Chapter 14.7, Problem 59E

Suppose that a scientist has reason to believe that two quantities x and y are related linearly, that is, y = mx + b, at least approximately, for some values of m and b. The scientist performs an experiment and collects data in the form of points (x₁, x₂), (x₂, y₂) ,…, (x_n, y_n) and then plots these points. The points don’t lie exactly on a straight line, so the scientist wants to find constants m and b so that the line y = mx + b “fits” the points as well as possible (see the figure).

Let d_i = y_i − (mx_i, + b) be the vertical deviation of the point (x_i, y_i) from the line. The method of least squares determines m and b so as to minimize ${\displaystyle {\sum }_{i=1}^n{d}_i^2}$, the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when

$m{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}+bn={\displaystyle \underset{i=1}{\overset{n}{\sum }}{y}_i}$

and $m{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i^2+b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}}={\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}{y}_i$