Hypothesis Testing of the Difference Between Two Population Proportions

This file is part of a program based on the Bio 4835 Biostatistics class taught at Kean University in Union, New Jersey.  The course uses the following text:
Daniel, W. W. 1999.  Biostatistics: a foundation for analysis in the health sciences.  New York: John Wiley and Sons.  
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 p1 p2 .  This permits the construction of a pooled estimate which is given by the following formula.

                pooled estimate formula
The standard error of the estimator is:

                standard error formula

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:       n1 = 55        x1 = 24        p-hat1 = .4364
Patients without hypertension:  n2 = 149      x2 = 36        p2 = .2416
                                                alpha = .05

(2) Assumptions

  • independent random samples from the populations

(3) Hypotheses

        H0 :  p1 less than or equal to p2

        HA :  p1 > p2

(4) Test statistic

The test statistic is z which is calculated as

                z score formula
        (a) Distribution of test statistic

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

        (b) Decision rule

With alpha = .05 the critical z score is 1.645.  We reject H0 if z > 1.645.

(5) Calculation of test statistic


(6) Statistical decision

Reject H0 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