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Used
to compare a proportion created by a random sample to a proportion originating from or
thought to represent the value for the entire population. As an example, to make sure your
random sample of 100 subjects is not biased regarding a person’s sex, you could
compare the proportion of women in the sample to the known proportion of women in the
underlying population as reported in census data or by some other reliable source.
Problem:
Historical data indicates that about 10% of your agency's clients believe they were given
poor service. Now under new management for six months, a random sample of 110 clients
found that 15% believe they were given poor service.
Pu
= .10
Ps
= .15
n
= 110
Assumptions
Independent
random sampling
Nominal
level data
Large
sample size
State
the Hypothesis
Ho:
There is no statistically significant difference between the historical proportion of
clients reporting poor service and the current proportion of clients reporting poor
service. 
If
2-tailed test
Ha:
There is a statistically significant difference between the historical proportion of
clients reporting poor service and the current proportion of clients reporting poor
service. 
If
1-tailed test
Ha:
The proportion of current clients reporting poor service is significantly greater than the
historical proportion of clients reporting poor service.
Set
the Rejection Criteria
Use
z-distribution
table to estimate critical value
If
2-tailed test, Alpha .05, Zcv = 1.96
If
1-tailed test, Alpha .05, Zcv = 1.65
Compute
the Test Statistic
Estimate
Standard Error
p
= population proportion
q
= 1 - p
n
= sample size
 
Test
Statistic
  
Decide
Results of Null Hypothesis
If
a 2-tailed test was used
Since
the test statistic of 1.724 did not meet or exceed the critical value of 1.96, there is
insufficient evidence to conclude there is a statistically significant difference between
the historical proportion of clients reporting poor service and the current proportion of
clients reporting poor service.
If
a 1-tailed test was used
Since
the test statistic of 1.724 exceeds the critical value of 1.65, you can conclude the
proportion of current clients reporting poor service is significantly greater than the
historical proportion of clients reporting poor service.
Software Output Example
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