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Regression Analysis Simple Regression

Regression Analysis Simple Regression. y = mx + b. y = a + bx. y = a + bx. where: y dependent variable (value depends on x) a y-intercept (value of y when x = 0) b slope (rate of change in ratio of delta y divided by delta x) x independent variable. Assumptions. Linearity

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Regression Analysis Simple Regression

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  1. Regression AnalysisSimple Regression

  2. y = mx + b y = a + bx

  3. y = a + bx where: ydependent variable(value depends on x) ay-intercept(value of y when x = 0) bslope (rate of change in ratio of delta y divided by delta x) x independentvariable

  4. Assumptions Linearity Independence of Error Homoscedasticity Normality

  5. Linearity The most fundamental assumption is that the model fits the situation [i.e.: the Y variable is linearly related to the value of the X variable].

  6. Independence of Error The error (residual) is independent for each value of X. [Residual = observed - predicted]

  7. Homoscedasticity The variation around the line of regression constant for all values of X.

  8. Normality The values of Y be normally distributed at each value of X.

  9. Linearity Independence Examine scatter plot of residuals versus fitted [Yhat] for evidence of nonlinearity Plot residuals in time order and look for patterns Diagnostic Checking

  10. Homoscedasticity Normality Examine scatter plots of residuals versus fitted [Yhat] and residuals vs time order and look for changing scatter. Examine histogram of residuals. Look for departures from normal curve. Diagnostic Checking

  11. Goal Develop a statistical model that can predict the values of a dependent (response) variable based upon the values of the independent (explanatory) variable(s).

  12. Goal

  13. Simple Regression A statistical model that utilizes onequantitativeindependent variable “X” to predict the quantitativedependent variable “Y.”

  14. Mini-Case Since a new housing complex is being developed in Carmichael, management is under pressure to open a new pie restaurant. Assuming that population and annual sales are related, a study was conducted to predict expected sales.

  15. Mini-Case(Descartes Pie Restaurants)

  16. Mini-Case • What preliminary conclusions can management draw from the data? • What could management expect sales to be if population of the new complex is approximately 18,000 people?

  17. Scatter Diagrams • The values are plotted on a two-dimensional graph called a “scatter diagram.” • Each value is plotted at its X and Y coordinates.

  18. Scatter Plot of Pieshop

  19. Types of Models No relationship between X and Y Positive linear relationship Negative linear relationship

  20. Method of Least Squares • The straight line that best fits the data. • Determine the straight line for which the differences between the actual values (Y) and the values that would be predicted from the fitted line of regression (Y-hat) are as small as possible.

  21. Measures of Variation Explained Unexplained Total

  22. Explained Variation Sum of Squares (Yhat - Ybar)2 due to Regression [SSR]

  23. Unexplained Variation Sum of Squares (Yobs - Yhat)2 Error [SSE]

  24. Total Variation Sum of Squares (Yobs - Ybar)2 Total [SST]

  25. H0: There is no linear relationship between the dependent variable and the explanatory variable

  26. Hypotheses H0:  = 0 H1:  0 or H0: No relationship exists H1: A relationship exists

  27. Analysis of Variance for Regression

  28. Standard Error of the Estimate sy.x -the measure of variability around the line of regression

  29. Relationship When null hypothesis is rejected, a relationship between Y and X variables exists.

  30. Coefficient of Determination R2 measures the proportion of variation that is explained by the independent variable in the regression model. R2 = SSR / SST

  31. Confidence interval estimates • True mean YX • Individual Y-hat

  32. Pieshop Forecasting

  33. Coefficient of Sanity

  34. Diagnostic Checking • H0retain or reject {Reject if p-value  0.05} • R2 (larger is “better”) • sy.x (smaller is “better”)

  35. Analysis of Variance for Regression for Pieshop

  36. Coefficient of Determination R2 = SSR / SST = 90.27 % thus, 90.27 percent of the variation in annual sales is explained by the population.

  37. Standard Error of the Estimate sy.x = 13.8293 with SSE = 1,530.0

  38. Regression Analysis[Simple Regression] *** End of Presentation *** Questions?

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