AP Statistics Section 3.2 A Regression Lines

1. AP Statistics Section 3.2 ARegression 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. 8 6 4 2 0 F a t G a i n (kg) -100 0 100 200 300 400 500 600 700 NEA (calories)

6. 8 6 4 2 0 F a t G a i n (kg) -100 0 100 200 300 400 500 600 700 NEA (calories)

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. 8 6 4 2 0 F a t G a i n (kg) -100 0 100 200 300 400 500 600 700 NEA (calories)

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. 8 6 4 2 0 F a t G a i n (kg) -100 0 100 200 300 400 500 600 700 NEA (calories)

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

24. 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. a b