1. M22- Regression & Correlation 1  Department of ISM, University of Alabama, 1992-2003 Lesson Objectives  Know what the equation of a straight line is, in terms of slope and y-intercept.  Learn how find the equation of the least squares regression line.  Know how to draw a regression line on a scatterplot.  Know how to use the regression equation to estimate the mean of Y for a given value of X.

2. M22- Regression & Correlation 2  Department of ISM, University of Alabama, 1992-2003 Best graphical tool for “seeing” the relationship between two quantitative variables. Use to identify: Patterns (relationships) Unusual data (outliers) Scatterplot

3. M22- Regression & Correlation 3  Department of ISM, University of Alabama, 1992-2003 Y X Y X Y X Y X Y Positive Linear Relationship Negative Linear Relationship Nonlinear Relationship, need to change the model No Relationship (X is not useful)

4. M22- Regression & Correlation 4  Department of ISM, University of Alabama, 1992-2003 Regression Analysis mechanics

5. M22- Regression & Correlation 5  Department of ISM, University of Alabama, 1992-2003 Equation of a straight line. Y = mx + b m = slope = “rate of change” b = the “y” intercept. Y = a + bx ^ b = slope a = the “y” intercept. Days of algebra Statistics form Y = estimate of the mean of Y for some X value. ^

6. M22- Regression & Correlation 6  Department of ISM, University of Alabama, 1992-2003  by “eyeball”.  by using equations by hand.  by hand calculator.  by computer: Minitab, Excel, etc. Equation of a straight line. How are the slope and y-intercept determined?

7. M22- Regression & Correlation 7  Department of ISM, University of Alabama, 1992-2003 Equation of a straight line. Y = a + bx ^ X-axis 0 rise run a “y” intercept b =

8. M22- Regression & Correlation 8  Department of ISM, University of Alabama, 1992-2003 Equation of a straight line. Y = a + bx ^ X-axis 0 rise run a “y” intercept b =

9. M22- Regression & Correlation 9  Department of ISM, University of Alabama, 1992-2003 Population: All ST 260 students Each value of X defines a subpopulation of “height” values. The goal is to estimate the true mean weight for each of the infinite number of subpopulations. Example 1: Y = Weight in pounds, X = Height in inches. Measure: Is height a good estimator of mean weight?

10. M22- Regression & Correlation 10  Department of ISM, University of Alabama, 1992-2003 Sample of n = 5 students Y = Weight in pounds, X = Height in inches. 1234512345 Ht Wt 73 175 68 158 67 140 72 207 62 115 Case Example 1: Step 1?

11. M22- Regression & Correlation 11  Department of ISM, University of Alabama, 1992-2003 DTDP

12. M22- Regression & Correlation 12  Department of ISM, University of Alabama, 1992-2003 100 120 140 160 180 200 220 6064687276 HEIGHT .    .... WEIGHT Where should the line go? X Y 73 175 68 158 67 140 72 207 62 115 X Y 73 175 68 158 67 140 72 207 62 115 Example 1

13. M22- Regression & Correlation 13  Department of ISM, University of Alabama, 1992-2003 page 615 Equation of Least Squares Regression Line Slope: y-intercept These are not the preferred computational equations.

14. M22- Regression & Correlation 14  Department of ISM, University of Alabama, 1992-2003 Basic intermediate calculations (x i - x)(y i - y)  (x i - x) 2  (y i - y) 2  1 2 3 = S xy = = S xx = = S yy = Numerator part of S 2 Look at your formula sheet

15. M22- Regression & Correlation 15  Department of ISM, University of Alabama, 1992-2003 1 2 3 = S xy = xy  ( x )( y )  n  = S xx = = S yy = y2y2 n  y)y) 2 ((  x2x2 n  x)x) 2 ((  Alternate intermediate calculations Look at your formula sheet Numerator part of S 2

16. 1234512345 Case x y Ht Wt 73 175 68 158 67 140 72 207 62 115 342 795  x  y xy Ht*Wt 12775 10744.. __.___ 54933  xy x 2 Ht 2 5329 4624.. _.___ 23470  x2x2 30625 24964.. _ _.___ 131263 y 2 Wt 2  y2y2 Example 1

17. M22- Regression & Correlation 17  Department of ISM, University of Alabama, 1992-2003 Intermediate Summary Values  xy  ( x )( y )  n  54933  (342)(795) 5  1 =  x2x2 n  x)x) 2 ((  2 23470 ( 342 ) 2 5   =  y2y2 n  y)y) 2 ((  3 131263 ( 795 ) 2 5   = Example 1

18. M22- Regression & Correlation 18  Department of ISM, University of Alabama, 1992-2003 Intermediate Summary Values Example 1 1 2 3 = 555.0 = 77.2 = 4858.0 Once these values are calculated, the rest is easy!

19. M22- Regression & Correlation 19  Department of ISM, University of Alabama, 1992-2003 Least Squares Regression Line where ^ Y = a + b X b  a  ybx 1 2 Prediction equation Estimated Slope Estimated Y - intercept

20. M22- Regression & Correlation 20  Department of ISM, University of Alabama, 1992-2003 Slope, for Weight vs. Height b  1 2 77.2 555 = = 7.189 Example 1

21. M22- Regression & Correlation 21  Department of ISM, University of Alabama, 1992-2003 Intercept, for Weight vs. Height a  b y x – 332.73 = = 795 5 y= 159 342 5 x == 68.4 = 159  a (+7.189) 68.4 Example 1

22. M22- Regression & Correlation 22  Department of ISM, University of Alabama, 1992-2003 Prediction equation ^ Y = a + b X Wt = – 332.73 + 7.189 Ht ^ Y = – 332.73 + 7.189 X ^^ Example 1

23. M22- Regression & Correlation 23  Department of ISM, University of Alabama, 1992-2003     100 120 140 160 180 200 220 6064687276 HEIGHT Y = – 332.7 + 7.189X ^  WEIGHT Example 1 Draw the line on the plot

24. M22- Regression & Correlation 24  Department of ISM, University of Alabama, 1992-2003    100 120 140 160 180 200 220 6064687276 HEIGHT  Y = – 332.7 + 7.189 60 ^ ^ Y = 98.64 X  Y = – 332.7 + 7.189 76 ^ ^ Y = 213.7 X WEIGHT Example 1 Draw the line on the plot

25. M22- Regression & Correlation 25  Department of ISM, University of Alabama, 1992-2003 What a regression equation gives you:  The “line of means” for the Y population.  A prediction of the mean of the population of Y-values defined by a specific value of X.  Each value of X defines a subpopulation of Y-values; the value of regression equation is the “least squares” estimate of the mean of that Y subpopulation.

26. M22- Regression & Correlation 26  Department of ISM, University of Alabama, 1992-2003 Example 2:Estimate the weight of a student 5’ 5” tall. Y = a + b X = – 332.73 + 7.189 X ^

27. M22- Regression & Correlation 27  Department of ISM, University of Alabama, 1992-2003    100 120 140 160 180 200 220 6064687276 HEIGHT  Y = – 332.7 + 7.189(65) = ^  WEIGHT Example 2

28. M22- Regression & Correlation 28  Department of ISM, University of Alabama, 1992-2003 Calculate your own weight. Why was your estimate not exact?    

29. M22- Regression & Correlation 29  Department of ISM, University of Alabama, 1992-2003 1. Calculate the least squares regression line. 2. Plot the data and draw the line through the data. 3. Predict Y for a given X. 4. Interpret the meaning of the regression line. Regression: Know How To:

30. M22- Regression & Correlation 30  Department of ISM, University of Alabama, 1992-2003

31. M22- Regression & Correlation 31  Department of ISM, University of Alabama, 1992-2003 Correlation

32. M22- Regression & Correlation 32  Department of ISM, University of Alabama, 1992-2003 Sample Correlation Coefficient, r A numerical summary statistic that measures the strength of the linear association between two quantitative variables.

33. M22- Regression & Correlation 33  Department of ISM, University of Alabama, 1992-2003 Notation: r = sample correlation.  = population correlation, “rho”. r is an “estimator” of 

34. M22- Regression & Correlation 34  Department of ISM, University of Alabama, 1992-2003 Interpreting correlation: -1.0  r  +1.0 r > 0.0 Pattern runs upward from left to right; “positive” trend. r < 0.0 Pattern runs downward from left to right; “negative” trend.

35. M22- Regression & Correlation 35  Department of ISM, University of Alabama, 1992-2003 Upward & downward trends: r > 0.0r < 0.0 Y X-axis Y Slope and correlation must have the same sign.

36. M22- Regression & Correlation 36  Department of ISM, University of Alabama, 1992-2003 All data exactly on a straight line: r = _____ Perfect positive relationship Perfect negative relationship Y X-axis Y

37. M22- Regression & Correlation 37  Department of ISM, University of Alabama, 1992-2003 r = _____________ Which has stronger correlation? Y X-axis Y

38. M22- Regression & Correlation 38  Department of ISM, University of Alabama, 1992-2003 r close to -1 or +1 means _________________________ linear relation. r close to 0 means _________________________ linear relation. "Strength": How tightly the data follow a straight line.

39. M22- Regression & Correlation 39  Department of ISM, University of Alabama, 1992-2003 r = ________________ Which has stronger correlation? Y X-axis Y

40. M22- Regression & Correlation 40  Department of ISM, University of Alabama, 1992-2003 Y X-axis Y Which has stronger correlation? Strong parabolic pattern! We can fix it. r = ________________

41. M22- Regression & Correlation 41  Department of ISM, University of Alabama, 1992-2003 Computing Correlation  by hand using the formula  using a calculator (built-in)  using a computer: Excel, Minitab,....

42. M22- Regression & Correlation 42  Department of ISM, University of Alabama, 1992-2003 Formula for Sample Correlation (Page 627)  2 3 1 r  S xy S yy S xx Look at your formula sheet

43. M22- Regression & Correlation 43  Department of ISM, University of Alabama, 1992-2003 Calculating Correlation 2 3 1 r =r = Look at your formula sheet Example 1; Weight versus Height = “Go to Slide 18 for values.”

44. M22- Regression & Correlation 44  Department of ISM, University of Alabama, 1992-2003 Positive Linear Relationship Example 6 Real estate data, previous section r =

45. M22- Regression & Correlation 45  Department of ISM, University of Alabama, 1992-2003 Negative Linear Relationship Example 7 AL school data, previous section r =

46. M22- Regression & Correlation 46  Department of ISM, University of Alabama, 1992-2003 No linear Relationship Example 9 Rainfall data, previous section r =

47. M22- Regression & Correlation 47  Department of ISM, University of Alabama, 1992-2003 Size of “r” does NOT reflect the steepness of the slope, “b”; but “r” and “b” must have the same sign. r = b  s x s y and = br s y s x  Comment 1:

48. M22- Regression & Correlation 48  Department of ISM, University of Alabama, 1992-2003 Changing the units of Y and X does not affect the size of r. Comment 2: Inchestocentimeters Poundstokilograms CelsiustoFahrenheit X to Z (standardized)

49. M22- Regression & Correlation 49  Department of ISM, University of Alabama, 1992-2003 Comment 3: High correlation does not always imply causation. Example: X = dryer temperature Y = drying time for clothes Causation: Changes in X actually do cause changes in Y. Consistency, responsiveness, mechanism

50. M22- Regression & Correlation 50  Department of ISM, University of Alabama, 1992-2003 Common Response Both X and Y change as some unobserved third variable changes. Comment 4: Example: In basketball, there is a high correlation between points scored and personal fouls committed over a season. Third variable is ___?

51. M22- Regression & Correlation 51  Department of ISM, University of Alabama, 1992-2003 Confounding The effect of X on Y is "hopelessly" mixed up with the effects of other variables on Y. Example: Is adult behavior most affected by environment or genetics? Comment 5:

52. M22- Regression & Correlation 52  Department of ISM, University of Alabama, 1992-2003 The end