1. Least-Squares Regression Section 3.3

2. Correlation measures the strength and direction of a linear relationship between two variables. How do we summarize the overall pattern of a linear relationship? Draw a line! Recall from 3.2:

4. Least-Squares Regression A method for finding a line that summarizes the relationship between two variables, but only in a specific setting.

5. Regression Line “Best-fit Line” A straight line that descirbes how a response variable y changes as an explanatory variable x changes. Predict y from x. Requires that we have an explanatory variable and a response variable.

6. Example 3.8, p. 150

7. Least-Squares Regression Line Because different people will draw different lines by eye on a scatterplot, we need a way to minimize the vertical distances.

8. Least-Squares Regression Line The LSRL of y on x is the line that makes the sum of the squares of the vertical distances of the data from the line as small as possible.

9. Equation of LSRL

11. Facts about LSRL:

12. LSRL in the Calculator

13. After you’ve entered data, STAT PLOT. ZoomStat (Zoom 9)

14. LSRL in the Calculator To determine LSRL: Press STAT, CALC, 8:LinReg(a+bx), Enter

15. LSRL in the Calculator To get the line to graph in your calculator: Press STAT, CALC 8:LinReg(a+bx) L 1, L 2, Y 1 Now look in Y =. Then look at your graph.

16. LSRL in the Calculator

17. To plot the line on the scatterplot by hand:

18. For Example: Smallest x = 0, Largest x = 52 Use these two x-values to predict y.

19. For Example: (0, 1.0892), (52, 10.9172)

20. Extrapolation Suppose that we have data on a child’s growth between 3 and 8 years of age. The least-squares regression line gives us the equation, where x represents the age of the child in years, and will be the predicted height in inches. What if you wanted to predict the height of a 25 year old girl? Would this equation be appropriate to use? NO!

21. Extrapolation Extrapolation is the use of a regression line for prediction far outside the domain of values of the explanatory variable x that you used to obtain the line or curve. Such predictions are often not accurate. That’s over 7’ 9” tall!

22. Practice Exercises Exercises 3.40, 3.41 p. 157