1. Topic 8 Correlation and Regression Analysis
Business Statistics
Topic 8 Correlation and Regression Analysis

2. Business Statistics:Topic 8
Learning Objectives
By the end of this topic you will be able to:
Measure the strength of association using correlation analysis
Use Linear Regression models
Set up the linear regression equation
Make predictions using the regression equation
Test goodness of fit of the regression model
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3. Business Statistics:Topic 8
Correlation analysis
A statistical technique used to measure the
strength of the relationship (correlation)
between two variables.
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4. Business Statistics:Topic 8
Examples
Is there any relationship between height & weight?
Is there any relationship between housing loan interest rates and the number of housing loans?
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5. Correlation coefficient
The index which defines the strength of the association between two given variables
Population correlation coefficient  (Rho)
Sample correlation coefficient r is used to estimate 
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6. Choosing Variables for Assessing
Dependent Variable is the one whose value is to be predicted - usually denoted by ‘Y’.
Independent Variable is the one whose value is used to make the prediction- usually denoted by ‘X’.
Assumption: The two variables are bivariate normally distributed.
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7. Business Statistics:Topic 8
Scatter Plot
A scatter diagram is a display in which the ordered pairs of measurements are plotted on a coordinate axes.
The chart portrays the relationship between the two variables of interest.
Plot of all (Xi, Yi) pairs
Suggests how the the two variables are related
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8. Business Statistics:Topic 8
Example
A well known company wants to assess:
How are ‘advertising cost’ and ‘sales’ related?
What is the dependent variable?
‘Sales’ chosen as ‘Y’ for the analysis
Advertising cost chosen as ‘X’ for the analysis
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9. Business Statistics:Topic 8
What Do you Notice?
Advertising and sales move together (positive)
The relationship appears to be linear
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10. Business Statistics:Topic 8
Scatter Plots
Positive Linear Relationship
Negative Linear Relationship
Non- Linear Relationship
No Relationship
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11. Sample Correlation Coefficient ‘r’
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12. Properties of the correlation coefficient
r always lies between -1 and +1.
r = 1, the two variables have perfect positive correlation. On a scatter plot, this means that the points all lie on a straight line with positive slope.
r = -1, the two variables have perfect negative correlation. On a scatter plot, this means that the points all lie on a straight line with negative slope
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13. Business Statistics:Topic 8
Properties
Closer to +1 strong positive linear relation
Closer to –1 strong negative linear relation
Closer to 0 weak linear relationship
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14. Scatter plot & ‘r’ values
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15. Population Correlation Coefficient 
Is used to measure the strength of relationship
between any two variables.
The sample correlation coefficient ‘r’ is an
estimate of 
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16. Testing for Correlation
Hypothesis
H0:  = 0 (There is no correlation)
H1:   0 (There is correlation)
Rule
Two sided ‘t’ test with  = .05 and degrees of
freedom n-2: t,n-2
Test Statistic:
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17. Example: WOM Insurance company
An insurance company manager is concerned with the health of female adults, since the company is prepared to give a reduced premium rate to those who have a certain level of fitness. In particular, he would like to investigate how their height is related to their weight, with a view to possibly using these measurements as a fitness criterion.
He selects a random sample of 12 adult females and measures both their height and weight. The results are shown in the following table:
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18. Example: Assessing Height & Weight
Since the value of ‘r’ is close to 1 (.82), there is a strong positive linear relationship between ‘height’ and ‘weight’.
r= 0.82 is an estimate of 
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19. Testing Correlation: Height & Weight
Test Statistic
Critical Value using  = .05 and degrees of freedom n-2 is
Decision: Reject H0
Conclusion: There is evidence that there is a correlation between Height & Weight
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20. Business Statistics:Topic 8
Regression Analysis
Regression Analysis is used to establish a mathematical equation between two related variables.
Predict variable (Y) based on the independent variable(X)
Explain the effect of the independent variable (X) on the dependent variable (Y)
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21. Sample Linear Regression Model
Y is the dependent variable
X is the independent variable
is the sample Y-intercept
is the sample slope coefficient
e is the residual or error
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22. Sample Linear Regression Function: SRF
Is used for predicting the Y value for a given X
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23. Sample Regression coefficients
is the value of Y when the value of X is
zero.
measures the change in the value of Y as a
result of a one-unit change in X.
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24. Sample Regression Coefficients
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25. Example: WOM Insurance Company
= b0 + b1 x =
= X
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26. Business Statistics:Topic 8
Slope
The slope b1 means that for each one unit of increase in X, the estimated Y value increases by 1.05 units
In this example, if the height increases by 1 cm, then the estimated weight increases by 1.05 kg.
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27. Coefficient of Determination
Measures the percentage of variation in Y that is explained by the independent variable X
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28. Example Height & Weight
r2 = 0.67
67% of the variation in the weight of people can be explained by the variability in the height of people
Also if r2 is close to 1, then the linear model fits the data well.
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29. Business Statistics:Topic 8
Estimation Using SRF
To estimate the weight if a person is 170 cm tall,
Put x=170, in the SRF
r2 = 0.67
As r2 is close to 1, the linear model fits in the data well. Hence the estimation is good
= b0 + b1 x =
= X
= *170
 72 kg
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30. Estimated Population Linear Regression Equation
Y is the dependent variable
X is the independent variable
is the population Y-intercept
is the population slope coefficient
is a point estimate of
is a point estimate of
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31. Estimated Regression Equation
Height & Weight
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32. Business Statistics:Topic 8
Testing the Slope
Hypothesis
H0: 1 = 0 (no linear dependence)
H1: 1  0 (linear dependence)
Rule
Two sided ‘t’ test with  = .05 and degrees of freedom n-2: t,n-2
Test Statistic:
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33. Testing the slope: Height & Weight Model
Test statistic
Critical value with  = .05 and degrees of freedom n-2=10 is 2.228
Decision: Reject H0
Conclusion: Evidence that height & weight have a linear relationship
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34. Business Statistics:Topic 8
Summary
In this topic you have discussed:
Assessing the strength of relationship between variables
The Linear Regression model
Estimation using the linear regression equation
Testing the estimation model
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