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Research Methods

William G. Zikmund

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- 1. Business Research Methods William G. Zikmund Chapter 23Bivariate 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 Measure ofMeasurement AssociationInterval and Correlation CoefficientRatio Scales Bivariate Regression
- 5. Type of Measure ofMeasurement AssociationOrdinal Scales Chi-square Rank Correlation
- 6. Type of Measure ofMeasurement Association Chi-Square Nominal 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 rxy variables, X and Y is .
- 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. Simple Correlation Coefficientrxy = ryx = ∑ ( X − X )(Y − Y ) i i ∑ ( Xi − X ) ∑ (Yi − Y ) 2 2
- 11. Simple Correlation Coefficient σ xy rxy = ryx = σ σ2 x 2 y
- 12. Simple Correlation Coefficient Alternative Method σ 2 x= Variance of X σ = Variance of Y 2 y σ xy= Covariance of X and Y
- 13. Y Correlation Patterns NO CORRELATION X .
- 14. Y Correlation Patterns PERFECT NEGATIVE CORRELATION - r= -1.0 X .
- 15. Correlation PatternsY A HIGH POSITIVE CORRELATION r = +.98 X .
- 16. Calculation of r − 6.3389r= (17.837)( 5.589 ) − 6.3389 = = −.635 99.712 Pg 629
- 17. Coefficient of Determination Explained variancer = 2 Total Variance
- 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 Var3Var1 1.0 0.45 0.31Var2 0.45 1.0 0.10Var3 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. Walkup’s First Laws of Statistics• 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!
- 23. Regression DICTIONARY GOING OR DEFINITION MOVING BACKWARDGoing back to previous conditions Tall men’s sons
- 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. 160 Y Scatter Diagram150 and Eyeball Forecast140 My line Your line1301201101009080 X 70 80 90 100 110 120 130 140 150 160 170 180 190 .
- 29. Regression Line and SlopeY130120110 ˆ ˆ ˆ Y = a + βX10090 ∆Yˆ80 ∆X 80 90 100 110 120 130 140 150 160 170 180 190 X .
- 30. 160 Y Least-Squares150 Regression Line140 Actual Y for Dealer 7130120110 Y “hat” for Dealer 7100 Y “hat” for Dealer 390 Actual Y for80 Dealer 3 70 80 90 100 110 120 130 140 150 160 170 180 190 X
- 31. Scatter Diagram of ExplainedY and Unexplained Variation130 Deviation not explained } {}120 Total deviation110 Deviation explained by the regression100 Y9080 80 90 100 110 120 130 140 150 160 170 180 190 X .
- 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∑ 2 e is minimumi =1 i
- 35. ei = Yi - ˆ Yi (The “residual”)Yi = actual value of the dependent variable ˆYi = estimated value of the dependent variable (Y hat)n = number of observationsi = number of the observation
- 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. Bivariate Regression ˆX a =Y −β ˆ
- 38. Bivariate Regressionˆ= n( ∑ XY ) − ( ∑ X )( ∑Y )β n( ∑ X ) − (∑ X ) 2 2
- 39. ˆβ = estimated slope of the line (the “regression coefficient”)ˆa = estimated intercept of the y axisY = dependent variableY = mean of the dependent variableX = independent variableX = mean of the independent variablen = number of observations
- 40. ˆ = 15(193,345) − 2,806,875β 15( 245,759 ) − 3,515,625 2,900,175 − 2,806,875 = 3,686,385 − 3,515,625 93,300 = = .54638 170,760
- 41. a = 99.8 − .54638(125)ˆ = 99.8 − 68.3 = 31.5
- 42. a = 99.8 − .54638(125)ˆ = 99.8 − 68.3 = 31.5
- 43. Y = 31.5 + .546( X ) ˆ = 31.5 + .546( 89 ) = 31.5 + 48.6 = 80.1
- 44. Y = 31.5 + .546( X ) ˆ = 31.5 + .546( 89 ) = 31.5 + 48.6 = 80.1
- 45. Dealer 7 (Actual Y value = 129) Y7 = 31.5 + .546(165) ˆ = 121.6Dealer 3 (Actual Y value = 80) Y3 = 31.5 + .546( 95) ˆ = 83.4
- 46. ˆei = Y9 − Y9 = 97 − 96.5 = 0 .5
- 47. Dealer 7 (Actual Y value = 129) Y7 = 31.5 + .546(165) ˆ = 121.6Dealer 3 (Actual Y value = 80) Y3 = 31.5 + .546( 95) ˆ = 83.4
- 48. ˆei = Y9 − Y9 = 97 − 96.5 = 0 .5
- 49. ˆ = 31.5 + .546(119 )Y9
- 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 inthe 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. Partitioning the Variance(Yi − Y ) = Yi ( ˆ −Y ) ( ˆ + Yi − Yi ) Deviation Deviation unexplained byTotal = explained by the + the regressiondeviation regression (Residual error)
- 54. Y = Mean of the total group ˆY = Value predicted with regression equationYi = Actual value
- 55. ∑ (Y − Y ) i 2 = ∑ (Yˆ − Y ) i 2 ( + ∑ Yi − Yi ˆ ) 2Total Unexplained Explainedvariation = + variation variationexplained (residual)
- 56. Sum of SquaresSSt = SSr + SSe
- 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 r 2 SSr SSer = 2 = 1− SSt SSt
- 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 3,398.49r = 2 = .875 3,882.4
- 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 30 20 Change in Ticket 10 Price 0 Change in-10 Player Salary-20 1995 1996 1997 1998 1999 2000 2001

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