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The plan

The plan. Practice – Correlation A straight line A regression equation Practice! A quicker way to compute a correlation. Practice. Interpret the following: 1) The correlation between vocational-interest scores at age 20 and at age 40 was .70. 2) Age and IQ is correlated -.16.

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The plan

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  1. The plan • Practice – Correlation • A straight line • A regression equation • Practice! • A quicker way to compute a correlation

  2. Practice • Interpret the following: • 1) The correlation between vocational-interest scores at age 20 and at age 40 was .70. • 2) Age and IQ is correlated -.16. • 3) The correlation between IQ and family size is -.30. • 4) The correlation between sexual promiscuity and dominance is .32. • 5) In a sample of males happiness and height is correlated .11.

  3. Sleeping and Happiness • You are interested in the relationship between hours slept and happiness. • 1) Make a scatter plot • 2) Guess the correlation • 3) Guess and draw the location of the regression line

  4. . . . . .

  5. Sleeping and Happiness • 4) Compute the correlation • Hours Slept M = 7.0 SD = 1.4 • Happiness M = 6.8 SD = 1.7

  6. Blanched Formula r = XY = 247 X = 7.0 Y = 6.8 Sx = 1.4 Sy = 1.7 N = 5

  7. Blanched Formula 247 r = XY = 247 X = 7.0 Y = 6.8 Sx = 1.4 Sy = 1.7 N = 5

  8. Blanched Formula 247 7.0 6.8 r = XY = 247 X = 7.0 Y = 6.8 Sx = 1.4 Sy = 1.7 N = 5

  9. Blanched Formula 247 7.0 6.8 5 .76 = 1.4 1.7 XY = 247 X = 7.0 Y = 6.8 Sx = 1.4 Sy = 1.7 N = 5

  10. . . . . . r = .76

  11. Remember this:Statistics Needed • Need to find the best place to draw the regression line on a scatter plot • Need to quantify the cluster of scores around this regression line (i.e., the correlation coefficient)

  12. Regression allows us to predict! . . . . .

  13. Straight Line Y = mX + b Where: Y and X are variables representing scores m = slope of the line (constant) b = intercept of the line with the Y axis (constant)

  14. Excel Example

  15. That’s nice but. . . . • How do you figure out the best values to use for m and b ? • First lets move into the language of regression

  16. Straight Line Y = mX + b Where: Y and X are variables representing scores m = slope of the line (constant) b = intercept of the line with the Y axis (constant)

  17. Regression Equation Y = a + bX Where: Y = value predicted from a particular X value a = point at which the regression line intersects the Y axis b = slope of the regression line X = X value for which you wish to predict a Y value

  18. Practice • Y = -7 + 2X • What is the slope and the Y-intercept? • Determine the value of Y for each X: • X = 1, X = 3, X = 5, X = 10

  19. Practice • Y = -7 + 2X • What is the slope and the Y-intercept? • Determine the value of Y for each X: • X = 1, X = 3, X = 5, X = 10 • Y = -5, Y = -1, Y = 3, Y = 13

  20. Finding a and b • Uses the least squares method • Minimizes Error Error = Y - Y  (Y - Y)2 is minimized

  21. . . . . .

  22. Error = Y - Y  (Y - Y)2 is minimized . Error = 1 . Error = .5 . . Error = -1 . Error = 0 Error = -.5

  23. Finding a and b • Ingredients • r value between the two variables • Sy and Sx • Mean of Y and X

  24. b b = r = correlation between X and Y SY = standard deviation of Y SX = standard deviation of X

  25. a a = Y - bX Y = mean of the Y scores b= regression coefficient computed previously X = mean of the X scores

  26. Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41

  27. Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41 . . . . .

  28. Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41 b =

  29. Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41 2.41 b = .88 1.50 1.41

  30. Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41 b = 1.5 a = Y - bX

  31. Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41 b = 1.5 0.1 = 4.6 - (1.50)3.0

  32. Regression Equation Y = a + bX Y = 0.1 + (1.5)X

  33. Y = 0.1 + (1.5)X . . . . .

  34. Y = 0.1 + (1.5)XX = 1; Y = 1.6 . . . . . .

  35. Y = 0.1 + (1.5)XX = 5; Y = 7.60 . . . . . . .

  36. Y = 0.1 + (1.5)X . . . . . . .

  37. Practice

  38. Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 r = -.57

  39. Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 r = -.57 b =

  40. Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 r = -.57 4.43 b = -.57 -1.17 2.16

  41. Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 b = -1.17 a = Y - bX

  42. Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 b = -1.17 21.52= 14.50 - (-1.17)6.0

  43. Regression Equation Y = a + bX Y = 21.52 + (-1.17)X

  44. Y = 21.52 + (-1.17)X . 22 20 . 18 16 . 14 . 12 10

  45. Y = 21.52 + (-1.17)X . . 22 20 . 18 16 . 14 . 12 10

  46. Y = 21.52 + (-1.17)X . . 22 20 . 18 16 . 14 . . 12 10

  47. Y = 21.52 + (-1.17)X . . 22 20 . 18 16 . 14 . . 12 10

  48. Mean Y = 108.57; Sy = 20.30Mean X = 4.29; Sx= 1.03

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