546 CHAPTER 10 Correlation and Regression YOUR TURN. Do Exercise 13 “Cars.” Explained and Unexplained Variation Assume that we have a sample of paired data having the following properties shown in Figure 10-7: ■ There is sufficient evidence to support the claim of a linear correlation between x and y. ■ The equation of the regression line is y n = 3 + 2x. ■ The mean of the y values is given by y = 9. ■ One of the pairs of sample data is x = 5 and y = 19. ■ The point (5, 13) is one of the points on the regression line, because substituting x = 5 into the regression equation of yn = 3 + 2x yields yn = 13. Explained deviation (y – y) y = 3 + 2x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 7 8 9 (5, 9) (5, 13) (5, 19) Total deviation (y – y) Unexplained deviation (y – y) y = 9 x y ˆ ˆ ˆ FIGURE 10-7 Total, Explained, and Unexplained Deviation INTERPRETATION The 95% prediction interval is 73.7 million tickets * y * 122 million tickets (which does contain the value of 90 million tickets that were actually sold in this particular lottery). This means that if we select some particular lottery with a jackpot of 625 million dollars 1x = 6252, we have 95% confidence that the limits of 73.7 million tickets and 122 million tickets contain the actual ticket sales in millions. That is a wide range of values. The prediction interval would be much narrower and our estimated number of tickets would be much better if the margin of error E was not so large (due to the small sample size and the large difference between the outlier jackpot of x = 625 million dollars and x = 214.8889 million dollars).
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