ch15regressionline

ch15regressionline - Chapter 15: Regression Line

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Chapter 15: Regression Line Earlier, each problem considered just one variable, say  x =weight, in kg.  Real-life problems, however, often involve several  variables.  e.g. a patient’s condition,  may be dependent on his  temperature,  x , his pulse,  u , and his blood pressure,  v . How to discover the relationship?  Let us just look at the simplest case of two variables,  x   and  y We want to know how  y  depends on  x .
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Example 1 : Let x = baldness, in % and y = assets, in $m. Suppose four men yield the following paired data: (a) Summarize the data into a linear model . (b) A 5 th  man is observed to be 60% bald.  Estimate his assets. ) , ( ), , ( ), , ( ), , ( 4 4 3 3 2 2 1 1 y x y x y x y x
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Solution : First, draw a scatter diagram to examine initially the relationship between x and y.
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Secondly, seeing that the data lie roughly along a straight line, we try using a linear model: where “a” is the “y-intercept” and “b” is the “gradient”. Note that while     is the actual asset of the  i- th person,             is the theoretical asset of the  i- th person when     is plugged  into equation (1). Thirdly, we propose to find the most suitable  and  b  so that  the line (1)                is closest to the data.  This is going to be the “line of best fit.” bx a y + = ˆ ) 1 ..( .......... ˆ bx a y + = i i bx a y + = ˆ i x i y
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To do this, we first set up n linear equations: We have more equations than the number of unknowns  two ( a  and  b ).  Multiply each equation by 1 ( the coefficient of  a ), and  add; Then multiply each equation by (the coefficient of  b ), and  add; ) 2 ....( .......... ) ( b x na y i i + = ) 3 ....( .......... ) ( ) ( 2 b x a x y x i i i i + =
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(2) and (3) are the two normal equations. To solve 
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ch15regressionline - Chapter 15: Regression Line

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