Bluman A.G. Elementary Statistics- A Step By Step Approach

Section 9–4 Testing the Difference Between Proportions 505where sp 2 and s 2 1 p 2are the variances of the proportions, q 1 1 p 1 , q 2 1 p 2 , and n 1and n 2 are the respective sample sizes.Since p 1 and p 2 are unknown, a weighted estimate of p can be computed by using theformulap n 1pˆ 1 n 2 pˆ 2n 1 n 2and q 1 p. This weighted estimate is based on the hypothesis that p 1 p 2 . Hence, p isa better estimate than either pˆ1 or pˆ2 , since it is a combined average using both pˆ1 and pˆ2 .Since pˆ1 X 1 n 1 and pˆ2 X 2 n 2 , p can be simplified top X 1 X 2n 1 n 2Finally, the standard error of the difference in terms of the weighted estimate iss ˆp1 ˆp 2 Ap q ¢ 1 n 1 1 n 2≤The formula for the test value is shown next.Formula for the z Test for Comparing Two Proportionswherez pˆ 1 pˆ 2 p 1 p 2 A p q ¢ 1 n 1 1 n 2≤p X 1 X 2n 1 n 2pˆ 1 X 1n 1q 1 p pˆ 2 X 2n 2This formula follows the formatTest value observed value expected valuestandard errorAssumptions for the z Test for Two Proportions1. The samples must be random samples.2. The sample data are independent of one another.3. For both samples np 5 and nq 5.Example 9–9Vaccination Rates in Nursing HomesIn the nursing home study mentioned in the chapter-opening Statistics Today, theresearchers found that 12 out of 34 small nursing homes had a resident vaccinationrate of less than 80%, while 17 out of 24 large nursing homes had a vaccination rate9–35