1. Pharmaceutical Statistics
Hypothesis Testing
Proportions

2. Hypothesis Testing: A Single Population Proportion
Testing hypotheses about population proportion is carried out in the same manner for means.
One sided and two sided tests may be made, depending on the question being asked.
When the sample size is sufficiently large, the central limit theorem is applied in the same manner as discussed before.

3. Hypothesis Testing: A Single Population Proportion
In a survey of injection drug users in a large city, it was found that 18 out of 423 were HIV positive. We wish to know if we can conclude that fewer than 5% of the injection drug users in the sampled population are HIV positive.
Data: p'= 18/423 =
Hypotheses: H0: p ≥ 0.05
HA: p  0.05

4. Hypothesis Testing: A Single Population Proportion
We conduct the test at the point of equality.
The standard error is the only thing that is different:
Where p0 and q0 are the proportions under testing and its complement respectively.
In our question:

5. Hypothesis Testing: A Single Population Proportion
The test statistic is:
In this case is z:

6. Hypothesis Testing: A Single Population Proportion
For the decision rule, let α=0.05, for a one (left) tailed test, the critical z value will be
Do not reject the null hypothesis since is within the non-rejection zone.
We can not conclude the proportion of HIV positive individuals in this population is less than 5%.

7. Hypothesis Testing: The Difference Between Two population proportions
The most frequent test employed relative to the difference between two population proportions is that their difference is zero.
Both one sided and two sided tests are possible.

8. Hypothesis Testing: The Difference Between Two population proportions
When the null hypothesis to be tested is:p1-p2=0, we are hypothesizing that the two population proportions are equal.
We use this justification to combine the results of the two samples to come up with a pooled estimate of the hypothesized common proportion.
Where x1 and x2 are the numbers (frequency) in the first and second samples, respectively, possessing the character of interest.

9. Hypothesis Testing: The Difference Between Two population proportions
This pooled estimate of p=p1=p2 is used in computing the standard error term.
In this case, the test statistic becomes:

10. Hypothesis Testing: The Difference Between Two population proportions
In a study of nutrition care in nursing homes, it was found that among 55 patients with hypertension, 24 were on Na restricted diets. Of 149 patients without hypertension, 36 were on Na restricted diets.
Can we conclude that in the sampled populations the proportion of patients on Na restricted diets is higher among patients with hypertension than among patients without hypertension.

11. Hypothesis Testing: The Difference Between Two population proportions
We assume that the patients in the study constitute independent random samples from populations of patients with and without hypertension.
H0: pH ≤ pNH or pH - pNH ≤ 0
HA: pH > pNH or pH - pNH > 0
Where pH is the proportion on Na restricted diets in the population of hypertensive patients and pNH is the proportion on Na restricted diets in the population of patients without hypertension.

12. Hypothesis Testing: The Difference Between Two population proportions
Let α=0.05, the critical value of z is Reject H0 if the calculated z is greater than

13. Hypothesis Testing: The Difference Between Two population proportions
Reject the Null hypothesis.
The proportion of patients on Na restricted diets in higher among hypertensive patients.
P=0.034