Sleeping and Happiness Hours slept (X) Happiness (Y) Pam 8 7 Jim 9 Dwight 5 4 Michael 6 Meredith 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

. . . . . r = .76

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)

Regression allows us to predict! . . . . .

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)

Excel Example

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

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)

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

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

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

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

. . . . .

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

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

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

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

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

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

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

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

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

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

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

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

Y = 0.1 + (1.5)X X = 1; Y = 1.6 . . . . . .

Y = 0.1 + (1.5)X X = 5; Y = 7.60 . . . . . . .

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