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

Hours slept

(X)

Happiness

(Y)

Pam 8 7

Jim 9 9

Dwight 5 4

Michael 6 8

Meredith 7 6

0

2

4

6

8

10

12

2 4 6 8 10

Hours Slept

Hap

pine

ss ..

.. .

Sleeping and Happiness

• 4) Compute the correlation

• Hours SleptM = 7.0SD = 1.4

• HappinessM = 6.8SD = 1.7

Hours slept

(X)

Happiness

(Y)

Pam 8 7

Jim 9 9

Dwight 5 4

Michael 6 8

Meredith 7 6

Blanched Formula

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5

r =

Blanched Formula

r =

247

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5

Blanched Formula

r =

247 6.87.0

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5

Blanched Formula

.76 =

247

1.4 1.7

56.87.0

XY = 247

X = 7.0

Y = 6.8

Sx = 1.4

Sy = 1.7

N = 5

0

2

4

6

8

10

12

2 4 6 8 10

Hours Slept

Hap

pine

ss ..

.. .

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!

0

2

4

6

8

10

12

2 4 6 8 10

Hours Slept

Hap

pine

ss ..

.. .

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

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

.

.. ..

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

.

.. ..

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

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

.

.. ..


b =


b =2.41

1.41.881.50


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

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

Y = 0.1 + (1.5)X

.

.. ..

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

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

.

.

.. ..

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

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

.

.

.

.. ..

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

Y = 0.1 + (1.5)X

.

.

.

.. ..

Practice

AggressionY

HappinessX

Mr. Blond 10 9

Mr. Blue 20 4

Mr. Brown 12 5

Mr. Pink 16 6

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

b =


b =4.43

2.16-.57-1.17


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

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

Regression Equation

Y = a + bX

Y = 21.52 + (-1.17)X

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10

Happiness

Agg

ress

ion

.

.

..

10

12

14

16

18

20

22

Y = 21.52 + (-1.17)X

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10

Happiness

Agg

ress

ion

.

.

..

10

12

14

16

18

20

22

Y = 21.52 + (-1.17)X

.

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10

Happiness

Agg

ress

ion

.

.

..

10

12

14

16

18

20

22

Y = 21.52 + (-1.17)X

.

.

Sales($ thousands)

Y

Advertising($ thousands)

X70 3

120 4110 3100 5140 6120 5100 4

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

XY = 3360

X = 4.29

Y = 108.57

Sx = 1.03

Sy = 20.30

N = 7

r =

Blanched Formula

.68 =

3360

2.16 4.43

714.5

XY = 3360

X = 4.29

Y = 108.57

Sx = 1.03

Sy = 20.30

N = 7

(4.29) (108.57)

(1.03) (20.30)

b =

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

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

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

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?


Y = 51.08 + (13.40)X


• 77.88 = 51.08 + (13.40)2

• $77, 880

Y = 51.08 + (13.40)X


• 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

SmileY

TalkX

Y2 X2 XY

Jerry 9 5Elan 2 1George 5 3Newman 4 4Kramer 3 2

Step 2: Square Y

SmileY

TalkX

Y2 X2 XY

Jerry 9 5 81Elan 2 1 4George 5 3 25Newman 4 4 16Kramer 3 2 9

Step 3: Square X

Smile Y

Talk X

Y2 X2 XY

Jerry 9 5 81 25 Elan 2 1 4 1 George 5 3 25 9 Newman 4 4 16 16 Kramer 3 2 9 4

Step 4: Multiply XY

Smile Y

Talk X

Y2 X2 XY

Jerry 9 5 81 25 45 Elan 2 1 4 1 2 George 5 3 25 9 15 Newman 4 4 16 16 16 Kramer 3 2 9 4 6

Step 5: Sum

SmileY

TalkX

Y2 X2 XY

Jerry 9 5 81 25 45Elan 2 1 4 1 2George 5 3 25 9 15Newman 4 4 16 16 16Kramer 3 2 9 4 6

23 15 135 55 84

Step 6: Plug in values

r =

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5


r =15

15

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5


r =23

23

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

15

15


r =84

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

23

23

15

15


r =84

55

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

23

23

15

15


r =84

135

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

55

23

23

15

15


r =84(5)

(5) (5)

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

13555

23

23

15

15

Step 7: Solve!

r =84(5)

(5) (5)

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

13555

23

23

15

15

Step 7: Solve!

r =23 225

84(5)

(5) (5)225 529

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

13555

2315

Step 7: Solve!

r =23

23

15

225

23

275 675

420 345

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

225 529

Step 7: Solve!

r =23

23

15

225

23420 345

50 146

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

Step 7: Solve!

r =23

23

15

225

23

146 507300

75

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

Step 7: Solve!

r =23

23

15

225

23

146 507300

75

85.44

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

Step 7: Solve!

.88 =23

23

15

225

23

146 507300

75

85.44

Y = 23 Y2 = 135

X =15 X2 = 55

XY = 84 N = 5

Practice

AlcoholX

Nose TouchesY

Norm 6 5

Cliff 8 1

Sam 4 4

Woody 2 9

Practice

AlchX

NoseY

X2 Y2 XY

Norm 6 5Cliff 8 1Sam 4 4Woody 2 9

Practice

AlchX

NoseY

X2 Y2 XY

Norm 6 5 36 25 30Cliff 8 1 64 1 8Sam 4 4 16 16 16Woody 2 9 4 81 18

20 19 120 123 72

Practice

r =

X = 20 X2 = 120

Y =19 Y2 = 123

XY = 72 N = 4

20

20

19

19

72

120 123

(4)

(4) (4)

Practice

r =

X = 20 X2 = 120

Y =19 Y2 = 123

XY = 72 N = 4

20

20

19

19120 123(4) (4)

-92

Practice

r =

X = 20 X2 = 120

Y =19 Y2 = 123

XY = 72 N = 4

20 19-92

80 131

Practice

-.90 =

X = 20 X2 = 120

Y =19 Y2 = 123

XY = 72 N = 4

20 19-92

80 131102.37

Practice

• Page 116 -- # 21

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