- Recognize when a one-sample [latex]z[/latex]-test or a two-sample [latex]z[/latex]-test is needed to answer a research question.
- Complete a two-sample [latex]z[/latex]-test for proportions from hypotheses to conclusions.
Conditions needed for a two-sample [latex]z[/latex]-test for proportions
When testing a claim that compares two populations, you must ensure that the two populations are independent. This is one of the conditions for a two-sample [latex]z[/latex]-test for proportions. How about sample size? What are the necessary sample sizes?
Recall that the Central Limit Theorem states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.
However, when we have two independent samples, and we need to compare the proportions between the two groups, we might have two different sample sizes and two different proportions. This means that the previous calculation [latex]np \geq 10[/latex] and [latex]n(1-p) \geq 10[/latex] will not work. We need something called pooled proportion [latex]\hat{p}_c[/latex]. The pooled proportion is a combined proportion calculated from the two independent samples in the hypothesis test. It is used when assuming that the true proportions in both groups are equal under the null hypothesis.
Conditions for Two-Sample [latex]z[/latex]-Test of Proportions
- Large Counts: For [latex]\hat{p}_c = \frac{x_1+x_2}{n_1+n_2}[/latex], check that:
- [latex]n_1\hat{p}_c \ge 10[/latex],
- [latex]n_2\hat{p}_c \ge 10[/latex],
- [latex]n_1(1-\hat{p}_c) \ge 10[/latex], and
- [latex]n_2(1-\hat{p}_c) \ge 10[/latex].
- Random Samples/Assignment: Check that the two samples are independent and random samples or that they come from randomly assigned groups in an experiment.
- 10%: Check that [latex]n_1<0.10(N_1)[/latex] and [latex]n_2<0.10(N_2)[/latex].