152 CHAPTER 3 Linear and Quadratic Functions Select two other points and complete the solution. Graph the line on the scatter plot obtained in Figure 6. Now Work PROBLEMS 11(b) AND (c) (b) Figure 9 shows the scatter plot with the graph of the line found in part (a). 3 Use a Graphing Utility to Find the Line of Best Fit The model obtained in Example 3 depends on the selection of points, which varies from person to person. So, the model that we found might be different from the model you found. Although the model in Example 3 appears to fit the data well, there may be a model that “fits them better.” Do you think your model fits the data better? Is there a line of best fit? As it turns out, there is a method for finding a model that best fits linearly related data (called the line of best fit).* *We do not discuss the underlying mathematics of lines of best fit in this text. Finding the Line of Best Fit for Linearly Related Data Use the data in Table 6 from Example 1. (a) Use a graphing utility to find the line of best fit that models the relation between on-base percentage and runs scored. (b) Graph the line of best fit on the scatter plot obtained in Example 1(b). (c) Interpret the slope. (d) Use the line of best fit to predict the number of runs a team will score if their on-base percentage is 32.2. EXAMPLE 4 Figure 10(a) Linear regression Solution (a) Graphing utilities contain built-in programs that find the line of best fit for a collection of points in a scatter plot. Executing the LINear REGression program on a TI-84 Plus CE provides the results shown in Figure 10(a). This output shows the equation = + y ax b, where a is the slope of the line and b is the y-intercept. The line of best fit that relates on-base percentage to runs scored may be expressed as the line = − y x 57.35 1120.75 The Model Figure 9 = − y x 55 1042.5 (31.5, 690) (33.1, 778) 0 29.5 30.0 30.5 31.0 31.5 32.0 32.5 33.0 33.5 34.0 On-base Percentage Runs Scored versus On-base Percentage in the National League Runs Scored 825 800 775 750 725 700 625 850 600 675 650 x y
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