1. AP Statistics Section 3.2 A Regression Lines

2. Linear relationships between two quantitative variables are quite common. Correlation measures the direction and strength of these relationships. Just as we drew a density curve to model the data in a histogram, we can summarize the overall pattern in a linear relationship by drawing a _______________ on the scatterplot. regression line

3. Note that regression requires that we have an explanatory variable and a response variable. A regression line is often used to predict the value of y for a given value of x.

4. Who:______________________________ What:______________________________ ______________________________ Why:_______________________________ When, where, how and by whom? The data come from a controlled experiment in which subjects were forced to overeat for an 8-week period. Results of the study were published in Science magazine in 1999. 16 healthy young adults Exp.-change in NEA (cal) Resp.-fat gain (kg) Do changes in NEA explain weight gain

5. NEA (calories) F a t G a i n (kg) -100 0 100 200 300 400 500 600 700 8642086420

6. NEA (calories) F a t G a i n (kg) -100 0 100 200 300 400 500 600 700 8642086420

7. Numerical summary: The correlation between NEA change and fat gain is r = _______

8. A least-squares regression line relating y to x has an equation of the form ___________ In this equation, b is the _____, and a is the __________. slope y-intercept

9. The formula at the right will allow you to find the value of b:

10. Once you have computed b, you can then find the value of a using this equation.

11. We can also find these values on our TI-83/84.

12. For this example, the LSL is or

13. Interpreting b: The slope b is the predicted _____________ in the response variable y as the explanatory variable x changes. rate of change

14. The slope b = -.0034 tells us that fat gain goes down by.0034 kg for each additional calorie of NEA.

15. You cannot say how important a relationship is by looking at how big the regression slope is.

16. Interpreting a: The y-intercept a = 3.505 kg is the fat gain estimated by the model if NEA does not change when a person overeats.

17. Model: Using the equation above, draw the LSL on your scatterplot.

18. NEA (calories) F a t G a i n (kg) -100 0 100 200 300 400 500 600 700 8642086420

19. TI 83/84 8:LinReg(a+bx) GRAPH

20. Prediction: Predict the fat gain for an individual whose NEA increases by 400 cal by: (a) using the graph ___________ (b) using the equation _________

21. NEA (calories) F a t G a i n (kg) -100 0 100 200 300 400 500 600 700 8642086420

22. Prediction: Predict the fat gain for an individual whose NEA increases by 400 cal by: (a) using the graph ___________ (b) using the equation _________

25. Predict the fat gain for an individual whose NEA increases by 1500 cal.

26. So we are predicting that this individual loses fat when he/she overeats. What went wrong? 1500 is way outside the range of NEA values in our data

27. Extrapolation is the use of a regression line for prediction outside the range of values of the explanatory variable x used to obtain the line. Such predictions are often not accurate.

28. abab