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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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Presentation Transcript
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

.

.

.

.

.

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

.

.

.

.

.

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)
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)

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

.

.

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

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

0.1 = 4.6 - (1.50)3.0

Regression Equation

Y = a + bX

Y = 0.1 + (1.5)X

b =

4.43

b =

-.57

-1.17

2.16

a = Y - bX

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

.

22

20

.

18

16

.

14

.

12

10

Y = 21.52 + (-1.17)X

.

.

22

20

.

18

16

.

14

.

12

10

Y = 21.52 + (-1.17)X

.

.

22

20

.

18

16

.

14

.

.

12

10

Y = 21.52 + (-1.17)X

.

.

22

20

.

18

16

.

14

.

.

12

10

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

b =

a = Y - bX

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
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
Step 6: Plug in values

r =

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 5

Step 6: Plug in values

15

r =

15

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 5

Step 6: Plug in values

15

23

r =

15

23

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 5

Step 6: Plug in values

84

15

23

r =

15

23

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 5

Step 6: Plug in values

84

15

23

r =

55

15

23

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 5

Step 6: Plug in values

84

15

23

r =

55

135

15

23

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 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 = 84 N = 5

Step 7: Solve!

(5)

84

15

23

r =

(5)

55

(5)

135

15

23

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 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 = 84 N = 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 = 84 N = 5

Step 7: Solve!

420

345

23

23

15

r =

50

23

146

225

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 5

Step 7: Solve!

75

23

23

15

r =

146

23

7300

50

225

Y = 23 Y2 = 135

X = 15 X2 = 55

XY = 84 N = 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 = 84 N = 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 = 84 N = 5

Practice

(4)

72

20

19

r =

(4)

120

(4)

123

20

19

X = 20 X2 = 120

Y = 19 Y2 = 123

XY = 72 N = 4

Practice

-92

20

19

r =

(4)

120

(4)

123

20

19

X = 20 X2 = 120

Y = 19 Y2 = 123

XY = 72 N = 4

Practice

-92

20

19

r =

80

131

X = 20 X2 = 120

Y = 19 Y2 = 123

XY = 72 N = 4

Practice

-92

20

19

-.90 =

102.37

80

131

X = 20 X2 = 120

Y = 19 Y2 = 123

XY = 72 N = 4

Practice
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