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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. - PowerPoint PPT Presentation
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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
.
.. ..
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
b =2.41
1.41.881.50
Mean Y = 4.6; SY = 2.41 r = .88Mean X = 3.0; SX = 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
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 =
Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 r = -.57
b =4.43
2.16-.57-1.17
Mean Y = 14.50; Sy = 4.43Mean X = 6.00; Sx= 2.16 r = -.57
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
.
.
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?
• 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
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
Step 6: Plug in values
r =15
15
Y = 23 Y2 = 135
X =15 X2 = 55
XY = 84 N = 5
Step 6: Plug in values
r =23
23
Y = 23 Y2 = 135
X =15 X2 = 55
XY = 84 N = 5
15
15
Step 6: Plug in values
r =84
Y = 23 Y2 = 135
X =15 X2 = 55
XY = 84 N = 5
23
23
15
15
Step 6: Plug in values
r =84
55
Y = 23 Y2 = 135
X =15 X2 = 55
XY = 84 N = 5
23
23
15
15
Step 6: Plug in values
r =84
135
Y = 23 Y2 = 135
X =15 X2 = 55
XY = 84 N = 5
55
23
23
15
15
Step 6: Plug in values
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
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