We would like to fit the quadratic curve y = 1 + 2x + 3x2 to a

We would like to fit the quadratic curve y = β1 + β2x + β3x2 to a set of points (x1, y1), (x2, y2), . . . , (xn, yn) by the method of least squares. To do this, let

(a) By setting the three first partial derivatives of h with respect to β1, β2, and β3 equal to 0, show that β1, β2, and β3 satisfy the following set of equations (called normal equations), all of which are sums going from 1 to n:

(b) For the data

Show that
a = ˆ’1.88, b = 9.86, and c = ˆ’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 is 6y = ˆ’1.88 + 9.86x ˆ’ 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).

Transcribed Image Text:

xi yi; = xǐ yi. (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, ΣΧί = 133.34, ΣΧι = 867.75, ΣΧ = 6197.21, ΣΧ-46,318.88, Σνί= 404.22. ΣΧΟǐ = 2138.38. and 2v-13.380.30