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William G. Zikmund, Barry J. Babin, Jon C. Carr, Mitch Griffin

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- 1. Business Research Methods William G. Zikmund Chapter 23 Bivariate Analysis: Measures of Associations
- 2. Measures of Association • A general term that refers to a number of bivariate statistical techniques used to measure the strength of a relationship between two variables.
- 3. Relationships Among Variables • Correlation analysis • Bivariate regression analysis
- 4. Type of Measurement Measure of Association Interval and Ratio Scales Correlation Coefficient Bivariate Regression
- 5. Type of Measurement Measure of Association Ordinal Scales Chi-square Rank Correlation
- 6. Type of Measurement Measure of Association Nominal Chi-Square Phi Coefficient Contingency Coefficient
- 7. Correlation Coefficient • A statistical measure of the covariation or association between two variables. • Are dollar sales associated with advertising dollar expenditures?
- 8. The Correlation coefficient for two variables, X and Y is xyr .
- 9. Correlation Coefficient • r • r ranges from +1 to -1 • r = +1 a perfect positive linear relationship • r = -1 a perfect negative linear relationship • r = 0 indicates no correlation
- 10. ( )( ) ( ) ( )∑ ∑ ∑ −− −− == 22 YYiXXi YYXX rr ii yxxy Simple Correlation Coefficient
- 11. 22 yx xy yxxy rr σσ σ == Simple Correlation Coefficient
- 12. = Variance of X = Variance of Y = Covariance of X and Y 2 xσ 2 yσ xyσ Simple Correlation Coefficient Alternative Method
- 13. X Y NO CORRELATION . Correlation Patterns
- 14. X Y PERFECT NEGATIVE CORRELATION - r= -1.0 . Correlation Patterns
- 15. X Y A HIGH POSITIVE CORRELATION r = +.98 . Correlation Patterns
- 16. Pg 629 ( )( )589.5837.17 3389.6− =r 712.99 3389.6− = 635.−= Calculation of r
- 17. Coefficient of Determination Variance variance2 Total Explained r =
- 18. Correlation Does Not Mean Causation • High correlation • Rooster’s crow and the rising of the sun – Rooster does not cause the sun to rise. • Teachers’ salaries and the consumption of liquor – Covary because they are both influenced by a third variable
- 19. Correlation Matrix • The standard form for reporting correlational results.
- 20. Correlation Matrix Var1 Var2 Var3 Var1 1.0 0.45 0.31 Var2 0.45 1.0 0.10 Var3 0.31 0.10 1.0
- 21. Walkup’s First Laws of Statistics • Law No. 1 – Everything correlates with everything, especially when the same individual defines the variables to be correlated. • Law No. 2 – It won’t help very much to find a good correlation between the variable you are interested in and some other variable that you don’t understand any better.
- 22. • Law No. 3 – Unless you can think of a logical reason why two variables should be connected as cause and effect, it doesn’t help much to find a correlation between them. In Columbus, Ohio, the mean monthly rainfall correlates very nicely with the number of letters in the names of the months! Walkup’s First Laws of Statistics
- 23. Going back to previous conditionsGoing back to previous conditions Tall men’s sonsTall men’s sons DICTIONARYDICTIONARY DEFINITIONDEFINITION GOING ORGOING OR MOVINGMOVING BACKWARDBACKWARD Regression
- 24. Bivariate Regression • A measure of linear association that investigates a straight line relationship • Useful in forecasting
- 25. Bivariate Linear Regression • A measure of linear association that investigates a straight-line relationship • Y = a + bX • where • Y is the dependent variable • X is the independent variable • a and b are two constants to be estimated
- 26. Y intercept • a • An intercepted segment of a line • The point at which a regression line intercepts the Y-axis
- 27. Slope • b • The inclination of a regression line as compared to a base line • Rise over run • D - notation for “a change in”
- 28. Y 160 150 140 130 120 110 100 90 80 70 80 90 100 110 120 130 140 150 160 170 180 190 X My line Your line . Scatter Diagram and Eyeball Forecast
- 29. 130 120 110 100 90 80 80 90 100 110 120 130 140 150 160 170 180 190 X Y . XaY βˆˆˆ += X∆ Yˆ∆ Regression Line and Slope
- 30. X Y 160 150 140 130 120 110 100 90 80 70 80 90 100 110 120 130 140 150 160 170 180 190 Y “hat” for Dealer 3 Actual Y for Dealer 7 Y “hat” for Dealer 7 Actual Y for Dealer 3 Least-Squares Regression Line
- 31. 130 120 110 100 90 80 80 90 100 110 120 130 140 150 160 170 180 190 X Y } } { Deviation not explained Total deviation Deviation explained by the regression Y . Scatter Diagram of Explained and Unexplained Variation
- 32. The Least-Square Method • Uses the criterion of attempting to make the least amount of total error in prediction of Y from X. More technically, the procedure used in the least-squares method generates a straight line that minimizes the sum of squared deviations of the actual values from this predicted regression line.
- 33. The Least-Square Method • A relatively simple mathematical technique that ensures that the straight line will most closely represent the relationship between X and Y.
- 34. Regression - Least-Square Method ∑= n i ie 1 2 minimumis
- 35. = - (The “residual”) = actual value of the dependent variable = estimated value of the dependent variable (Y hat) n = number of observations i = number of the observation ie iY iYˆ iY iYˆ
- 36. The Logic behind the Least- Squares Technique • No straight line can completely represent every dot in the scatter diagram • There will be a discrepancy between most of the actual scores (each dot) and the predicted score • Uses the criterion of attempting to make the least amount of total error in prediction of Y from X
- 37. XYa βˆˆ −= Bivariate Regression
- 38. ( ) ( )( ) ( ) ( )22 ˆ ∑∑ ∑∑∑ − − = XXn YXXYn β Bivariate Regression
- 39. = estimated slope of the line (the “regression coefficient”) = estimated intercept of the y axis = dependent variable = mean of the dependent variable = independent variable = mean of the independent variable = number of observations βˆ X Y n aˆ Y X
- 40. ( ) ( ) 625,515,3759,24515 875,806,2345,19315ˆ − − =β 625,515,3385,686,3 875,806,2175,900,2 − − = 760,170 300,93 = 54638.=
- 41. ( )12554638.8.99ˆ −=a 3.688.99 −= 5.31=
- 42. ( )12554638.8.99ˆ −=a 3.688.99 −= 5.31=
- 43. ( )XY 546.5.31ˆ += ( )89546.5.31 += 6.485.31 += 1.80=
- 44. ( )XY 546.5.31ˆ += ( )89546.5.31 += 6.485.31 += 1.80=
- 45. ( )165546.5.31ˆ 129)valueY(Actual7Dealer 7 += = Y 6.121= ( )95546.5.31ˆ )80valueY(Actual3Dealer 3 += = Y 4.83=
- 46. 99 ˆYYei −= 5.9697 −= 5.0=
- 47. ( )165546.5.31ˆ 129)valueY(Actual7Dealer 7 += = Y 6.121= ( )95546.5.31ˆ )80valueY(Actual3Dealer 3 += = Y 4.83=
- 48. 99 ˆYYei −= 5.9697 −= 5.0=
- 49. ( )119546.5.31ˆ 9 +=Y
- 50. F-Test (Regression) • A procedure to determine whether there is more variability explained by the regression or unexplained by the regression. • Analysis of variance summary table
- 51. Total Deviation can be Partitioned into Two Parts • Total deviation equals • Deviation explained by the regression plus • Deviation unexplained by the regression
- 52. “We are always acting on what has just finished happening. It happened at least 1/30th of a second ago.We think we’re in the present, but we aren’t. The present we know is only a movie of the past.” Tom Wolfe in The Electric Kool-Aid Acid Test .
- 53. ( ) ( ) ( )iiii YYYYYY ˆˆ −+−=− Partitioning the Variance Total deviation = Deviation explained by the regression Deviation unexplained by the regression (Residual error) +
- 54. = Mean of the total group = Value predicted with regression equation = Actual value Y Yˆ iY
- 55. ( ) ( ) ( )∑∑∑ −+−=− 222 ˆˆ iiii YYYYYY Total variation explained = Explained variation Unexplained variation (residual) +
- 56. SSeSSrSSt += Sum of Squares
- 57. Coefficient of Determination r2 • The proportion of variance in Y that is explained by X (or vice versa) • A measure obtained by squaring the correlation coefficient; that proportion of the total variance of a variable that is accounted for by knowing the value of another variable
- 58. Coefficient of Determination r2 SSt SSe SSt SSr r −== 12
- 59. Source of Variation • Explained by Regression • Degrees of Freedom – k-1 where k= number of estimated constants (variables) • Sum of Squares – SSr • Mean Squared – SSr/k-1
- 60. Source of Variation • Unexplained by Regression • Degrees of Freedom – n-k where n=number of observations • Sum of Squares – SSe • Mean Squared – SSe/n-k
- 61. r2 in the Example 875. 4.882,3 49.398,32 ==r
- 62. Multiple Regression • Extension of Bivariate Regression • Multidimensional when three or more variables are involved • Simultaneously investigates the effect of two or more variables on a single dependent variable • Discussed in Chapter 24
- 63. Correlation Coefficient, r = .75 Correlation: Player Salary and Ticket Price -20 -10 0 10 20 30 1995 1996 1997 1998 1999 2000 2001 Change in Ticket Price Change in Player Salary

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