We observe that . That is, the predicted value at the sample mean is the sample mean of the dependent variable . This implies that the least-squares estimated line passes through the point . This point is at the intersection of the two dashed lines plotted on the graph in part (b) .
(d) The values of the least squares residuals, computed from , are:
161.71429
24−2.57143
3112.14286
49−2.14286
513−0.42857
6171.28571
(e) Their sum is and their sum of squares is
(f)
Exercise 2.4
(a) If the simple linear regression model becomes
(b) Graphically, setting implies the mean of the simple linear regression model passes through the origin (0, 0).
(c) To save on subscript notation we set The sum of squares function becomes
Figure xr2.4(a) Sum of squares for
The minimum of this function is approximately 25 and occurs at approximately The significance of this value is that it is the least-squares estimate.
(d) To find the value of b that minimizes we obtain
Setting this derivative equal to zero, we have
or
Exercise 2.4 (Continued)
Thus, the least-squares estimate is
which agrees with the approximate value of 2.7 that we obtained geometrically.
(e)
Figure xr2.4(b) Observations and fitted line
The fitted regression line is plotted in Figure xr2.4 (b). Note that the point does not lie on the fitted line in this instance.
(f) The least squares residuals, obtained from are:
Their sum is Note this value is not equal to zero as it was for
(g)