Halo Halo

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1/25/2013

After this, you should be able to:




Ronan Sierra Santos
MBA, Ed.D (ABD)
Twitter: @iRemainRonan









A scatter plot (or scatter diagram) is used to
show the relationship between two variables

Calculate and interpret the simple
correlation between two variables
Determine whether the correlation is
significant
Calculate and interpret the simple linear
regression equation for a set of data
Recognize regression analysis applications
for purposes of prediction and description
Determine whether a regression model is
significant

Linear relationships

Curvilinear relationships

y

y

Correlation analysis is used to measure
strength of the association (linear
relationship) between two variables

◦ Only concerned with strength of the
relationship

x

x

y

y

◦ No causal effect is implied
x

x

(continued)
Strong relationships
y

y

y

x
y

x

x
y

y

x

(continued)
No relationship

Weak relationships

x

x

1

1/25/2013

(continued)




The population correlation coefficient ρ
(rho) measures the strength of the
association between the variables





The sample correlation coefficient r is
an estimate of ρ and is used to measure
the strength of the linear relationship in
the sample observations



Sample correlation coefficient,
algebraic equivalent:

y

y

y



Unit free
Range between -1 and 1
The closer to -1, the stronger the negative
linear relationship
The closer to 1, the stronger the positive
linear relationship
The closer to 0, the weaker the linear
relationship

n xy   x y

r

x

r = -1
y

x

x

r = -.6

[n( x 2 )  ( x )2 ][n( y 2 )  ( y )2 ]

r=0

y
where:

r = +.3

x

r = +1

r = Sample correlation coefficient
n = Sample size
x = Value of the independent variable
y = Value of the dependent variable

x

(continued)

Tree
Height

Trunk
Diamete
r

y

x

xy

y2

x2

35

8

280

1225

64

50

49

9

441

2401

81

40

27

7

189

729

49

30

33

6

198

1089

36

Tree
Height,
y 70

r

n xy   x  y

[n(  x 2 )  (  x) 2 ][n(  y 2 )  (  y) 2 ]

60



8(3142)  (73)(321)
[8(713)  (73)2 ][8(14111)  (321)2 ]

 0.886

20

10

60

13

780

3600

169

21

7

147

441

49

45

11

495

2025

12

612

2601

 =321

 =73

r = 0.886 → relatively strong positive
linear association between x and y

121

51

 =3142  =14111

0
0

2

4

6

8

10

Trunk Diameter, x

12

14

144
 =713

2

1/25/2013



Regression analysis is used to:
◦ Predict the value of a dependent variable based
on the value of at least one independent
variable
◦ Explain the impact of changes in an
independent variable on the dependent
variable







Dependent variable: the variable we wish to
explain

Only one independent variable, x
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