During the lecture using the least squares method for the Simple Linear Regression Model we have derived the following normal equations: N N, D Yi=Nbo+bi ) X i=1 i=1 N N N Y XYi=bo) Xi+b) X7, i=1 i=1 i=1 where by and b; are the least squares estimators of 5y and f;, respectively. These estimators, the corresponding regression line ?; = by + b1 X;, and model residuals e, =Y, f}, for i = 1,..., N satisfy the foowing properties: (1) The sum of the residuals is zero: N ) i=1 2 (2) The sum of the square residuals E;i] e; is minimized, i.e., for all ay R and a; R, N Ea NV X ) = (%-a,gal .,',). =1 =1 (3) The sum of the observed values Y; equals the sum of the fitted values ?g (4) The sum of the residuals weighted by the predictors Xj is zero N :E::;K}Bizz 0. i=1 (5) The sum of the residuals weighted by the fitted value of the response variables Y; is N Zi:iieg==0. =1 (6) The regression line always goes through the point (X z Y), where X = % E;*:] X; and Y= % Zi\\;l Y. Zero Problem 1 (25p) (Derivation of the Properties of the Least Squares Solution) Please derive mathematically the Properties (1)-(6) listed above using the assumptions of the Linear Regression Model and the Normal Equations