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- Practice – Correlation
- A straight line
- A regression equation
- Practice!
- A quicker way to compute a correlation

- 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.

- 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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- 4) Compute the correlation
- Hours Slept
M = 7.0

SD = 1.4

- Happiness
M = 6.8

SD = 1.7

r =

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5

247

r =

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5

247

7.0

6.8

r =

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5

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

- 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)

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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)

- How do you figure out the best values to use for m and b ?
- First lets move into the language of regression

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)

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

- 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 = -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

- 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

- Ingredients
- r value between the two variables
- Sy and Sx
- Mean of Y and X

b =

r = correlation between X and Y

SY = standard deviation of Y

SX = standard deviation of X

a = Y - bX

Y = mean of the Y scores

b= regression coefficient computed previously

X = mean of the X scores

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b =

2.41

b =

.88

1.50

1.41

a = Y - bX

0.1 = 4.6 - (1.50)3.0

Y = a + bX

Y = 0.1 + (1.5)X

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b =

4.43

b =

-.57

-1.17

2.16

a = Y - bX

21.52= 14.50 - (-1.17)6.0

Y = a + bX

Y = 21.52 + (-1.17)X

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22

20

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18

16

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14

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12

10

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22

20

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18

16

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14

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12

10

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22

20

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18

16

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14

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12

10

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22

20

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18

16

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14

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12

10

- 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)?

r =

XY = 3360

X = 4.29

Y = 108.57

Sx = 1.03

Sy = 20.30

N = 7

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

b =

a = Y - bX

51.08 = 108.57 - (13.40)4.29

Y = a + bX

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?

- How much money would likely be earned if an advertiser spent $2,000?
- 77.88 = 51.08 + (13.40)2
- $77, 880

- How much money would likely be earned if an advertiser spent $10,000?
- 185.08 = 51.08 + (13.40)10
- $185,080

- 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

r =

r =

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

15

r =

15

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

15

23

r =

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

84

15

23

r =

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

84

15

23

r =

55

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

84

15

23

r =

55

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

(5)

84

15

23

r =

(5)

55

(5)

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

(5)

84

15

23

r =

(5)

55

(5)

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

(5)

84

15

23

r =

(5)

55

(5)

135

23

225

529

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

420

345

23

23

15

r =

275

23

225

675

529

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

420

345

23

23

15

r =

50

23

146

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

75

23

23

15

r =

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

75

23

23

15

r =

85.44

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

75

23

23

15

.88 =

85.44

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

(4)

72

20

19

r =

(4)

120

(4)

123

20

19

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4

-92

20

19

r =

(4)

120

(4)

123

20

19

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4

-92

20

19

r =

80

131

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4

-92

20

19

-.90 =

102.37

80

131

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4

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