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, y₂), (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, y;) from the line. The method of least squares determines m and b so as to minimize 

the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when

Thus the line is found by solving these two equations in the two unknowns m and b. 

Transcribed Image Text:

y. 0 (X₁,Y₁). (xi, Yi) di{L mx¡ + b X