Introductory Biostatistics

2. Choose a significance levela, a common choice being 0.05.


3. Calculate thezscore

z¼

p 2 p 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pð 1 pÞð 1 =n 1 þ 1 =n 2 Þ

p

wherepis the pooled proportion, defined by

p¼

x 1 þx 2

n 1 þn 2

an estimate of the common proportion underH 0.

4. Refer to the table for standard normal distribution (Appendix B) for
   selecting a cut point. For example, if the choice ofais 0.05, the rejection
   region is determined by:
   (a) For the one-sided alternativeHA:p 2 >p 1 ,zb 1 :65.
   (b) For the one-sided alternativeHA:p 2 <p 1 ,za 1 :65.
   (c) For the two-sided alternativeHA:p 10 p 2 ,za 1 :96 orzb 1 :96.

What we are doing here follows the same format used in previous sections.

The basic term ofp 2 p 1 measures the di¤erencebetween the two samples,

Its expected hypothesized value (i.e., underH 0 ) is zero.

The denominator ofzis the standard error ofp 2 p 1 , a measure of how

goodp 2 p 1 is as an estimate ofp 2 p 1.

Therefore,zmeasures the number of standard errors thatp 2 p 1 , theevi-

dence, is away from its hypothesized value.

In the two-sided form, the square of thezscore, denotedX^2 , is more often
used. The test is referred to as thechi-square test. The test statistic can also be
obtained using the shortcut formula

X^2 ¼

ðn 1 þn 2 Þ½x 1 ðn 2 x 2 Þx 2 ðn 1 x 1 ފ^2

n 1 n 2 ðx 1 þx 2 Þðn 1 þn 2 x 1 x 2 Þ

and the null hypothesis is rejected at the 0.05 level when

X^2 b 3 : 84

It should be noted that in the general case, with data in a 2
2 table (Table
6.4), the chi-square statistic above is simply

X^2 ¼

ðaþbþcþdÞðadbcÞ^2

ðaþcÞðbþdÞðaþbÞðcþdÞ

its denominator being the product of the four marginal totals.

214 COMPARISON OF POPULATION PROPORTIONS