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 1 : REGRESSION ANALYSIS M.Ravishankar [ And it’s application in Business ]
 2 : Introduction. . . Father of Regression Analysis Carl F. Gauss (1777-1855). contributions to physics, Mathematics & astronomy. The term “Regression” was first used in 1877 by Francis Galton.
 3 : 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.
 4 : Regression types. . . Simple Regression: single explanatory variable Multiple Regression: includes any number of explanatory variables.
 5 : 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.
 6 : Regression Analysis. . . Linear Regression: straight-line relationship Form: y=mx+b Non-linear: implies curved relationships logarithmic relationships
 7 : Regression Analysis. . . Cross Sectional: data gathered from the same time period Time Series: Involves data observed over equally spaced points in time.
 8 : 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
 9 : Types of Regression Models. . .
 10 : 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
 11 : 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
 12 : Sample Data
 13 :
 14 : Output. . . The regression equation is:
 15 : Graphical Presentation . . . House price model: scatter plot and regression line Slope = 0.10977 Intercept = 98.248
 16 : 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
 17 : Interpretation of the Slope Coefficient, b1 b1 measures the estimated change in the average value of Y as a result of a one-unit 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
 18 : Estimated Regression Equation: Example: House Prices Predict the price for a house with 2000 square feet
 19 : 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
 20 : 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
 21 : 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
 22 : 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
 23 : 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
 24 : Output. . . 58.08% of the variation in house prices is explained by variation in square feet
 25 : 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
 26 : 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
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