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

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Hours Slept

Hap

pine

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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!

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Hours Slept

Hap

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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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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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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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eY = 0.1 + (1.5)X

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eY = 0.1 + (1.5)XX = 1; Y = 1.6

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eY = 0.1 + (1.5)XX = 5; Y = 7.60

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eY = 0.1 + (1.5)X

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