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

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Presentation on theme: "Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10.1 - 1 Section 10-4 Variation and Prediction Intervals."— Presentation transcript:

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

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

8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved (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 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 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 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 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 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 =  y = 6.35  xy = b 0 = b 1 = s e = n - 2  y 2 - b 0  y - b 1  xy s e = 6 – – ( )(6.35) – ( )(9.4575) Example: s e = = 0.123

14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved 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) 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 E = t  2 s e + n(  x 2 ) – (  x) 2 n(x0 – x)2n(x0 – x) n E = (2.776)( ) 6(9.77) – (6.50) 2 6(2.25 – ) E = (2.776)( )( ) = 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 Example: Construct the confidence interval. y – E < y < y + E 2.16 – < y < < y < 2.60 ^ ^

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


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