CK-12 Probability and Statistics - Advanced

8.4. Testing a Hypothesis for Dependent and Independent Samples http://www.ck12.org

When solving for the test statistic in large samples, we use the formula:

z=

(p 1 −p 2 )−(P 1 −P 2 )

sp 1 −p 2

where:

p 1 andp 2 =the observed sample proportions

P 1 andP 2 =the hypothesized population proportions

sp 1 −p 2 =the standard error of the difference between independent proportions

Similar to the standard error of the difference between independent samples, we need to do a bit of work to calculate
the standard error of the difference between independent proportions(sp 1 −p 2 ). To calculate this statistic, we use the
formula:

sp 1 −p 2 =

√

pq

(

1

n 1

+

1

n 2

)

where:

p=

f 1 +f 2

n 1 +n 2

q= 1 −p

f 1 =frequency of success in the first sample

f 2 =frequency of success in the second sample

Example:

Suppose that we are interested in finding out which particular city is more is more satisfied with the services provided
by the city government. We take a survey and find the following results:

TABLE8.8:

Number Satisfied City 1 City 2

Yes 122 84

No 78 66

Sample Size n 1 = 200 n 2 = 150

Proportion who said Yes 0. 61 0. 56

Is there a statistical difference in the proportions of citizens that are satisfied with the services provided by the city
government? Use a.05 level of significance.

Solution:

First, we establish the null and alternative hypotheses:

H 0 :P 1 =P 2

Ha:P 16 =P 2