1. transformations Remember scatterplots from CH3Remember scatterplots from CH3 Insert data L1(x),L2,(y) in your calculatorInsert data L1(x),L2,(y) in your calculator 8: Linreg(a +bx) L1,L2,Y1 ….(write down a,b,r,r 2 )8: Linreg(a +bx) L1,L2,Y1 ….(write down a,b,r,r 2 ) Check the scatterplotCheck the scatterplot Check the Residual Plot L1, RESIDCheck the Residual Plot L1, RESID Curved pattern = not a good fitCurved pattern = not a good fit Random pattern = good fitRandom pattern = good fit

2. Transformations If your Linear Model x,y is not Appropriate…..If your Linear Model x,y is not Appropriate….. There are a few options to try…..There are a few options to try….. Exponential Model x,Log(y)Exponential Model x,Log(y) Power Model Log(x), Log(y)Power Model Log(x), Log(y) Try the above options in that order, check r 2 and the residual plot,……if r 2 is high and the residual plot looks good then you have found a suitable modelTry the above options in that order, check r 2 and the residual plot,……if r 2 is high and the residual plot looks good then you have found a suitable model CAUTION…Real data may not have a perfect model….sometimes you have to settle on “good enough”CAUTION…Real data may not have a perfect model….sometimes you have to settle on “good enough”

3. Baseball Salaries Ballplayers have been signing very large contracts. The highest salaries (in millions of dollars per season) for some notable players are given in the following table.Ballplayers have been signing very large contracts. The highest salaries (in millions of dollars per season) for some notable players are given in the following table.

4. PlayerYearSalary (millions $) Nolan Ryan19801.0 George Foster19822.0 Kirby Puckett19903.0 Jose Canseco19904.7 Roger Clemens19915.3 Ken Griffey Jr.19968.5 Albert Belle199711.0 Pedro Martinez199812.5 Mike Piazza199912.5 Mo Vaughn199913.3 Kevin Brown199915.0 Carlos Delgado200117.0 Alex Rodriguez200122.0 Manny Ramirez200422.5 Alex Rodriguez200526.0

5. Year VS SALARY R 2 is high, however the scatterplot appears to have a curved pattern. A linear model may not be appropriate.

6. Year vs Log(salary) This is an exponential model. R 2 is very high and the scatterplot shows no curvature. This appears to be a good fit for this data. Make sure to check the residual plot to make sure.

7. Residual Plot This residual plot shows no curved pattern and the residuals are randomly scattered above and below the axis…this shows that your model is a good fit.

8. Exponential model Log(salary) = -109.133 + 0.05516YEARLog(salary) = -109.133 + 0.05516YEAR Make a prediction using your model for a salary in 2006.Make a prediction using your model for a salary in 2006. About 33 million a yearAbout 33 million a year

9. Life expectancy The following data is Life Expectancy for white males in the United States every decade during the last century (1 = 1900 to 1910, 2 = 1911 to 1920, etc.). Create a model to predict future increases in life expectancy.The following data is Life Expectancy for white males in the United States every decade during the last century (1 = 1900 to 1910, 2 = 1911 to 1920, etc.). Create a model to predict future increases in life expectancy. Decade12345678910 Life Exp.48.654.459.762.166.567.46870.772.774.9 Log(life) = 1.685 + 0.18497Log(Decade) Make a prediction using the above model for the life expectancy of the decade we are currently in. Power Model

10. Use log inverse 76.6683567676.66835676 About 76 to 77 yearsAbout 76 to 77 years