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Applied Statistics Handbook
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Comparing a Population Proportion to a Sample Proportion (Z-test)

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


Google

 

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