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1 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman C.M. Pascual S TATISTICS Section 9c Variation and Prediction Intervals

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2 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions

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3 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions Total Deviation from the mean of the particular point ( x, y ) the vertical distance y - y, which is the distance between the point ( x, y ) and the horizontal line passing through the sample mean y

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4 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions Total Deviation from the mean of the particular point ( x, y ) the vertical distance y - y, which is the distance between the point ( x, y ) and the horizontal line passing through the sample mean y Explained Deviation the vertical distance y - y, which is the distance between the predicted y value and the horizontal line passing through the sample mean y ^

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5 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions Total Deviation from the mean of the particular point ( x, y ) the vertical distance y - y, which is the distance between the point ( x, y ) and the horizontal line passing through the sample mean y Explained Deviation the vertical distance y - y, which is the distance between the predicted y value and the horizontal line passing through the sample mean y Unexplained Deviation the vertical distance y - y, which is the vertical distance between the point ( x, y ) and the regression line. (The distance y - y is also called a residual, as defined in Section 9-3.) ^ ^ ^

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6 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Figure 9-9 Unexplained, Explained, and Total Deviation

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7 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Total deviation ( y - y ) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Unexplained deviation ( y - y ) Explained deviation ( y - y ) (5, 19) (5, 13) (5, 9) y = 3 + 2 x ^ y = 9 ^ ^ y x 0123456789 Figure 9-9 Unexplained, Explained, and Total Deviation

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8 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman ( y - y ) = ( y - y ) + (y - y ) (total deviation) = (explained deviation) + (unexplained deviation) ^ ^

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9 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman ( y - y ) = ( y - y ) + (y - y ) (total deviation) = (explained deviation) + (unexplained deviation) (total variation) = (explained variation) + (unexplained variation) ( y - y ) 2 = ( y - y ) 2 + (y - y) 2 ^ ^ ^ ^ Formula 9-5

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10 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition Coefficient of determination the amount of the variation in y that is explained by the regression line

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11 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition r 2 = explained variation total variation Coefficient of determination the amount of the variation in y that is explained by the regression line

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12 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition r 2 = explained variation. total variation r 2 = b SS xy /Ss yy Where b = SS xy /SS xx Coefficient of determination the amount of the variation in y that is explained by the regression line

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13 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Prediction Intervals Definition Standard error of estimate

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14 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman a measure of the differences (or distances) between the observed sample y values and the predicted values y that are obtained using the regression equation Prediction Intervals ^ Definition Standard error of estimate

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15 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Standard Error of Estimate

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16 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Standard Error of Estimate s e = ( y - y ) 2 n - 2 y 2 - b 0 y - b 1 xy s e = n - 2 or Formula 9-6 ^

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17 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Standard Error of Estimate s e = ( y - y ) 2 n - 2 ^ = [SS yy – b SS xy /(n – 2)] 0.5

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18 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example 3 Using the problem in Chapter 9, determine the standard error (Se) and coefficient of determination (r 2 ) Solution: Getting the following data: SSxx = 362 ; SSxy = 22.7 ; SSyy = 2.78 b = 0.063 Solving for standard error (Se) Se = [(SSyy-b*SSxy)/(n-2)] 0.5 Se = [((2.78 – (0.063)(22.7))/(8-2) ] 0.5 = 1.59

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19 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example 3 Solution: Getting the following data: SSxx = 362 ; SSxy = 22.7 ; SSyy = 2.78 b = 0.063 Solving for Coefficient of determination (r 2 ) r 2 =(b*SSxy)/(SSyy) = (0.063*22.7)/(2.78) = 0.514

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