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REGRESSION ANALYSIS M.Ravishankar [ And it’s application in Business ] 

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Introduction. . . Father of Regression Analysis Carl F. Gauss (17771855).
contributions to physics, Mathematics & astronomy.
The term “Regression” was first used in 1877 by Francis Galton.


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Regression Analysis. . . It is the study of the relationship between variables.
It is one of the most commonly used tools for business analysis.
It is easy to use and applies to many situations.


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Regression types. . . Simple Regression: single explanatory variable
Multiple Regression: includes any number of explanatory variables.


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Dependant variable: the single variable being explained/ predicted by the regression model
Independent variable: The explanatory variable(s) used to predict the dependant variable.
Coefficients (ß): values, computed by the regression tool, reflecting explanatory to dependent variable relationships.
Residuals (e): the portion of the dependent variable that isn’t explained by the model; the model under and over predictions.


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Regression Analysis. . .
Linear Regression: straightline relationship
Form: y=mx+b
Nonlinear: implies curved relationships
logarithmic relationships


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Regression Analysis. . . Cross Sectional: data gathered from the same time period
Time Series: Involves data observed over equally spaced points in time.


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Simple Linear Regression Model. . . Only one independent variable, x
Relationship between x and y is described by a linear function
Changes in y are assumed to be caused by changes in x


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Types of Regression Models. . .


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The sample regression line provides an estimate of the population regression line Estimated Regression Model. . . Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) y value Independent variable The individual random error terms ei have a mean of zero 

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Simple Linear Regression Example. . . A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)
A random sample of 10 houses is selected
Dependent variable (y) = house price in $1000s
Independent variable (x) = square feet




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Output. . . The regression equation is: 

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Graphical Presentation . . . House price model: scatter plot and regression line Slope
= 0.10977 Intercept
= 98.248 

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Interpretation of the Intercept, b0 b0 is the estimated average value of Y when the value of X is zero (if x = 0 is in the range of observed x values)
Here, no houses had 0 square feet, so b0 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet 

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Interpretation of the Slope Coefficient, b1 b1 measures the estimated change in the average value of Y as a result of a oneunit change in X
Here, b1 = .10977 tells us that the average value of a house increases by .10977($1000) = $109.77, on average, for each additional one square foot of size 

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Estimated Regression Equation: Example: House Prices Predict the price for a house with 2000 square feet 

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Example: House Prices Predict the price for a house with 2000 square feet: The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850 

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Coefficient of determination Coefficient of Determination, R2 Note: In the single independent variable case, the coefficient of determination is
where:
R2 = Coefficient of determination
r = Simple correlation coefficient 

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R2 = +1 Examples of Approximate R2 Values y x y x R2 = 1 R2 = 1 Perfect linear relationship between x and y:
100% of the variation in y is explained by variation in x 

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Examples of Approximate R2 Values y x y x 0 < R2 < 1 Weaker linear relationship between x and y:
Some but not all of the variation in y is explained by variation in x 

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Examples of Approximate R2 Values R2 = 0 No linear relationship between x and y:
The value of Y does not depend on x. (None of the variation in y is explained by variation in x) y x R2 = 0 

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Output. . . 58.08% of the variation in house prices is explained by variation in square feet 

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Standard Error of Estimate. . . The standard deviation of the variation of observations around the regression line is estimated by Where
SSE = Sum of squares error
n = Sample size
k = number of independent variables in the model 

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The Standard Deviation of the Regression Slope The standard error of the regression slope coefficient (b1) is estimated by where:
= Estimate of the standard error of the least squares slope
= Sample standard error of the estimate 

