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

Question:

We would like to fit the quadratic curve y = Î²1 + Î²2x + Î²3x2 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 Î²1, Î²2, and Î²3 equal to 0, show that Î²1, Î²2, and Î²3 satisfy the following set of equations (called normal equations), all of which are sums going from 1 to n:

We would like to fit the quadratic curve y =

(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).