k_ppt- simple regression and correlation

8/8/2019 K_PPT- Simple Regression and Correlation

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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

k_ppt- simple regression and correlation

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