Question

Bonus question 7. Suppose that a scientist has reason to believe that two quantities and y are related linearity, that is, y = mc + b, at least approximately, for some values of m and b. The scientist performs an experiment and collects data in the form of points (1,1), (22, y2),..., nyn), and then plots these points. The points don't lie exactly on a straight line, so the scientist wants to find the constants m and b so that the line ymx+b "fits" the points as well as possible. Let d, =% - (mzi + b) be the vertical deviation of the point ( y) from the line. The method of least squares determines m and b so as to minimise the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when Through solving these two equations for m and b we may find the line which best fits this data.

Answer 1

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,y1), (x2,y2),...........(xn,yn), and then plots these points.

Boros question 7. scientist hoy reason to Suppose that a believe tot two quantities de and y are sebated lineceonly, that is, y = netb, at least approximately, for some values of m and b. m2 to ²a, a {anya 1 Consider the function, - ffm, b) = { dat f (m, b) = mg +6 - (- mby boy The objective us to minimize the function f. The function, fm, b) point (m, b) minimizes the Then of zo

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Ho - bn so 1 1 :n 2 ч, sm , + Hence, the rouixed calations for the two lines 29-m{xs und and So, tnb and 992-