10-5 Nonlinear Regression 565 TABLE 10-7 Population (in millions) of the United States Year 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 2020 Coded Year 123456789101112 Population 5 10 17 31 50 76 106 132 179 227 281 335 TABLE 10-8 Models for the Population Data Model R2 Equation Quadratic 0.9995 y = 2.75x2 - 5.80x + 9.66 Exponential 0.9573 y = 5.7611.45x2 Power 0.9779 y = 3.26x1.79 Finding the Best Population Model EXAMPLE 1 Table 10-7 lists the population of the United States for different 20-year intervals. Find a mathematical model for the population size, then predict the size of the U.S. population in the year 2040. SOLUTION First, we “code” the year values by using 1, 2, 3, . . . , instead of 1800, 1820, 1840, . . . . The reason for this coding is to use values of x that are much smaller and much less likely to cause computational difficulties. 1. Look for a pattern in the graph. Examine the pattern of the data values in the TI-83>84 Plus display (shown in the margin), and compare that pattern to the generic models shown earlier in this section. The pattern of those points is clearly not a straight line, so we rule out a linear model. Good candidates for the model appear to be the quadratic, exponential, and power functions. 2. Find and compare values of R2. The TI-83>84 display for the quadratic model is shown in the margin. For the quadratic model, R2 = 0.9995 (rounded), which is quite high. Table 10-8 includes this result with results from two other potential models. In comparing the values of the coefficient R2, it appears that the quadratic model is best because it has the highest value of 0.9995. If we select the quadratic function as the best model, we conclude that the equation y = 2.75x2 - 5.80x + 9.66 best describes the relationship between the year x (coded with x = 1 representing 1800, x = 2 representing 1820, and so on) and the population y (in millions). Based on its R2 value of 0.9995, the quadratic model appears to be best, but the other values of R2 are also quite high. Our general knowledge of population growth might suggest that the exponential model is most appropriate. (With a constant birth rate and no limiting factors, population will grow exponentially.) To predict the U.S. population for the year 2040, first note that the year 2040 is coded as x = 13 (see Table 10-7). Substituting x = 13 into the quadratic model of y = 2.75x2 - 5.80x + 9.66 results in y = 399, which indicates that the U.S. population is estimated to be 399 million in the year 2040. TI-83, 84 Plus continued Clinical Trial Cut Short What do you do when you’re testing a new treatment and, before your study ends, you find that it is clearly effective? You should cut the study short and inform all participants of the treatment’s effectiveness. This happened when hydroxyurea was tested as a treatment for sickle cell anemia. The study was scheduled to last about 40 months, but the effectiveness of the treatment became obvious and the study was stopped after 36 months. (See “Trial Halted as Sickle Cell Treatment Proves Itself,” by Charles Marwick, Journal of the American Medical Association, Vol. 273, No. 8.)
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