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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com 1.2 Characterizing a Set of Measurements: Graphical Methods 5 selection of these items is somewhat at the discretion of the person who is involved in the construction. Although they are arbitrary, a few guidelines can be very helpful in selecting the intervals. Points of subdivision of the axis of measurement should be chosen so that it is impossible for a measurement to fall on a point of division. This eliminates a source of confusion and is easily accomplished, as indicated in Figure 1.1. The second guideline involves the width of each interval and consequently, the minimum number of intervals needed to describe the data. Generally speaking, we wish to obtain information on the form of the distribution of the data. Many times the form will be mound-shaped, as illustrated in Figure 1.2. (Others prefer to refer to distributions such as these as bellshaped, or normal.) Using many intervals with a small amount of data results in little summarization and presents a picture very similar to the data in their original form. The larger the amount of data, the greater the number of included intervals can be while still presenting a satisfactory picture of the data. We suggest spanning the range of the data with from 5 to 20 intervals and using the larger number of intervals for larger quantities of data. In most real-life applications, computer software (Minitab, SAS, R, S+, JMP, etc.) is used to obtain any desired histograms. These computer packages all produce histograms satisfying widely agreed-upon constraints on scaling, number of intervals used, widths of intervals, and the like. Some people feel that the description of data is an end in itself. Histograms are often used for this purpose, but there are many other graphical methods that provide meaningful summaries of the information contained in a set of data. Some excellent references for the general topic of graphical descriptive methods are given in the references at the end of this chapter. Keep in mind, however, that the usual objective of statistics is to make inferences. The relative frequency distribution associated with a data set and the accompanying histogram are sufficient for our objectives in developing the material in this text. This is primarily due to the probabilistic interpretation that can be derived from the frequency histogram, Figure 1.1. We have already stated that the area of a rectangle over a given interval is proportional to the fraction of the total number of measurements falling in that interval. Let’s extend this idea one step further. If a measurement is selected at random from the original data set, the probability that it will fall in a given interval is proportional to the area under the histogram lying over that interval. (At this point, we rely on the layperson’s concept of probability. This term is discussed in greater detail in Chapter 2.) For example, for the data used to construct Figure 1.1, the probability that a randomly selected measurement falls in the interval from 2.05 to 2.45 is .5 because half the measurements fall in this interval. Correspondingly, the area under the histogram in Figure 1.1 over the interval from FIGURE 1.2 Relative frequency distribution Relative Frequency 0 2.05 2.25 2.45 2.65 2.85 3.05

www.downloadslide.com 6 Chapter 1 What Is Statistics? 2.05 to 2.45 is half of the total area under the histogram. It is clear that this interpretation applies to the distribution of any set of measurements—a population or a sample. Suppose that Figure 1.2 gives the relative frequency distribution of profit (in millions of dollars) for a conceptual population of profit responses for contracts at specified settings of the independent variables (size of contract, measure of competition, etc.). The probability that the next contract (at the same settings of the independent variables) yields a profit that falls in the interval from 2.05 to 2.45 million is given by the proportion of the area under the distribution curve that is shaded in Figure 1.2. Exercises 1.2 Are some cities more windy than others? Does Chicago deserve to be nicknamed “The Windy City”? Given below are the average wind speeds (in miles per hour) for 45 selected U.S. cities: 8.9 12.4 8.6 11.3 9.2 8.8 35.1 6.2 7.0 7.1 11.8 10.7 7.6 9.1 9.2 8.2 9.0 8.7 9.1 10.9 10.3 9.6 7.8 11.5 9.3 7.9 8.8 8.8 12.7 8.4 7.8 5.7 10.5 10.5 9.6 8.9 10.2 10.3 7.7 10.6 8.3 8.8 9.5 8.8 9.4 Source: The World Almanac and Book of Facts, 2004. a b c d Construct a relative frequency histogram for these data. (Choose the class boundaries without including the value 35.1 in the range of values.) The value 35.1 was recorded at Mt. Washington, New Hampshire. Does the geography of that city explain the magnitude of its average wind speed? The average wind speed for Chicago is 10.3 miles per hour. What percentage of the cities have average wind speeds in excess of Chicago’s? Do you think that Chicago is unusually windy? 1.3 Of great importance to residents of central Florida is the amount of radioactive material present in the soil of reclaimed phosphate mining areas. Measurements of the amount of 238 U in 25 soil samples were as follows (measurements in picocuries per gram): .74 6.47 1.90 2.69 .75 .32 9.99 1.77 2.41 1.96 1.66 .70 2.42 .54 3.36 3.59 .37 1.09 8.32 4.06 4.55 .76 2.03 5.70 12.48 Construct a relative frequency histogram for these data. 1.4 The top 40 stocks on the over-the-counter (OTC) market, ranked by percentage of outstanding shares traded on one day last year are as follows: 11.88 6.27 5.49 4.81 4.40 3.78 3.44 3.11 2.88 2.68 7.99 6.07 5.26 4.79 4.05 3.69 3.36 3.03 2.74 2.63 7.15 5.98 5.07 4.55 3.94 3.62 3.26 2.99 2.74 2.62 7.13 5.91 4.94 4.43 3.93 3.48 3.20 2.89 2.69 2.61 a b Construct a relative frequency histogram to describe these data. What proportion of these top 40 stocks traded more than 4% of the outstanding shares?

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