We would like to fit the quadratic curve y = 1 + 2x + 3x2 to a set of points (x1, y1)

Chapter 7, Problem 7.6-10

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We would like to fit the quadratic curve y = 1 + 2x + 3x2 to a set of points (x1, y1), (x2, y2), ... , (xn, yn) by the method of least squares. To do this, let h(1, 2, 3) = n i=1 (yi 1 2xi 3x2 i ) 2. (a) By setting the three first partial derivatives of h with respect to 1, 2, and 3 equal to 0, show that 1, 2, and 3 satisfy the following set of equations (called normal equations), all of which are sums going from 1 to n: 1n + 2 xi + 3 x2 i = yi; 1 xi + 2 x2 i + 3 x3 i = xi yi; 1 x2 i + 2 x3 i + 3 x4 i = x2 i yi. (b) For the data (6.91, 17.52) (4.32, 22.69) (2.38, 17.61) (7.98, 14.29) (8.26, 10.77) (2.00, 12.87) (3.10, 18.63) (7.69, 16.77) (2.21, 14.97) (3.42, 19.16) (8.18, 11.15) (5.39, 22.41) (1.19, 7.50) (3.21, 19.06) (5.47, 23.89) (7.35, 16.63) (2.32, 15.09) (7.54, 14.75) (1.27, 10.75) (7.33,17.42) (8.41, 9.40) (8.72, 9.83) (6.09, 22.33) (5.30, 21.37) (7.30, 17.36) n = 25, xi = 133.34, x2 i = 867.75, x3 i = 6197.21, x4 i = 46,318.88, yi = 404.22, xiyi = 2138.38, and x2 i yi = 13,380.30. Show that a = 1.88, b = 9.86, and c = 0.995. (c) Plot the points and the linear regression line for these data. (d) Calculate and plot the residuals. Does linear regression seem to be appropriate? (e) Show that the least squares quadratic regression line is6y = 1.88 + 9.86x 0.995x2. (f) Plot the points and this least squares quadratic regression curve on the same graph. (g) Plot the residuals for quadratic regression and compare this plot with that in part (d).