1. 11/26/2015 V. J. Motto 1 Chapter 1: Linear Models V. J. Motto M110 Modeling with Elementary Functions 1.5 Best-Fit Lines and Residuals

2. Working Example Consider the data show below Let’s enter this data searching for model. Don’t forget to clear your calculator first! 11/26/2015V. J. Motto 2

3. Here is what we have… 11/26/2015V. J. Motto 3

4. Observation The calculator tells us that the linear model is y = 0.3x + 2.8 The linear correlation coefficient r = 0.55. This is a weak positive correlation. The model only accounts for 30% of the variance. This is not a good model! 11/26/2015V. J. Motto 4

5. Analysis Observe how far the points of the “line of best fit” do not coincide with the actual data points! Let’s calculate these errors using the formula: actual value -line of best fit value; that is 11/26/2015V. J. Motto 5

6. The Sum of Square of Error Let’s calculate these errors using the formula: line of best fit value – actual value; that is for each point in the data we find 11/26/2015V. J. Motto 6

7. Method of Least Squares The calculator uses an algorithm called “the method of least squares” to determine the equation of the “line of best fit.” This algorithm guarantees that the line of best is the line whose sum of squares of the errors is the smallest (the least). 11/26/2015V. J. Motto 7

8. Some Observations The Line of Best fit Always passes through the point : It has the smallest SSE of any line that passes through the data field. You can always find a line of best fit. However, you must make judgment whether it is a good model or not. 11/26/2015V. J. Motto 8

9. Another Line Consider the line y = 0.8x + 1 which also goes through the data field. What is its SSE? 11/26/2015V. J. Motto 9