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

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

3 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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

4 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 ^

5 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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.) ^ ^ ^

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

7 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Total deviation ( y - y ) Unexplained deviation ( y - y ) Explained deviation ( y - y ) (5, 19) (5, 13) (5, 9) y = x ^ y = 9 ^ ^ y x Figure 9-9 Unexplained, Explained, and Total Deviation

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

9 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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

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

11 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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

12 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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

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

14 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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

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

16 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 ^

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

18 Chapter 9. Section 9-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 = 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

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