Introductory Statistics. Lecture 17, Sections 9.1, 9.2

A Confidence Interval for the Difference of two Proportions Math 140 Introductory Statistics „ A very common and important situation involves taking two samples independently from two different populations with the goal of estimating the size of the difference between the proportion of successes in one population and the proportion of successes in the other. Lecture 17 Sections 9.1, 9.2 1 2 Example Intuitive Formula „ A recent poll of 29,700 U.S. households „ A confidence interval for the difference of two proportions, p1 − p2, where p1 is the proportion of successes in the first population and p2 is the proportion of successes in the second population: found that 63% owned a pet. The percentage in 1994 was 56%. [Source: American Pet Products Manufacturers Association, www.appma.org.] „ The two populations are the households in (pb1 ¡ pb2 ) § z ¤ ¢ standard error of(pb1 ¡ pb2 ) the United States in 1994 and the households now. The question you will investigate is: „ “What was the change in the percentage of U.S. households that own a pet?” pb11 and pb22 are the proportions of successes in the two samples. „ In our example, the 95% confidence interval for the difference between the proportion of U.S. households that own pets now and the proportion that owned pets in 1994: (0:63 ¡ 0:56) § 1:96 ¢ standard error of(pb1 ¡ pb2 ) 3 4 Standard Error of the Difference The Formula „ The standard errors of the distributions of pb1 „ The confidence interval for the difference, p1 − p2 , of the proportion of successes in one and pb2 can be estimated respectively as: population and the proportion of successes in the second population is, s pb1 (1 ¡ pb1 ) pb2 (1 ¡ pb2 ) (pb1 ¡ pb2 ) § z ¤ ¢ + n1 n2 s s pb1 (1 ¡ pb1 ) and n1 pb2 (1 ¡ pb2 ) n2 „ And if we assume the proportion samples are independent, then the standard error of the difference can be approximated as „ Where pb11 is the proportion of successes in a random sample of size n1 taken from the first population, and pb2 is the proportion of successes in a random sample of size n2 taken from the second population. (Sample sizes do not need to be equal.) s pb1 (1 ¡ pb1 ) pb2 (1 ¡ pb2 ) + n1 n2 5 6 Conditions for use Example: A Difference in Pet Ownership? „ The conditions that must be met in order to „ A recent pet ownership survey found that use this formula are that 63% of the 29,700 U.S. households sampled own a pet. A 1994 survey, taken by the same organization, found that 56% of the 6,786 U.S. households sampled owned a pet. Find and interpret a 95% confidence interval for the difference between the proportion of U.S. households that own a pet now and the proportion of U.S. households that owned a pet in 1994. the two samples are taken randomly and independently from two populations. „ each population is at least 10 times as large as its sample size. b1, n1(1−p b1), n2p b 2, and n2(1−pb2) are all „ n1p „ at least 5. 7 8 The Formula for the Confidence Interval Wireless Versus Wired „ The Mobile Difference is a report aimed at describing how behaviors differ among those embracing wireless connectivity and those who are still happy with traditional desktop computing. Developed by the Pew Internet and American Life Project, the study used random digit dialing to select samples that covered various types of technology users within two main classes, those “motivated by mobility” and those for whom “stationary media will do.” Here, four types are studied, two from each main class, with their definitions: „ A survey of 1200 households found that 45% of households own a dog, 34% own a cat, and 20% own both. „ „ „ a. What percentage of households in the survey own neither a cat nor a dog? b. Are owning a dog and owning a cat independent events? c. Should you use the techniques of this section to estimate the difference in the percentage of households that own a dog and the percentage that own a cat? Explain. 9 10 Wireless Versus Wired Wireless Versus Wired „ Digital Collaborators (DC). With the most tech assets, „ With regard to getting health information online, Digital Collaborators use them to work with and share their creations with others. They are enthusiastic … and confident in how to manage digital devices and information. „ Media Movers (MM). Media Movers have a wide range of online and mobile habits, and they are bound to find or create an information nugget, such as a digital photo, and pass it on. „ With regard to getting health information online, estimate the difference between the population proportions for DC and MM types in a 90% confidence interval. Interpret this interval. estimate the difference between the population proportions for DC and MM types in a 90% confidence interval. Interpret this interval. 11 12 Significance Test for the Difference of two Proportions Components of a Significance Test for the Difference of two Proportions „ Often times we need to decide which is the „ See pages 435-436 in your book. greater of two proportions, or whether we can assume they are the same. For example. Is snowboarding or skiing more likely to result in a serious injury? „ Does a new treatment for AIDS result in fewer deaths than an old treatment? „ Is Reggie Jackson’s World Series record so much better than his play during the regular season that the difference can’t reasonably be attributed to chance? „ 13 14 Are men more likely to be left-handed? Are men more likely to be left-handed? „ Using a sampling procedure that is typical of „ The survey just described found that 113 of 1067 randomly selected males were lefthanded while 92 of 1170 randomly selected females were left-handed. Perform a significance test to determine whether males are more likely to be left-handed than females. many such surveys, the NHANES survey did not randomly sample 1067 males and then randomly sample 1170 females to ask about handedness. It randomly (by a somewhat complex design) selected a large sample of adults from across the country and then sorted out the males and females. The resulting two samples can still be considered to be independent and random, as long as the sample sizes are large. 15 16