sleeping and happiness you are interested in the relationship between hours slept and happiness. 1)...
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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
Hours slept
(X)
Happiness
(Y)
Pam 8 7
Jim 9 9
Dwight 5 4
Michael 6 8
Meredith 7 6
0
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10
12
2 4 6 8 10
Hours Slept
Hap
pine
ss ..
.. .
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!
0
2
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12
2 4 6 8 10
Hours Slept
Hap
pine
ss ..
.. .
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
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1 2 3 4 5
Talk
Smil
e
.
.. ..
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Talk
Smil
e
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Error = 1
Error = -1Error = .5
Error = -.5Error = 0
Error = Y - Y
(Y - Y)2 is minimized
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 = .88Mean X = 3.0; SX = 1.41
SmileY
TalkX
Jerry 9 5
Elan 2 1
George 5 3
Newman 4 4
Kramer 3 2
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1 2 3 4 5
Talk
Smil
e
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.. ..
Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41
b =
Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41
b =2.41
1.41.881.50
Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41
a = Y - bX
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
Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 1.41 b = 1.5
Regression Equation
Y = a + bX
Y = 0.1 + (1.5)X
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Talk
Smil
eY = 0.1 + (1.5)X
.
.. ..
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1 2 3 4 5
Talk
Smil
eY = 0.1 + (1.5)XX = 1; Y = 1.6
.
.
.. ..
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1 2 3 4 5
Talk
Smil
eY = 0.1 + (1.5)XX = 5; Y = 7.60
.
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0
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1 2 3 4 5
Talk
Smil
eY = 0.1 + (1.5)X
.
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.. ..