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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

  • 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.

  • 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.


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


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Sleeping and Happiness

  • 4) Compute the correlation

  • Hours Slept

    M = 7.0

    SD = 1.4

  • Happiness

    M = 6.8

    SD = 1.7


Blanched Formula

r =

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5


Blanched Formula

247

r =

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5


Blanched Formula

247

7.0

6.8

r =

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5


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


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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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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 = Y - Y

 (Y - Y)2 is minimized

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Error = 1

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Error = .5

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.

Error = -1

.

Error = 0

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 = .88Mean X = 3.0; SX = 1.41


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

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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

2.41

b =

.88

1.50

1.41


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

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


Regression Equation

Y = a + bX

Y = 0.1 + (1.5)X


Y = 0.1 + (1.5)X

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

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

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Y = 0.1 + (1.5)X

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Practice


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


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

b =


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

4.43

b =

-.57

-1.17

2.16


Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 b = -1.17

a = Y - bX


Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 b = -1.17

21.52= 14.50 - (-1.17)6.0


Regression Equation

Y = a + bX

Y = 21.52 + (-1.17)X


Y = 21.52 + (-1.17)X

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22

20

.

18

16

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14

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12

10


Y = 21.52 + (-1.17)X

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.

22

20

.

18

16

.

14

.

12

10


Y = 21.52 + (-1.17)X

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.

22

20

.

18

16

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14

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.

12

10


Y = 21.52 + (-1.17)X

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22

20

.

18

16

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14

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.

12

10


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


Practice

  • How much money would likely be earned if an advertiser spent $2,000 (i.e., x = 2)?

  • How much money would likely be earned if an advertiser spent $10,000 (i.e., x = 10)?


Blanched Formula

r =

XY = 3360

X = 4.29

Y = 108.57

Sx = 1.03

Sy = 20.30

N = 7


Blanched Formula

3360

14.5

(4.29) (108.57)

7

.68 =

2.16

4.43

(1.03) (20.30)

XY = 3360

X = 4.29

Y = 108.57

Sx = 1.03

Sy = 20.30

N = 7


Mean Y = 108.57; Sy = 20.30Mean X = 4.29; Sx= 1.03 r = .68

b =


Mean Y = 108.57; Sy = 20.30Mean X = 4.29; Sx= 1.03 b = 13.40

a = Y - bX


Mean Y = 108.57; Sy = 20.30Mean X = 4.29; Sx= 1.03 b = 13.40

51.08 = 108.57 - (13.40)4.29


Regression Equation

Y = a + bX

Y = 51.08 + (13.40)X


Y = 51.08 + (13.40)X

  • How much money would likely be earned if an advertiser spent $2,000?

  • How much money would likely be earned if an advertiser spent $10,000?


Y = 51.08 + (13.40)X

  • How much money would likely be earned if an advertiser spent $2,000?

  • 77.88 = 51.08 + (13.40)2

  • $77, 880


Y = 51.08 + (13.40)X

  • How much money would likely be earned if an advertiser spent $10,000?

  • 185.08 = 51.08 + (13.40)10

  • $185,080


A “quick” step backwards


Blanched Formula

  • Good way to calculate r if the means and standard deviations are already provided.

  • It is very time consuming to calculate these statistics if they are not already provided

  • If means and standard deviations are not given, use the raw-score formula


Raw-Score Formula

r =


Step 1: Set up table


Step 2: Square Y


Step 3: Square X


Step 4: Multiply XY


Step 5: Sum


Step 6: Plug in values

r =

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 6: Plug in values

15

r =

15

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 6: Plug in values

15

23

r =

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 6: Plug in values

84

15

23

r =

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 6: Plug in values

84

15

23

r =

55

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 6: Plug in values

84

15

23

r =

55

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 6: Plug in values

(5)

84

15

23

r =

(5)

55

(5)

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 7: Solve!

(5)

84

15

23

r =

(5)

55

(5)

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 7: Solve!

(5)

84

15

23

r =

(5)

55

(5)

135

23

225

529

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 7: Solve!

420

345

23

23

15

r =

275

23

225

675

529

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 7: Solve!

420

345

23

23

15

r =

50

23

146

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 7: Solve!

75

23

23

15

r =

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 7: Solve!

75

23

23

15

r =

85.44

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Step 7: Solve!

75

23

23

15

.88 =

85.44

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5


Practice


Practice


Practice


Practice

(4)

72

20

19

r =

(4)

120

(4)

123

20

19

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4


Practice

-92

20

19

r =

(4)

120

(4)

123

20

19

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4


Practice

-92

20

19

r =

80

131

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4


Practice

-92

20

19

-.90 =

102.37

80

131

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4


Practice

  • Page 116 -- # 21


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