Find the \(95 \%\) confidence interval for the difference of proportions for the data in Example 9-9.

Example 9-9

In the nursing home study mentioned in the chapter-opening Statistics Today, the researchers found that 12 out of 34 randomly selected small nursing homes had a resident vaccination rate of less than \(80 \%\), while 17 out of 24 randomly selected large nursing homes had a vaccination rate of less than \(80 \%\). At \(\alpha=0.05\), test the claim that there is no difference in the proportions of the small and large nursing homes with a resident vaccination rate of less than \(80 \%\).