Chapter 10: Inference for Two Populations
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.
Like the case of differences in sample means, we construct a sampling distribution for differences in sample proportions:
[latex]p'A=\frac{X_{A}}{n_{A}}[/latex] and [latex]p'\frac{X_{B}}{X_{B}}[/latex] are the sample proportions for the two sets of data in question [latex]X_{A}[/latex] and [latex]X_{B}[/latex].
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]z=\frac{(p'_{A}-p'_{B})-(p_{a}-p_{b})}{\sqrt{P_{C}(1-P_{C})(\frac{1}{n_{a}}+\frac{1}{n_{B}})}}[/latex]
