1. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 1 Section 10-4 Variation and Prediction Intervals

2. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 2 Key Concept In this section we present a method for constructing a prediction interval, which is an interval estimate of a predicted value of y.

3. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 3 Definition Assume that we have a collection of paired data containing the sample point (x, y), that y is the predicted value of y (obtained by using the regression equation), and that the mean of the sample y-values is y. ^

4. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 4 Definition The total deviation of ( x, y ) is the vertical distance y – y, which is the distance between the point ( x, y ) and the horizontal line passing through the sample mean y.

5. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 5 Definition The explained deviation is the vertical distance y – y, which is the distance between the predicted y- value and the horizontal line passing through the sample mean y. ^

6. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 6 Definition The unexplained deviation is 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 10-3.) ^ ^

7. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 7 Figure 10-7 Unexplained, Explained, and Total Deviation

8. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 8 (total deviation) = (explained deviation) + (unexplained deviation) ( y - y ) = ( y - y ) + (y - y ) ^ ^ (total variation) = (explained variation) + (unexplained variation)  ( y - y ) 2 =  ( y - y ) 2 +  (y - y) 2 ^^ Formula 10-5 Relationships

9. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 9 Definition r2 =r2 = explained variation. total variation The value of r 2 is the proportion of the variation in y that is explained by the linear relationship between x and y. Coefficient of determination is the amount of the variation in y that is explained by the regression line.

10. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 10 A prediction interval, is an interval estimate of a predicted value of y. Definition

11. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 11 The standard error of estimate, denoted by s e is 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. Definition ^

12. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 12 Standard Error of Estimate s e = or s e =  y 2 – b 0  y – b 1  xy n – 2 Formula 10-6  ( y – y ) 2 n – 2 ^

13. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 13 Use Formula 10-6 to find the standard error of estimate s e for the paired pizza/subway fare data listed in Table 10-1in the Chapter Problem. n = 6  y 2 = 9.2175  y = 6.35  xy = 9.4575 b 0 = 0.034560171 b 1 = 0.94502138 s e = n - 2  y 2 - b 0  y - b 1  xy s e = 6 – 2 9.2175 – (0.034560171)(6.35) – (0.94502138)(9.4575) Example: s e = 0.12298700 = 0.123

14. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 14 y - E < y < y + E ^ ^ Prediction Interval for an Individual y where E = t   2 s e n(x2)n(x2) – (  x) 2 n(x0 – x)2n(x0 – x)2 1 + + 1 n x 0 represents the given value of x t   2 has n – 2 degrees of freedom

15. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 15 E = t  2 s e + n(  x 2 ) – (  x) 2 n(x0 – x)2n(x0 – x)2 1 + 1 n E = (2.776)(0.12298700) 6(9.77) – (6.50) 2 6(2.25 – 1.0833333) 2 1 + 1 6 E = (2.776)(0.12298700)(1.2905606) = 0.441 Example: For the paired pizza/subway fare costs from the Chapter Problem, we have found that for a pizza cost of $2.25, the best predicted cost of a subway fare is $2.16. Construct a 95% prediction interval for the cost of a subway fare, given that a slice of pizza costs $2.25 (so that x = 2.25). +

16. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 16 Example: Construct the confidence interval. y – E < y < y + E 2.16 – 0.441 < y < 2.16 + 0.441 1.72 < y < 2.60 ^ ^

17. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 17 Recap In this section we have discussed:  Explained and unexplained variation.  Coefficient of determination.  Standard error estimate.  Prediction intervals.