lec7.2 - 189 Lecture 19: CHAPTER 7: CORRELATION AND SIMPLE...

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189 Lecture 19: CHAPTER 7: CORRELATION AND SIMPLE LINEAR REGRESSION The Least-Squares Line (Section 7.2, page 517) Last class, we took a look at regression analysis which uses information about x to draw some type of conclusion concerning y . The linear model is: i i i x y 1 0 where i y is called the dependent variable , i x is called the independent variable , 0 and 1 are the regression coefficients , and i is called the error . To compute the equation of the least-squares line: x y 1 0 ˆ ˆ ˆ  we must determine the values for the slope 1 ˆ and the intercept 0 ˆ that minimize the sum of the squared residuals n i i e 1 2 . As shown in my graph above, the residual is defined as the vertical distance from each data point ( i i y x , ) to the predicted point ( i i y x ˆ , ) on the least-squares line, given by i i i y y e ˆ . The point is, we want to minimize the sum of the squared residuals. How do we do this? We first solve for i e from our linear model:
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190
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191 Therefore, the equation of the least-squares line (or fitted value) is x y 1 0 ˆ ˆ ˆ  ; where Sxx Sxy 1 ˆ and x y 1 0 ˆ ˆ Example: Consider the previous example (from last lecture) of scores on two class quizzes. Sample data is (1,5), (1,4), (2,4), (4,2), (5,1), (5,2), (2,2), (4,4). In our calculation of 717 . 0 r , we also found that 3 x , 3 y , 20 xx S , 14 yy S and 12 xy S Find the equation of the least squares line: Scatter Plot:
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Note: The least squares line only fits the points used in its calculation. Do not be tempted to extend it below the smallest x- value or above the largest x-value. New observations with x-values outside this range may not fall anywhere close to the fitted line. So if a situation requires you to extend the line, do so with caution. Example: The data below gives the cost estimates and actual costs (in millions) of a random sample of 10 construction projects at a large industrial facility. Estimate(x) Actual(y) 44.277 51.174 2.737 9.683 7.004 14.827 22.444 22.159 18.843 26.537 46.514 50.281 3.165 15.550 21.327 23.896 42.337 50.144 7.737 13.567 The least squares regression line for this bivariate data is x y 937658 . 49228
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lec7.2 - 189 Lecture 19: CHAPTER 7: CORRELATION AND SIMPLE...

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