ࡱ> HJEFGu bjbj `wjjd)lLLLLPPPd%%%8"&Tv&d@h6'6(L(L(x(9991TIT$ l9P95.999;;LLL(x([ ;;;;;;9|LL(Px(;;9;;* ;;eEp JPL(,*' A[Uxd%}9.t0@ 4Y:4;;ddLLLL Chapter 11 Correlation and Regression Estimation of Relationship Statistics as a Core Knowledge of Management The social scientist will acquire his dignity and his strength when he has worked out his method. He will do that by turning into opportunity the need of the great society for instruments of analysis by which an invisible and most stupendously difficult environment can be made intelligible. (Walter Lippman in Public Opinion) Issue: Understanding the population using multivariate data. Regression The framework: Identify one variable as the dependent variable. The rest are independent variables (predictors, explanatory variables, etc.) Goal of data analysis: Forecasting Estimation of the effect Examples for bivariate data: Income and Vacation Expenditures Target Population: families living a city Sales Area and Sales Target Population: All stores of a chain Price and Unit Sales for a New Shampoo Formula Target Population: All test settings Distance to the UW Campus and Apartment Rent Target Population: All apartments in Seattle Four Steps for Regression Scatterplot and Correlation Coefficient Least Squares Regression Equation Specification of Population Regression and Data Collection Model Statistical Inference on the Population Regression Coefficients Scatterplot Display of Bivariate Data Y = Salary vs. X =Years Experience (n = 6 employees)  We learn from the plot: Increasing (b) Linear (c) Not exact , but strong (Correlation r = 0.8667) Example 1: Mortgage Rates & Fees Y = Interest Rate vs. X = Loan Fee  Description of the Relationship (Correlation is r = 0.441) Example 2: The Stock Market Y = Todays vs. X = Yesterdays Percent Change  (Correlation is r = 0.18) Example 3: Maximizing Yield Y = Output Yield vs. X =Temperature for an industrial process.  (a) (b) (c) (Correlation r = 0.0155) Example 4 : Telecommunications Y = Circuit Miles vs. X = Investment (lower left) For telecommunications firms  Special Feature: Unequal Variability Variability is stabilized by taking logarithms. Correlation r = 0.820 after the log transformation Example 5: Cost and Quantity Y = Cost vs. X = Number Produced  Special Feature: An outlier is visible: The correlation coefficient is unreliable Scatterplot and Correlation Coefficient Computer Demonstration  Computing the Sample Covariance and Correlation Coefficient, r Formula (see text page 439)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Example: n = 6 SDX = SDY = Covariance = The Least-Squares Regression Line Y=a+bX  the regression line summarizes scatterplot for predicting Y from X least squares line: the line that minimizes the sum of squared deviations (actual Y - predicted Y) computing the slope and the Intercept slope  EMBED Equation.2  = 15.318 intercept  EMBED Equation.DSMT4  = 1.673 Excel Regression Menu Output   Performance of the Regression Line Checking by Individual Data Points Predicted Value Residual = Data Value Predicted Value Overall Measures Standard Error of Estimate R-Squared or Adjusted R-squared 1. Predicted Values and Residuals Predicted Value comes from Least-Squares Line Mary (with 20 years of experience) has predicted salary 15.32+1.673(20) = 48.8 Residual = actual Y - predicted Y Marys residual = 55 48.8 = 6.2 She earns about $6,200 more than the predicted salary for a person with 20 years of experience A person who earns less than predicted will have a negative residual  2. Understanding the SE of Estimate & R-Squared   2 (a): Computing the Standard Error of Estimate Approximate size of prediction errors (residuals)  EMBED Equation.DSMT4  Example (Salary vs. Experience) Se = 6.52  Interpretation Actual salaries are about 6.52 (i.e., $6,520) away from the predicted salaries. About 68% of the data are within one standard error of estimate of the least-squares line   2 (b): Computing the R-Squared, Adjusted R-Squared  Prediction MethodUnexplained SSUnexplained Variance  Sample Mean  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Regression  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Ratio  EMBED Equation.DSMT4  EMBED Equation.DSMT4  R-Squared Unadjusted / Adjusted   EMBED Equation.DSMT4  EMBED Equation.DSMT4  Population Regression as Model of Data Collection Population Regression Y = a + bX + e a + bX Population relationship, on average e Randomness of individuals e follows N(0, s)  Inference for the Population Slope, b Standard Error of the Slope  EMBED Unknown  Approximately how far the observed slope b is from the population slope, b. Example (Salary vs. Experience) Sb = 95% Confidence Interval for b. Test of Significance of the Slope Why Test? Illusory Regression seem to show a relationship, when, in fact, the population is just random. 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