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# The plan - PowerPoint PPT Presentation

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

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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

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

### 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)

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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)

### 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

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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

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b =

2.41

b =

.88

1.50

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

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

### 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

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### 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)?

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

### 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

### 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

r =

r =

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

15

r =

15

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

15

23

r =

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

84

15

23

r =

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

84

15

23

r =

55

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

84

15

23

r =

55

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

(5)

84

15

23

r =

(5)

55

(5)

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

(5)

84

15

23

r =

(5)

55

(5)

135

15

23

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

(5)

84

15

23

r =

(5)

55

(5)

135

23

225

529

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

420

345

23

23

15

r =

275

23

225

675

529

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

420

345

23

23

15

r =

50

23

146

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

75

23

23

15

r =

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

75

23

23

15

r =

85.44

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

75

23

23

15

.88 =

85.44

146

23

7300

50

225

Y = 23Y2 = 135

X =15X2 = 55

XY = 84N = 5

(4)

72

20

19

r =

(4)

120

(4)

123

20

19

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4

-92

20

19

r =

(4)

120

(4)

123

20

19

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4

-92

20

19

r =

80

131

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4

-92

20

19

-.90 =

102.37

80

131

X = 20X2 = 120

Y =19Y2 = 123

XY = 72N = 4

### Practice

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