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This lecture is from Statistics. Key important points are: Association Between Two Variables, Regression Analysis, Covariance and Correlation Coefficient, Three Degrees of Correlation, Degree of Relatedness, Correlation Coefficient, Strong Negative Linear Relationship, Correlation Coefficient, Strong Positive Linear Relationship, Closer the Correlation
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Uploaded on 01/29/2013
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As a decision maker many times we are interested in the relationship between two variables.
Two descriptive measures of the relationship
between two variables are covariance and correlation
coefficient.
Regression Analysis explains both nature and the
strength of the relationship between variables.
Just because two variables are highly correlated, it
does not mean that one variable is the cause of the
other. High correlation represent the strength of the
relationship.
Correlation is a measure of the degree of relatedness
of variables and do not explain the cause and effect
relationship.
Correlation Coefficient
Values near +1 indicate a strong positive linear
relationship.
Values near -1 indicate a strong negative linear
relationship.
The coefficient can take on values between -1 and +1.
The closer the correlation is to zero, the weaker the
relationship.
Three Degrees of Correlation
r < 0 r > 0
r = 0
x y
( x (^) i − x ) ( yi − y ) (^ x^ i −^ x^ )(^ yi^ − y )
Average
Std. Dev.
Total
Example: Golfing Study
xy xy x y
s r s s
i i xy
x x y y s n
∑
Example: Golfing Study
Year Amount Spent on Research and Development (Millions)
Annual Profit (Millions)
1990 2 20
1991 3 25
1992 5 34
1993 4 30
1994 11 40
1995 5 31
The vice president for R&D wants an equation for predicting
annual profit from the amount budgeted for R&D.
y = β 0 + β 1 x + ε
where:
The simple linear regression model is:
The equation that describes how y is related to x and
an error term is called the regression model.
Simple Linear Regression Equation
n The simple linear regression equation is:
Simple Linear Regression Equation
n Negative Linear Relationship
y
x
is negative
Intercept^ Regression line
Simple Linear Regression Equation
n No Relationship
y
x
is 0
Regression line Intercept
Regression Model y = β 0 + β 1 x + ε Regression Equation E ( y ) = β 0 + β 1 x
Unknown Parameters
β 0 , β 1
Sample Data: x y
x 1 y 1
.. .. x (^) n y (^) n
b 0 and b 1 provide estimates of β 0 and β 1
Estimated Regression Equation
Sample Statistics b 0 , b 1
y ˆ = b 0 (^) + b x 1
developed by minimizing the sum of squared error.
min (^) ∑ ( y (^) i − y ^ i )
2
where:
yi = observed value of the dependent variable
for the i th observation ^ yi = estimated value of the dependent variable
for the i th observation
