This file is part of a program based on the Bio 4835 Biostatistics class taught
at

Daniel, W. W. 1999. Biostatistics: a foundation for analysis in the
health sciences.

The file follows this text very closely and readers are encouraged to consult
the text for further information.

**D) Hypothesis testing of the difference between
two population proportions**

It is frequently important to test the difference
between two population proportions. Generally we would test = . This permits the construction of a
pooled estimate which is given by the following formula.

The standard error of the estimator is:

Example 7.6.1

In a study of patients on sodium-restricted diets, 55
patients with hypertension were studied. Among these, 24 were on
sodium-restricted diets. Of 149 patients without hypertension, 36 were on
sodium-restricted diets. We would like to know if we can conclude that,
in the sampled population, the proportion of patients on sodium-restricted
diets is higher among patients with hypertension than among patients without
hypertension.

(1) Data

Patients with hypertension: = 55
= 24 = .4364

Patients without hypertension: = 149 =
36 = .2416

=
.05

(2) Assumptions

- independent random samples from the populations

(3) Hypotheses

: p1 p2

: p1 > p2

(4) Test statistic

The test statistic is z which is calculated as

(a) Distribution of test statistic

If the null hypothesis is true, the test statistic approximately follows the
standard normal distribution.

(b) Decision rule

With = .05 the critical z score is
1.645. We reject if z > 1.645.

(5) Calculation of test statistic

(6) Statistical decision

Reject because 2.71 > 1.645

(7) Conclusion

The proportion of patients on sodium restricted diets among hypertensive
patients is higher than in nonhypertensive patients.

p = .0034