
Example - Continued
Critical Value Method:
7. Because the test statistic does not fall in the critical region, we fail
to reject the null hypothesis.
8. We fail to reject the claim that 93% computers have antivirus
software. We conclude that there is not sufficient sample
evidence to warrant rejection of the claim that 93% of computers
have antivirus programs.
Example - Continued
Confidence Interval Method:
The claim of p = 0.93 can be tested at the α = 0.05 level
of significance with a 95% confidence interval.
Using the methods of Section 7-2, we get:
0.929 < p < 0.971
This interval contains p = 0.93, so we do not have
sufficient evidence to warrant the rejection of the claim
that 93% of computers have antivirus programs.
Part 2
Exact Method for Testing Claims
about a Proportion pˆ
Testing Claims
Using the Exact Method
We can get exact results by using the binomial probability
distribution.
Binomial probabilities are a nuisance to calculate manually,
but technology makes this approach quite simple.
Also, this exact approach does not require that np ≥ 5 and
nq ≥ 5 so we have a method that applies when that
requirement is not satisfied.
To test hypotheses using the exact binomial distribution,
use the binomial probability distribution with the P-value
method, use the value of p assumed in the null hypothesis,
and find P-values as follows:
Testing Claims Using
the Exact Method
Left-tailed test:
The P-value is the probability of getting x or fewer
successes among n trials.
Right-tailed test:
The P-value is the probability of getting x or more
successes among n trials.
Testing Claims Using
the Exact Method
Two-tailed test:
If pˆ  p , the P-value is twice the probability of
getting x or more successes
If pˆ  p , the P-value is twice the probability of
getting x or fewer successes
Example
In testing a method of gender selection, 10 randomly
selected couples are treated with the method, and 9 of the
babies are girls.
Use a 0.05 significance level to test the claim that with
this method, the probability of a baby being a girl is
greater than 0.75.
Example - Continued
We will test
H 0 : p  0.75
H 1 : p  0.75
using technology to find probabilities in a binomial
distribution with p = 0.75.
Because it is a right-tailed test, the P-value is the
probability of 9 or more successes among 10 trials,
assuming p = 0.75.
Example - Continued
The accompanying STATDISK display shows exact
binomial probabilities.
Example - Continued
The probability of 9 or more successes is 0.2440252,
which is the P-value of the hypothesis test.
The P-value is high (greater than 0.05), so we fail to reject
the null hypothesis.
There is not sufficient evidence to support the claim that
with the gender selection method, the probability of a girl
is greater than 0.75.