Chapter 9: Hypothesis Testing with Two Samples
9.3 Comparing Two Independent Population Proportions
Learning Objectives
By the end of this section, the student should be able to
- Conduct and interpret hypothesis tests for two population proportions
Hypothesis tests for two population proportions
When conducting a hypothesis test that compares two independent population proportions, the following characteristics should be present:
- The two independent samples are simple random samples that are independent.
- The number of successes is at least five, and the number of failures is at least five, for each of the samples.
- Growing literature states that the population must be at least ten or 20 times the size of the sample. This keeps each population from being over-sampled and causing incorrect results.
Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions.
The difference of two proportions follows an approximate normal distribution. Generally, the null hypothesis states that the two proportions are the same. That is, H0: pA = pB. To conduct the test, we use a pooled proportion, pc.
[latex]{p}_{c}=\frac{{x}_{A}+{x}_{B}}{{n}_{A}+{n}_{B}}[/latex]
[latex]{{P}^{\prime }}_{A}-{{P}^{\prime }}_{B}\sim N\left[0,\sqrt{{p}_{c}\left(1-{p}_{c}\right)\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}\right][/latex]
[latex]z=\frac{\left({{p}^{\prime }}_{A}-{{p}^{\prime }}_{B}\right)-\left({p}_{A}-{p}_{B}\right)}{\sqrt{{p}_{c}\left(1-{p}_{c}\right)\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}}[/latex]
Example
Two types of medication for hives are being tested to determine if there is a difference in the proportions of adult patient reactions. Twenty out of a random sample of 200 adults given medication A still had hives 30 minutes after taking the medication. Twelve out of another random sample of 200 adults given medication B still had hives 30 minutes after taking the medication. Test at a 1% level of significance.
Solution
The problem asks for a difference in proportions, making it a test of two proportions.
Let A and B be the subscripts for medication A and medication B, respectively. Then pA and pB are the desired population proportions.
Random Variable: P′A – P′B = difference in the proportions of adult patients who did not react after 30 minutes to medication A and to medication B.
H0: pA = pB
pA – pB = 0
Ha: pA ≠ pB
pA – pB ≠ 0
The words “is a difference” tell you the test is two-tailed.
Distribution for the test: Since this is a test of two binomial population proportions, the distribution is normal:
[latex]{p}_{c}=\frac{{x}_{A}+{x}_{B}}{{n}_{A}+{n}_{B}}=\frac{20+12}{200+200}=0.08\text{ }1–{p}_{c}=0.92[/latex]
[latex]{{P}^{\prime }}_{A}–{{P}^{\prime }}_{B}~N\left[0,\sqrt{\left(0.08\right)\l