uppose that a scientist has reason to believe that two quantities and are related

Chapter 14, Problem 53

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uppose that a scientist has reason to believe that two quantities 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.) m b y mx Suppose that a scientist has reason to believe that two quantitiesand are related linearly, that is, , at leastapproximately, for some values of and . The scientist performsan experiment and collects data in the form of points, , , and then plots these points. Thepoints dont lie exactly on a straight line, so the scientist wantsto find constants and so that the line fits thepoints as well as possible. (See the figure.)m b y mx bx1, y1 x2, y2 . . . , xn, yn m bx y y mx b23p q r 1 Pp q rP 2pq 2pr 2rqLLet be the vertical deviation of the pointfrom the line. The method of least squares determinesand so as to minimize , the sum of the squares ofthese deviations. Show that, according to this method, the lineof best fit is obtained whenThus, the line is found by solving these two equations in thetwo unknowns and . (See Section 1.2 for a further discussionand applications of the method of least squares.)