SECTION 8.4 Testing the Difference Between Proportions 449 Sample Statistics for Seat Belt Use Washington Vermont n1 = 200 np 1 = 0.930 x1 = 186 n2 = 250 np 2 = 0.888 x2 = 222 A Two-Sample z-Test for the Difference Between Proportions A study of 200 randomly selected people driving vehicles in Washington and 250 randomly selected people driving vehicles in Vermont shows that 93.0% of people driving vehicles in Washington and 88.8% of people driving vehicles in Vermont wear seat belts while driving. At a = 0.10, can you reject the claim that the proportion of people who wear seat belts while driving is the same for Washington and Vermont? (Adapted from National Highway Traffic Safety Administration) SOLUTION The samples are random and independent. Also, the weighted estimate of p1 and p2 is p = x1 + x2 n1 + n2 = 186 + 222 200 + 250 = 408 450 ≈ 0.9067 and the value of q is q = 1 - p ≈ 1 - 0.9067 = 0.0933. Because n1p ≈ 20010.90672, n1q ≈ 20010.09332, n2p ≈ 25010.90672, and n2q ≈ 250 0.09332 are at least 5, you can use a two-sample z@test. The claim is “the proportion of people who wear seatbelts while driving is the same for Washington and Vermont.” So, the null and alternative hypotheses are H0: p1 = p2 (Claim) and Ha: p1 ≠ p2. Because the test is two-tailed and the level of significance is a = 0.10, the critical values are -z0 = -1.645 and z0 = 1.645. The rejection regions are z 6 -1.645 and z 7 1.645. The standardized test statistic is z = 1 np 1 - np 22 - 1p1 - p22 B p qa 1 n1 + 1 n2b ≈ 10.930 - 0.8882 - 0 B 10.9067210.09332a 1 200 + 1 250b ≈ 1.52. The figure below shows the location of the rejection regions and the standardized test statistic z. Because z is not in the rejection region, you fail to reject the null hypothesis. 1 − = 0.90 z ≈ 1.52 α α= 0.05 1 2 α= 0.05 1 2 z 0 −1 −3 −2 1 2 3 z0 = 1.645 −z0 = −1.645 Interpretation There is not enough evidence at the 10% level of significance to reject the claim that the proportion of people who wear seatbelts while driving is the same for Washington and Vermont. TRY IT YOURSELF 1 Consider the results of the study discussed on page 417. At a = 0.05, can you support the claim that there is a difference between the proportion of people who use yoga who are 40- to 49-year-olds and the proportion of people who do not use yoga who are 40- to 49-year-olds? Answer: Page A41 Study Tip To find x1 and x2, use x1 = n1 np 1 and x2 = n2 np 2. See TI-84 Plus steps on page 465. EXAMPLE 1
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