Lecture 42 Section Fri, Nov 13, 2009

Transcription

1 Koer on Lecture 42 Section Koer HampdenSydney College Fri, Nov 13, 2009

2 Outline Koer on on 4 5 6

3 Outline Koer on on 4 5 6

4 Koer on Exercise 13.2, page 821. Data were gared to estimate regression line for a model where a gymnast s score (y) depends on gymnast s physical condition as measured by his or her weight (x). estimated regression line is given by ŷ = x. Complete following sentence: Every additional pound that a gymnast weighs drops predicted score by about points, on average.

5 Koer on Exercise 13.2, page 821. slope 1.9 tells how fast ŷ changes with respect to x. That is, slope represents change in ŷ for an increase of 1 in x. refore, if a gymnast gains one pound, n we expect his score to drop by 1.9 points.

6 Outline Koer on on 4 5 6

7 Least Squares Koer on equation of regression line is of form ŷ = a bx. b is slope of regression line. a is yintercept. We need to find coefficients a and b from data.

8 Least Squares Koer on formula for b is b = formula for a is (x x)(y y) (x x) 2 a = y bx.

9 Least Squares Example (Strong Positive Association) Koer on

10 Least Squares Example (Weak Positive Association) Koer on

11 Least Squares Example (Strong Negative Association) Koer on

12 Least Squares Example (Weak Negative Association) Koer on

13 Least Squares Example (No Association) Koer on

14 Least Squares Koer on An alternate formula for b is b = n xy x y n x 2 ( x) 2. This is one that uses.

15 Example Koer Example ( ) Consider again data set on x y x x y y (x x) 2 (y y) 2 (x x)(y y)

16 Example Koer Example ( ) Compute x deviations. on x y x x y y (x x) 2 (y y) 2 (x x)(y y)

17 Example Koer Example ( ) Compute y deviations. on x y x x y y (x x) 2 (y y) 2 (x x)(y y)

18 Example Koer Example ( ) Compute squared x deviations. on x y x x y y (x x) 2 (y y) 2 (x x)(y y)

19 Example Koer Example ( ) Compute squared y deviations. on x y x x y y (x x) 2 (y y) 2 (x x)(y y)

20 Example Koer Example ( ) Compute product of x and y deviations. on x y x x y y (x x) 2 (y y) 2 (x x)(y y)

21 Example Koer Example ( ) Find sums. on x y x x y y (x x) 2 (y y) 2 (x x)(y y)

22 Example Koer on Example ( ) Compute coefficients from formula. equation is b = = 1.1. a = 15 (1.1)(7) = 7.3. ŷ = x.

23 Example Koer Example ( ) Consider yet again data set on x y x 2 y 2 xy

24 Example Koer Example ( ) Square x. on x y x 2 y 2 xy

25 Example Koer Example ( ) Square y. on x y x 2 y 2 xy

26 Example Koer Example ( ) Find xy. on x y x 2 y 2 xy

27 Example Koer Example ( ) Add up columns. on x y x 2 y 2 xy

28 Second Formula Koer on Compute coefficients from alternate formula. equation is (8)(1005) (56)(120) b = (8)(542) (56) 2 = = 1.1. a = 15 (1.1)(7) = 7.3. ŷ = x.

29 Example Koer on second method is usually easier if you are doing it by hand. By eir method, we get equation ŷ = x.

30 Outline Koer on on 4 5 6

31 on Koer on (2Var Stats) Enter 2Var Stats L 1,L 2. Press ENTER. calculator reports that n = 8 Σx = 56 Σx 2 = 542 Σy = 120 Σy 2 = 2006 Σxy = 1005 Use 2Var Stats to get basic summations. n use formulas.

32 on Koer on (LinReg(abx)) Put x values in L 1. Put y values in L 2. Select STAT > CALC > LinReg(abx) (item #8). Press Enter. LinReg(abx) appears in display. Enter L 1,L 2. Press ENTER. Or, use LinReg(abx) function.

33 on Koer on (LinReg(abx)) following appear in display. title LinReg. equation y=abx. value of a. value of b. value of r 2 (to be discussed later). value of r (to be discussed later).

34 on Koer on Graphing Follow procedure for using LinReg(abx), except... Enter LinReg(abx) L 1,L 2, Press VARS > YVARS > Function > Y 1. Now display shows LinReg(abx) L 1,L 2,Y 1 Press ENTER. Press ZOOM > ZoomStat to draw graph. Press Y= to see regression equation.

35 Outline Koer on on 4 5 6

36 Koer on Press VARS > YVARS > Function > Y 1. Press ENTER. Press (. Enter value of x. Press ). Press ENTER. value predicted by model appears. Once function is entered as Y 1, it is easy to interpolate and extrapolate.

37 Outline Koer on on 4 5 6

38 Example Koer on Example ( ) Find equation of regression line for schooldistrict data on freelunch participation rate graduation rate. Let x be freelunch participation. Let y be graduation rate.

39 Example Koer on Example ( ) District Grad. District Grad. Amelia King and Queen Caroline King William Charles City Louisa Chesterfield New Kent Colonial Hgts Petersburg Cumberland Powhatan Dinwiddie Prince George Goochland Richmond Hanover Sussex Henrico West Point Hopewell

40 Example Koer on Example ( ) regression equation is ŷ = x.

41 Example Koer on Example ( )

42 Example Koer on Example ( )

43 Example Koer on Example (Predicting ŷ) What graduation rate would we predict in a district if we knew that freelunch participation rate was 50%? Calculate ŷ(50) = (50) = model predicts a graduation rate of 66.3%.

44 Example Koer on Example (Predicting ŷ)

45 Example Koer on Example (Predicting ŷ)

46 Outline Koer on on 4 5 6

47 Koer on Read Section , page Let s Do It! Exercises 3(bc), 4(bcd), 5(b), 6, page 821.