chapter6m211 - Math 211 Introduction to Statistics Chapter...

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Math 211 Introduction to Statistics Chapter 6 Curve Fitting and the Method of Least Squares Relationship between variables Example 1. Weights of adults depend on some degree on their heights. The circumferences of circles depend on their radii. The pressure of a given mass of gas depends on its temperature and volume. This relation can be expressed in mathematical form by determining an equation that connects the variables. Curve Fitting To determine an equation, Step1 . Collect data that shows corresponding values of the variables (for example; x denotes the height and denotes the weight.) Then choose a sample of N individuals N x x x ,..., , 2 1 , and . ,..., , 2 1 N y y y Step2. plot points ( 29 ( 29 ( 29 N N y x y x y x , ,...... , , 2 , 2 1 on a rectangular coordinate system. The resulting figure is called ‘ SCATTER DIAGRAM’. From the scatter diagram, it is often possible to see a curve approximating data. Such a curve is called ‘ APPROXIMATE CURVE’ . In general, the problem of finding equations of approximating curves that fit given set of data is called ‘ CURVE FITTING’. . Sonuc Zorlu Lecture Notes 1
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Math 211 Introduction to Statistics Equations of Approximating Curves Straight line : X a a Y 1 0 + = For any two points ( 29 1 1 , y x and ( 29 2 2 , y x , the equation of the straight line is ( 29 . 1 1 2 1 2 1 x X x x y y y Y - - - = - Parabola or quadratic curve : 2 2 1 0 X a X a a Y + + = n-degree curve : N N X a X a X a a Y + + + + = ...... 2 2 1 0 The method of Least Squares Definition. {Best fitting curve} Of all curves approximating a given set of data points, the curve having the property that 2 2 2 2 1 ..... N D D D + + + is minimum is called a ‘BEST FITTING CURVE’ . The Least Squares Line The least squares line approximating the set of points ( 29 1 1 , y x , ( 29 2 2 , y x ,…., ( 29 N N y x , has the equation X a a Y 1 0 + = , REGRESSION LINE of Y on X , where the constants 0 a and 1 a are determined by solving the equation, + = + = 2 1 0 1 0 X a X a XY X a N a Y (X is independent variable). The above equations are NORMAL EQUATIONS for the least squares line. The constants 0 a and 1 a can be obtained using, ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 0 - - = X X N XY X X Y a ( 29 ( 29 ( 29 ( 29 ( 29 2 2 1 - - = X X N Y X XY N a Sonuc Zorlu Lecture Notes 2
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Math 211 Introduction to Statistics
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This note was uploaded on 08/12/2011 for the course AS 211 taught by Professor Yuceltandogdu during the Spring '11 term at Eastern Mediterranean University.

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chapter6m211 - Math 211 Introduction to Statistics Chapter...

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