11. Comparing Proportions
In Section 9.4 we covered a one-sample test for proportions using the approximation to the binomial distribution. Here we want to compare a proportion in one population with in another population, a two-sample test for proportions, also using the approximation to the binomial distribution. Define
where and are the number of items of interest in the samples from the two populations and and are their sample sizes. Also define the corresponding , , and . The hypotheses we want to test is
which is equivalent to
If , and are all then the appropriate normal distribution will provide a good approximation to the relevant binomial distribution and we can use the following test statistic to test the hypotheses
are the proportions of items of interest and not of interest in the two samples combined.
Example 11.1 : In a nursing home study we are interested in the proportions of nursing homes that have vaccination rates of less than 80. The two populations we want to compare are small nursing homes and large nursing homes. In a sample of 34 small nursing homes, 12 were found to have a vaccination rate of less than 80. In a sample of 24 large nursing homes, 17 were found to have a vaccination rate of less than 80. At is there a difference in the proportions of small and large nursing homes with vaccination rates of less than 80?
0. Data reduction.
First define: population 1 = small nursing homes and population 2 = large nursing homes. Then compute the proportions:
2. Critical statistic.
Use Table F, the last () line in the column for a two-tailed test at :
3. Test statistic.
There is enough evidence, from a proportions test at to support the observation that large nursing homes have worse vaccination rates than small nursing homes. Make sure your parents end up in a small nursing home. (Note that rejection of in a one-tail test allows us to believe the direction of difference given by the sample data.)