Solved: Suppose that a scientist has reason to believe that two quan tities and are

Chapter 14, Problem 55

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Suppose that a scientist has reason to believe that two quan tities and are related linearly, that is, , at least approximately, for some values of and . The scientist performs an experiment and collects data in the form of points , , , and then plots these points. The points dont lie exactly on a straight line, so the scientist wants to find constants and so that the line fits the points as well as possible (see the figure). Let be the vertical deviation of the point from the line. The method of least squares determines and 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 and . (See Section 1.2 for a further discussion and applications of the method of least squares.)

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