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Basic Business Statistics:
Concepts & Applications
Simple Linear Regression
& Correlation
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Regression Analysis
Managerial decisions often are based on the relationshipbetween two or more variables .For eg Afterconsidering the relationship between advertising
expenditures and sales ,a marketing manager mightattempt to predict sales for a given level ofadvertising expenditures.
What is Regression analysis ?
Regression analysis establishes the nature of therelationship between various variables throughdevelopment of a mathematical function.
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Y Xi i i! F F I0 1
Linear Regression Model
Relationship Between Variables Is a
Linear Function
Dependent
(Response)
Variable
Independent
(Explanatory)
Variable
SlopeY-InterceptRandom
Error
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Ii= Random Error
Y
X
PopulationLinear Regression Model
Observed
Value
Observed Value
Q F FYX iX! 0 1
Y Xi i i! F F I0 1
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ei= Random
Error
Y
X
SampleLinear Regression Model
Observed Value
Unsampled
Observation
Y b b X ei i i! 0 1
Y b b X i i! 0 1
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Ordinary Least Squares
1. Best Fit Means Difference Between
Actual Values (Yi) & Predicted Values
( ) Are a Minimum
But Positive Differences Off-Set Negative
2. OLS Minimizes the Sum of the
Squared Differences (or Errors)
Yi
Y Yei i
i
n
ii
n
!! !
2
1
2
1
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e2
Y
X
e1 e3
e4
Ordinary Least SquaresGraphically
Y b b X ei i i! 0 1
Y b b X i i! 0 1
OLS Minimizes e e e e eii
n
2
1
1
222
3
24
2! !
PredictedValue
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Y b b X
b
X Y nXY
X n X
b Y b X
i i
i i
i
n
ii
n
!
!
!
!
!
0 1
11
2 2
1
0 1
Coefficient Equations
Sample Regression
Equation
Sample Slope
Sample Y-Intercept
n = # (Xi, Yi) Pairs
Average Xis,
Then Square
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Example (Ad -Sales)
Youre a marketing analyst for Hasbro
Toys. You gather the following data:
Ad Sales (Units)1 1
2 1
3 2
4 25 4
What is the relationship
between sales & advertising?
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0
1
2
34
0 1 2 3 4 5
Scatter DiagramExample: Ad-Sales
Sales
Advertising
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Example (Ad $-Sales)Solution Table
Xi Yi Xi2
Yi2
XiYi
1 1 1 1 12 1 4 1 2
3 2 9 4 6
4 2 16 4 85 4 25 16 20
15 10 55 26 37
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Example (Ad -Sales)Calculations
b
X Y nXY
X n X
b Y b X
Y X
i i
i
n
i
i
n
i i
1
1
2 2
1
0 1
37 5 3 2
55 5 0 70
2 0 70 3 0 10
0 10 0 70
!
!
!
! ! !
!
!
!
.
. .
. .
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Example (Ad -Sales)Interpretation
1. Slope (b1)
Sales Volume (Y) Is Expected to Increase
by 0.7 Units for Each Re1 Increase inAdvertising (X)
2. Y-Intercept (b0)
Average Value of Sales Volume (Y
) Is-.10 Units When Advertising (X) Is 0
Difficult to Explain to Marketing Manager
Expect Some Sales Without Advertising
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Correlation Coefficient
1. Answer How Strong Is the LinearRelationship Between 2 Variables?
2. Coefficient of Correlation Used Population correlation coefficient denoted
V (Rho) Values range from -1 to +1
Measures degree of association
3. Used Mainly for Understanding and forinterpretaion Coeff. Of Determination isused.
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Measures of Variationin Regression
1. Total Sum of Squares (SST)
Measures variation of observed Yi around
the mean,DY
2. Explained Variation (SSR)
Variation due to relationship between
X & Y
3. Unexplained Variation (SSE)
Variation due to other factors
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Y
X
DY
Xi
Variation Measures
Total Sum of
Squares (Yi- Y)2
Unexplained Sum of
Squares (Yi - Yi)2^
Explained Sum of
Squares (Yi - Y)2^
Yi
Y b b X i i! 0 1
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Proportion of Variation Explained by
Relationship Between X & Y
Coefficient of Determination
0e r2 e 1
r
b Y b X Y n Y
Y n Y
i i i
i
n
i
n
i
i
n
2
0 1
2
11
2
1
2
! !
!
!!
!
Explained Variation
Total Vari ation
SSR
SST
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Interpretation of r square is useful
Rsquare falls between -1
and +1
.It measures the amount of variation which
is explained by its relation to he
variable or by the regression line .
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Coefficients of Determination (r2)and Correlation (r)
r2 = 1, r2 = 1,
r2 = .8, r2 = 0,Y
Yi= b
0+b1Xi
X
^
Y
Yi= b
0+b1Xi
X
^
Y
Yi= b
0+b1Xi
X
^
Y
Yi= b
0+b1Xi
X
^
r = +1 r = -1
r = +0.9 r = 0
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Corr Cofficient calculations
By using the formula r square is found to be equal to0.8167.
The Coefficient of correlation is - 0.9037. the sign of r is
determined by the sign of the slope of theregression line .
81% of the variation in sales is explained by the variationin the advt amount .The remaining 19%of thevariation is due to factors other than Advt Amount
.the reg line is being evaluated as good and can beused for predictive purposes . Suppose if Rs 6000 isspent on advertising the sales is estimated to be4100 units.
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Pearson Product-Moment Coefficient
of Correlation
SampleCoefficient of Correlation
r
X X Y Y
X X Y Y
i i
i
n
i
i
n
i
i
n
!
!
!
! !
1
2
1
2
1
Coefficient of Determination
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