7-2 Estimating a Population Mean 335 With x = 8.8057 g and E = 0.215589 g, we construct the confidence interval as follows: x - E 6 m 6 x + E 8.8057 - 0.215589 6 m 6 8.8057 + 0.215589 8.5901g 6 m 6 9.0213g 1rounded to four decimal places2 INTERPRETATION We are 95% confident that the limits of 8.5901 g and 9.0213 g actually do contain the value of the population mean m. If we were to collect many different random samples of 6 Reese’s Peanut Butter Cup Miniatures and find the mean weight of each sample, about 95% of the corresponding confidence intervals should contain the mean weight of all such peanut butter cups. We noted earlier that the sample is from a bag of 38 cups and the bag is labeled as containing a total weight of 340.2 g, so the mean weight of a cup should be 340.2 g>38 = 8.953 g. The confidence interval limits do contain the desired mean of 8.953 g, so the package appears to have been filled with candy in an acceptable way. (It would have been better to obtain a simple random sample of six Reese’s miniature peanut butter cups from six different bags from different regions of the country, but the author didn’t have time for that—he needed to rotate his car tires.) YOUR TURN. Do Exercise 13 “Archeology.” Bootstrap Resampling for Constructing Confidence Intervals Section 7-4 describes the method of bootstrap resampling for constructing a confidence interval estimate of a population parameter. The basic approach is to use technology such as Statdisk to “resample” the sample data many times (such as 1000), then use the sorted list of 1000 results to find the confidence interval. If we repeat Example 2 using the bootstrap resampling method, here is a typical result: 8.6551 g 6 m 6 8.9468 g. Because of the randomness used in the procedure, the resulting confidence interval may differ somewhat. EXAMPLE 3 Critical Thinking: Sales of Vinyl Records Listed below are sales (millions) of vinyl LP units in the United States. The sales numbers are listed in order by year beginning with 1993. a. Use the listed sample data to construct a 95% confidence interval estimate of the population mean. b. Is the confidence interval method described in this section a good tool for gaining insight into the nature of the sample data? c. Apart from the confidence interval, is there some other tool that would be better for gaining insight into the nature of the sample data? What is most notable about the sales numbers? 0.3 0.6 0.8 1.1 1.1 1.4 1.4 1.5 1.2 1.3 1.4 1.2 0.9 0.9 1.0 1.9 2.5 2.8 3.9 4.6 6.1 9.2 11.9 13.1 14.3 16.8 SOLUTION a. Using the methods of this chapter, the 95% confidence interval is found to be 2.02 million 6 m 6 5.92 million. continued Estimating Crowd Size There are sophisticated methods of analyzing the size of a crowd. Aerial photographs and measures of people density can be used with reasonably good accuracy. However, reported crowd size estimates are often simple guesses. After the Boston Red Sox won the World Series for the first time in 86 years, Boston city officials estimated that the celebration parade was attended by 3.2 million fans. Boston police provided an estimate of around 1 million, but it was admittedly based on guesses by police commanders. A photo analysis led to an estimate of around 150,000. Boston University Professor Farouk ElBaz used images from the U.S. Geological Survey to develop an estimate of at most 400,000. MIT physicist Bill Donnelly said that “it’s a serious thing if people are just putting out any number. It means other things aren’t being vetted that carefully.”
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