10-2 Regression 537 From Figure 10-6, we see that the residuals are -5, 11, -13, and 7, so the sum of their squares is 1-522 + 112 + 1-1322 + 72 = 364 We can visualize the least-squares property by referring to Figure 10-6, where the squares of the residuals are represented by the shaded square areas. The sum of the shaded square areas is 364, which is the smallest sum possible. Use any other straight line, and the shaded squares will combine to produce an area larger than the combined shaded area of 364. Fortunately, we need not deal directly with the least-squares property when we want to find the equation of the regression line. Calculus has been used to build the least-squares property into Formulas 10-3 and 10-4. Because the derivations of these formulas require calculus, we don’t include the derivations in this text. Residual Plots In this section and the preceding section we listed simplified requirements for the effective analyses of correlation and regression results. We noted that we should always begin with a scatterplot, and we should verify that the pattern of points is approximately a straight-line pattern. We should also consider outliers. A residual plot can be another helpful tool for analyzing correlation and regression results and for checking the requirements necessary for making inferences about correlation and regression. DEFINITION A straight line satisfies the least-squares property if the sum of the squares of the residuals is the smallest sum possible. DEFINITION A residual plot is a scatterplot of the 1x, y2 values after each of the y-coordinate values has been replaced by the residual value y - yn (where yn denotes the predicted value of y). That is, a residual plot is a graph of the points 1x, y - yn2. To construct a residual plot, draw a horizontal reference line through the residual value of 0, then plot the paired values of 1x, y - yn2. Because the manual construction of residual plots can be tedious, the use of technology is strongly recommended. Usefulness of a Residual Plot ■ A residual plot helps us determine whether the regression line is a good model of the sample data. ■ A residual plot helps us to check the requirement that for different values of x, the corresponding y values all have the same standard deviation. continued
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