336 CHAPTER 7 Estimating Parameters and Determining Sample Sizes b. A check of the requirements will show that the sample is small 1n … 302 and the data do not appear to be from a population having a normal distribution, so the requirements are not satisfied. The confidence interval is not a good tool for gaining insight into the nature of the data. c. Because the sales numbers are listed in order by year, a time series graph should be helpful in revealing the nature of the data. The accompanying time series graph clearly shows this notable feature: There is a distinct pattern of increasing sales over recent years. This shows that the population is changing over time, but the confidence interval does not reveal that trend. This also shows that instead of blindly applying statistical methods, we should always think about what we are doing! YOUR TURN. Do Exercise 19 “Mercury in Sushi.” Finding a Point Estimate and Margin of Error E from a Confidence Interval Technology and journal articles often express a confidence interval in a format such as 110.0, 30.02. The sample mean x is the value midway between those limits, and the margin of error E is one-half the difference between those limits (because the upper limit is x + E and the lower limit is x - E, the distance separating them is 2E). Point estimate of m: x = (upper confidence limit) + (lower confidence limit ) 2 Margin of error: E = (upper confidence limit) - (lower confidence limit) 2 For example, the confidence interval 110.0, 30.02 yields x = 20.0 and E = 10.0. Using Confidence Intervals to Describe, Explore, or Compare Data In some cases, confidence intervals might be among the different tools used to describe, explore, or compare data sets, as in the following example. EXAMPLE 4 Second-Hand Smoke Figure 7-6 shows graphs of confidence interval estimates of the mean cotinine level in each of three samples: (1) people who smoke; (2) people who don’t smoke but are exposed to tobacco smoke at home or work; (3) people who don’t smoke and are not exposed to smoke. (These confidence intervals are based on samples of data taken from Data Set 15 “Passive and Active Smoke” in Appendix B.) Because cotinine is produced by the body when nicotine is absorbed, cotinine is a good indication of nicotine intake. Figure 7-6 helps us see the effects of second-hand smoke. In Figure 7-6, we see that the confidence interval for smokers does not overlap the other confidence intervals, so it appears that the mean cotinine level of smokers is different from that of the other two groups. The two nonsmoking groups have confidence Benchmarking Survey Questions Pew Research Center sometimes includes “benchmarking questions” designed to help confirm the accuracy of survey data. Benchmarking questions often involve lifestyle characteristics for which government statistics are known. For example, a survey might include the question, “Are you married?”. The U.S. Census Bureau reports that 48% of adults in the United States are married, so the responses to the survey question can be compared to the benchmark of 48%. Serious questions would be raised about a survey in which the percentage of respondents saying they are married differs from 48% by a significant amount. One common solution is to use weighting so that underrepresented groups are correctly accounted for. th f
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