Lecture 46 Section Tue, Apr 15, 2008

Transcription

1 ar Koer ar Lecture 46 Section Koer Hampden-Sydney College Tue, Apr 15, 2008

2 Outline ar Koer

3 ar Koer We are now ready to calculate least-squares regression line. formulas are a bit daunting, but will do heavy lifting for us. Once we find regression line, we are n in a position to use it to make predictions.

4 Least Squares ar Koer equation of regression line is of form ŷ = a + bx. b is slope of regression line. a is y-intercept. We need to find coefficients a and b from data.

5 Least Squares ar Koer formula for b is or b = formula for a is (x x)(y y) (x x) 2 b = n xy x y n x 2 ( x) 2. a = y bx.

6 Least Squares ar Koer It so happens that regression line passes through center (x, y) of scatterplot. That means that point (x, y) satisfies equation ŷ = a + bx. That is, y = a + bx. Thus, a = y bx.

7 First Formula ar Koer Consider again data set x y x x y y

8 First Formula ar Koer Compute x and y deviations. x y x x y y

9 First Formula ar Koer Compute squared deviations. x x y y (x x) 2 (y y) 2 (x x)(y y)

10 First Formula ar Koer Compute squared deviations. x x y y (x x) 2 (y y) 2 (x x)(y y)

11 First Formula ar Koer Find sums. x x y y (x x) 2 (y y) 2 (x x)(y y)

12 First Formula ar Koer Compute coefficients from first formula. Compute b: n compute a: b = = 1.1. a = 15 (1.1)(7) = 7.3. equation is ŷ = x.

13 First Formula ar Koer Compute coefficients from first formula. Compute b: n compute a: b = = 1.1. a = 15 (1.1)(7) = 7.3. equation is ŷ = x.

14 First Formula ar Koer Compute coefficients from first formula. Compute b: n compute a: b = = 1.1. a = 15 (1.1)(7) = 7.3. equation is ŷ = x.

15 First Formula ar Koer Compute coefficients from first formula. Compute b: n compute a: b = = 1.1. a = 15 (1.1)(7) = 7.3. equation is ŷ = x.

16 Second Formula ar Koer Consider yet again data set x y x 2 y 2 xy

17 Second Formula ar Koer Square x and y and find xy. x y x 2 y 2 xy

18 Second Formula ar Koer Add up columns. x y x 2 y 2 xy

19 Second Formula ar Koer Compute coefficients from second formula. Compute b: b = (8)(1005) (56)(120) (8)(542) (56) 2 = = 1.1. n compute a as before: equation is a = 15 (1.1)(7) = 7.3. ŷ = x.

20 Second Formula ar Koer Compute coefficients from second formula. Compute b: b = (8)(1005) (56)(120) (8)(542) (56) 2 = = 1.1. n compute a as before: equation is a = 15 (1.1)(7) = 7.3. ŷ = x.

21 Second Formula ar Koer Compute coefficients from second formula. Compute b: b = (8)(1005) (56)(120) (8)(542) (56) 2 = = 1.1. n compute a as before: equation is a = 15 (1.1)(7) = 7.3. ŷ = x.

22 Second Formula ar Koer Compute coefficients from second formula. Compute b: b = (8)(1005) (56)(120) (8)(542) (56) 2 = = 1.1. n compute a as before: equation is a = 15 (1.1)(7) = 7.3. ŷ = x.

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

24 ar Koer On, we could use 2-Var Stats to get basic summations. Enter 2-Var Stats L 1,L 2. Press ENTER. calculator reports that n = 8 Σx = 56 Σx 2 = 542 Σy = 120 Σy 2 = 2006 Σxy = 1005 n use formulas.

25 ar Koer Or we can use LinReg function. Put x values in L 1. Put y values in L 2. Select STAT > CALC > LinReg(a+bx) (item #8). Press Enter. LinReg(a+bx) appears in display. Enter L 1,L 2. Press Enter.

26 ar Koer following appear in display. title LinReg. equation y=a+bx. value of a. value of b. value of r 2 (to be discussed later). value of r (to be discussed later).

27 ar Koer To graph regression line along with scatterplot, after selecting LinReg(a+bx), Enter L 1,L 2,Y 1. (That is, add Y 1.) Press Enter. Press ZOOM > ZoomStat to draw graph. To see regression equation, Press Y=.

28 Participation ar Koer Find equation of regression line for school-district data on free-lunch participation rate graduation rate. Let x be free-lunch participation. Let y be graduation rate.

29 es s ar Koer District Grad. Amelia Caroline Charles City Chesterfield Colonial Hgts Cumberland Dinwiddie Goochland Hanover Henrico Hopewell

30 es s ar Koer District Grad. King and Queen King William Louisa New Kent Petersburg Powhatan Prince George Richmond Sussex West Point

31 Participation ar Koer regression equation is ŷ = x.

32 Scatter Plot ar Koer

33 Scatter Plot with ar Koer

34 Predicting y ar Koer What graduation rate would we predict in a district if we knew that free-lunch participation rate was 50%? Calculate ŷ(50) = (50) = We predict a graduation rate of 66.3%.

35 Predicting y ar Koer What graduation rate would we predict in a district if we knew that free-lunch participation rate was 50%? Calculate ŷ(50) = (50) = We predict a graduation rate of 66.3%.

36 Predicting y ar Koer What graduation rate would we predict in a district if we knew that free-lunch participation rate was 50%? Calculate ŷ(50) = (50) = We predict a graduation rate of 66.3%.

37 Predicting y ar Koer

38 Predicting y ar Koer

39 Slope of ar Koer first formula for slope b of regression line is (x x)(y y) b =. (x x) 2 Let us consider numerator (x x)(y y). We will see why it is positive when trend is upwards, negative when trend is downwards, and zero when re is no trend.

40 Slope of ar Koer Consider free-lunch (x) graduation rate (y) data. average of x is average of y is Use se values to divide scatterplot into four quadrants.

41 Scatter Plot ar Koer

42 Scatter Plot ar Koer

43 Slope of ar Koer se two lines meet in center of scatterplot. For every point to right of center, x x > 0 and for every point to left, x x < 0.

44 Slope of ar Koer Similarly, for every point above of center, y y > 0 and for every point below center, y y < 0.

45 Slope of ar Koer refore, if a point is in upper-right quadrant or lower-left quadrant, n (x x)(y y) > 0. If a point is in upper-left quadrant or lower-right quadrant, n (x x)(y y) < 0.

46 Scatter Plot ar Koer

47 Slope of ar Koer In this example, it is clear that negative values dominate. refore, (x x)(y y) < 0 and regression line has negative slope.

48 Slope of ar Koer Had positive values dominated, n (x x)(y y) < 0 and slope would be negative. Had positive and negative values balanced, n (x x)(y y) = 0 and slope would be zero.

49 Variation in Model ar Koer re is a very simple relationship between variation in observed y values and variation in predicted y values.

50 Observed y and Predicted y ar Koer SST = Variation in Observed y SSR = Variation in Predicted y

51 Variation in Observed y ar Koer variation in observed y is measured by SST (same as SSY). For graduation rate data (L 2 ), SST =

52 Variation in Predicted y ar Koer variation in predicted y is measured by SSR. For predicted graduation rate data, let L 3 = Y 1 (L 1 ). SSR = SSE = Residual Sum of Squares It turns out that SST = SSE + SSR. That is,

53 Sum Squared Error ar Koer In example, SST SSR = = If we compute sum of squared residuals directly, we get SSE =

54 Explaining Variability ar Koer In equation SST = SSE + SSR, SSR is amount of variability in y that is explained by model. SSE is amount of variability in y that is not explained by model.

55 Explaining Variability ar Koer In last example, how much variability in graduation rate is explained by model (by free-lunch participation)?

56 ar Koer formulas for regression line ŷ = a + bx are and b = n xy x y n x 2 ( x) 2 a = y bx. function LinReg(a+bx) will do calculations. If we store regression equation in Y 1, n we can also draw it on and use to make predictions.