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Praxis Weekend Business Analytics

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- 1. Correlation and Regression Analysis: Learning Objectives • Explain the purpose of regression analysis and the meaning of independent versus dependent variables. • Compute the equation of a simple regression line from a sample of data, and interpret the slope and intercept of the equation. • Estimate values of Y to forecast outcomes using the regression model. • Understand residual analysis in testing the assumptions and in examining the fit underlying the regression line. • Compute a standard error of the estimate and interpret its meaning. • Compute a coefficient of determination and interpret it.
- 2. Correlation • Correlation is a measure of the degree of relatedness of variables. • Coefficient of Correlation (r) - applicable only if both variables being analyzed have at least an interval level of data.
- 3. Three Degrees of Correlation r<0 r>0 r=0
- 4. Degree of Correlation • The term (r) is a measure of the linear correlation of two variables – The number ranges from -1 to 0 to +1 Positive correlation: as one variable increases, the other variable increases Negative correlation: as one variable increases, the other one decreases No correlation: the value of r is close to 0 – Closer to +1 or -1, the higher the correlation between two variables
- 5. Pearson Product-Moment Correlation Coefficient
- 6. Regression Analysis • Regression analysis is the process of constructing a mathematical model or function that can be used to predict or determine one variable by another variable or variables.
- 7. Simple Regression Analysis • Bivariate (two variables) linear regression -- the most elementary regression model – dependent variable, the variable to be predicted, usually called Y – independent variable, the predictor or explanatory variable, usually called X – Usually the first step in this analysis is to construct a scatter plot of the data • Nonlinear relationships and regression models with more than one independent variable can be explored by using multiple regression models
- 8. Regression Models • Deterministic Regression Model - - produces an exact output: ˆ y 0 1 x • Probabilistic Regression Model ˆ y 0 1 x • 0 and 1 are population parameters • 0 and 1 are estimated by sample statistics b0 and b1
- 9. Equation of the Simple Regression Line
- 10. A typical regression line Y ϴ X
- 11. Least Squares Analysis • Least squares analysis is a process whereby a regression model is developed by producing the minimum sum of the squared error values • The vertical distance from each point to the line is the error of the prediction. • The least squares regression line is the regression line that results in the smallest sum of errors squared.
- 12. Least Squares Analysis X X Y Y XY nXY b X n X X X 2 1 2 2 Y X b Y b X n b n 0 1 1 X Y XY n X 2 X n 2
- 13. Least Squares Analysis SSXY X X Y Y SSXX b1 X X 2 X X Y XY n 2 X 2 n SSXY SSXX Y X b Y b X n b n 0 1 1
- 14. Airlines Cost Data include the costs and associated number of passengers for twelve 500-mile commercial airline flights using Boeing 737s during the same season of the year. Number of Passengers 61 63 67 69 70 74 76 81 86 91 95 97 Cost ($1,000) 4,280 4,080 4,420 4,170 4,480 4,300 4,820 4,700 5,110 5,130 5,640 5,560
- 15. Number of Passengers x x2 61 63 67 69 70 74 76 81 86 91 95 97 x Cost ($1,000) y 4.28 4.08 4.42 4.17 4.48 4.30 4.82 4.70 5.11 5.13 5.64 5.56 3,721 3,969 4,489 4,761 4,900 5,476 5,776 6,561 7,396 8,281 9,025 9,409 = 930 y = 56.69 x 2 = 73,764 xy 261.08 257.04 296.14 287.73 313.60 318.20 366.32 380.70 439.46 466.83 535.80 539.32 xy = 4,462.22
- 16. SS XY XY SS XX X b1 b0 2 X Y n ( X ) 2 n 4,462 .22 (930 )( 56 .69 ) 68 .745 12 (930 ) 2 73,764 1689 12 SS XY 68 .745 .0407 SS XX 1689 Y n b1 X n ˆ Y 1.57 .0407 X 56 .69 930 (. 0407 ) 1.57 12 12
- 17. Residual Analysis
- 18. Residual Analysis: Airline Cost Example Number of Passengers X 61 63 67 69 70 74 76 81 86 91 95 97 Cost ($1,000) Y Predicted Value ˆ Y Residual ˆ Y Y 4.28 4.08 4.42 4.17 4.48 4.30 4.82 4.70 5.11 5.13 5.64 5.56 4.053 4.134 4.297 4.378 4.419 4.582 4.663 4.867 5.070 5.274 5.436 5.518 .227 -.054 .123 -.208 .061 -.282 .157 -.167 .040 -.144 .204 .042 (Y Yˆ ) .001
- 19. Residual Analysis: Airline Cost Example Outliers: Data points that lie apart from the rest of the points. They can produce large residuals and affect the regression line.
- 20. Using Residuals to Test the Assumptions of the Regression Model • The assumptions of the regression model – The model is linear – The error terms have constant variances – The error terms are independent – The error terms are normally distributed
- 21. Using Residuals to Test the Assumptions of the Regression Model • The assumption that the regression model is linear does not hold for the residual plot shown above • In figure (a) below the error variance is greater for smaller values of x and smaller for larger values of x and vice-versa in figure (b) below. This is a case of heteroscedasiticity.
- 22. Standard Error of the Estimate • Residuals represent errors of estimation for individual points. • A more useful measurement of error is the standard error of the estimate. • The standard error of the estimate, denoted by se, is a standard deviation of the error of the regression model.
- 23. Standard Error of the Estimate Sum of Squares Error SSE Standard Error of the Estimate Y Y 2 Y b0 Y b1 XY 2 SSE Se n 2
- 24. Determining SSE for the Airline Cost Data Example Number of Passengers X Cost ($1,000) Y Residual ˆ Y Y ˆ (Y Y ) 2 61 63 67 69 70 74 76 81 86 91 95 97 4.28 4.08 4.42 4.17 4.48 4.30 4.82 4 .70 5.11 5.13 5.64 5.56 .227 -.054 .123 -.208 .061 -.282 .157 -.167 .040 -.144 .204 .042 .05153 .00292 .01513 .04326 .00372 .07952 .02465 .02789 .00160 .02074 .04162 .00176 (Y ˆ Y ) .001 (Y ˆ Y ) 2 =.31434 Sum of squares of error = SSE = .31434
- 25. • The coefficient of determination is the proportion of variability of the dependent variable (y) accounted for or explained by the independent variable (x) • The coefficient of determination ranges from 0 to 1. • An r 2 of zero means that the predictor accounts for none of the variability of the dependent variable and that there is no regression prediction of y by x. • An r 2 of 1 means perfect prediction of y by x and that 100% of the variability of y is accounted for by x.
- 26. SSYY Y Y Y 2 Y 2 2 n SSYY exp lained var iation un exp lained var iation SSYY SSR SSE SSR SSE 1 SSYY SSYY SSR 2 r SSYY SSE 1 SSYY SSE 1 2 Y 2 Y n
- 27. SSE 0.31434 Y 270.9251 56.69 3.11209 Y 2 SSYY 2 n SSE r 1 SSYY .31434 1 3.11209 .899 2 2 12 89.9% of the variability of the cost of flying a Boeing 737 is accounted for by the number of passengers.
- 28. Exercise in R: Linear Regression Open URL: www.openintro.org Go to Labs in R and select 7 - Linear Regression

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