HAPTER 16 Confidence Intervals: The Basics EXAMPLE 16.1 Body Mass Index of Young Men . . . . . . . . . . . . . . . . . . An NHANES report gives data for 936 men aged 20-20 years." The mean BMI of these 936 men was x = 27.2. On the basis of this sample, we want to estimate mean BMI it in the population of all 23.2 million American men in this age group To match the "simple conditions," we will treat the NHANES sample as an SR'S from a Normal population, and we will assume that we know that the standard devia. tion o = 11.6. (The sample standard deviation for these 936 men is 11.63 kg/m'. For purposes of the example, we round this to 11.6 and act as though this is the popul;. tion standard deviation o.) Here is the reasoning of statistical estimation in a nutshell: 1. To estimate the unknown population mean BMI /, use the mean x = 27.2 of the random sample. We don't expect x to be exactly equal to /, so we want to say how accurate this estimate is. 2. We know the sampling distribution of x. In repeated samples, x has the Normal 16 distribution with mean / and standard deviation o / Vn . So the average BMI of an SRS of 936 young men has standard deviation 11.6 =0.4 (rounded off) 1936 3. The 95 part of the 68-95-99.7 rule for Normal distributions says that x is within 2 standard deviations of the mean u in 95% of all samples. The standard deviation is 0.4, so 2 standard deviations is 0.8. Thus, for 95% of all samples of size 936, the distance between the sample mean x and the population mean / is WORLD less than 0.8. If we estimate that / lies somewhere in the interval from x - 0.8 to x + 0.8, we'll be right for 95% of all possible samples. For this particular sample, at this interval is . The nist x - 0.8 = 27.2 - 0.8= 26.4 91amphibnoo slamia" y of his to omic x + 0.8=27.2 +0.8=28.0 th, 4. Because we got the interval 26.4 to 28.0 from a method that captures the popula- plied: tion mean for 95% of all possible samples, we say that we are 95% confident that one the mean BMI / of all young men is some value in that interval-no lower than 26.4 and no higher than 28.0. The big idea is that the sampling distribution of x tells us how close to the sample mean x is likely to be. Statistical estimation just turns that informa- tion around to say how close to x the unknown population mean / is likely to interval for M. be. We call the interval of numbers between the values x 1 0.8 a 95% confidenceice, Table C gives N n = 1.645 x 2 \\ 2 m = 43.3 Because 43 patrons will give a slightly larger margin of error than desired, and 44 patrons will give a slightly smaller margin of error, we must observe 44 patrons. Always round up to the next higher whole number when finding n. APPLY YOUR KNOWLEDGE 18.12 Body Mass Index of Young Men. Example 16.1 (page 368) assumed that the body mass index (BMD) of all American young men follows a Normal distribution with standard deviation = 11.6 kg/m'. How large a sample would be needed to estimate the mean BMI / in this population to within 1 1 with 95% confidence? 18.13 Number Skills of Eighth-Graders. Suppose that scores on the mathemat ics part of the National Assessment of Educational Progress (NAEP) test for eighth-grade students follow a Normal distribution with standard deviation o = 40. You want to estimate the mean score within 1 with 90% confidence. How large an SRS of scores must you choose? Studies: The Power of