828 CHAPTER 12 Statistics Linear Regression Let’s now turn to regression. Linear regression is the process of determining the linear relationship between two variables. Our goal is to describe this relationship using a linear equation. Under certain conditions, if we are given a value for one variable we can use this linear equation to predict the corresponding value of the second variable. Using a set of bivariate data, we will determine the equation of the line of best fit. The line of best fit is also called the regression line, or the least squares line. The line of best fit is the line such that the sum of the squares of the vertical distances from the line to the data points (on the scatter diagram) is a minimum, as shown in Fig. 12.43. In Fig. 12.43, the line of best fit minimizes the sum of the squares of d1 through d .8 To determine the equation of the line of best fit, y mx b, = + we must determine m and then b. Recall from Section 6.6 that the slope–intercept form of a straight line is y mx b, = + where m is the slope and b (0, ) is the y-intercept. The formulas for determining m and b are as follows. x y x2 y2 xy 1 250 1 62,500 250 2 230 4 52,900 460 3 200 9 40,000 600 4 210 16 44,100 840 5 140 25 19,600 700 6 120 36 14,400 720 7 210 49 44,100 1470 8 100 64 10,000 800 9 90 81 8100 810 10 85 100 7225 850 55 1635 385 302,925 7500 Now try Exercise 19 = Σ − Σ Σ Σ − Σ Σ − Σ = − − − = − ≈ − ≈ − r n xy x y n x x n y y ( ) ( )( ) ( ) ( ) ( ) ( ) 10(7500) (55)(1635) 10(385) (55) 10(302,925) (1635) 14,925 825 356,025 14,925 17,138.28 0.871 2 2 2 2 2 2 From Table 12.9, for n 10 = and 0.05, α= we get 0.632. Since 0.871 0.871 − = and 0.871 0.632, > a correlation exists. The correlation is negative, which indicates that the longer the time period, the smaller is the amount of drug remaining. 7 y x d1 d2 d3 d4 d5 d6 d7 d8 Figure 12.43 The Line of Best Fit The equation of the line of best fit is = + = Σ − Σ Σ Σ − Σ = Σ − Σ y mx b m n xy x y n x x b y m x n , where ( ) ( )( ) ( ) ( ) and ( ) 2 2
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