Research Methods William G. Zikmund, Ch22

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Research Methods
William G. Zikmund

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Research Methods William G. Zikmund, Ch22

  1. 1. BusinessResearch Methods William G. Zikmund Chapter 22: Bivariate Analysis - Tests of Differences
  2. 2. Common Bivariate Tests Differences among Differences betweenType of Measurement three or more two independent groups independent groups Independent groups: One-way Interval and ratio t-test or Z-test ANOVA
  3. 3. Common Bivariate Tests Differences among Differences betweenType of Measurement three or more two independent groups independent groups Mann-Whitney U-test Ordinal Kruskal-Wallis test Wilcoxon test
  4. 4. Common Bivariate Tests Differences among Differences betweenType of Measurement three or more two independent groups independent groups Z-test (two proportions) Nominal Chi-square test Chi-square test
  5. 5. Type of Differences betweenMeasurement two independent groups Nominal Chi-square test
  6. 6. Differences Between Groups• Contingency Tables• Cross-Tabulation• Chi-Square allows testing for significant differences between groups• “Goodness of Fit”
  7. 7. Chi-Square Test (Oi − Ei )²x² = ∑ Ei x² = chi-square statistics Oi = observed frequency in the ith cell Ei = expected frequency on the ith cell
  8. 8. Chi-Square Test Ri C jEij = nRi = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample size
  9. 9. Degrees of Freedom(R-1)(C-1)=(2-1)(2-1)=1
  10. 10. Degrees of Freedom d.f.=(R-1)(C-1)
  11. 11. Awareness of Tire Manufacturer’s Brand Men Women TotalAware 50 10 60Unaware 15 2540 65 35 100
  12. 12. Chi-Square Test: Differences Among Groups Example (50 − 39) 2 (10 − 21) 2X = 2 + 39 21 (15 − 26) 2 ( 25 − 14) 2 + + 26 14
  13. 13. χ = 3.102 + 5.762 + 4.654 + 8.643 = 2χ = 22.161 2
  14. 14. d . f . = ( R − 1)(C − 1)d . f . = (2 − 1)(2 − 1) = 1 X2=3.84 with 1 d.f.
  15. 15. Type of Differences betweenMeasurement two independent groupsInterval and t-test or ratio Z-test
  16. 16. Differences Between Groups when Comparing Means• Ratio scaled dependent variables• t-test – When groups are small – When population standard deviation is unknown• z-test – When groups are large
  17. 17. Null Hypothesis About MeanDifferences Between Groups µ −µ 1 2 OR µ −µ =0 1 2
  18. 18. t-Test for Difference of Means mean 1 - mean 2t= Variability of random means
  19. 19. t-Test for Difference of Means Χ1 − Χ 2 t= S X1 − X 2 X1 = mean for Group 1 X2 = mean for Group 2 SX1-X2 = the pooled or combined standard error of difference between means.
  20. 20. t-Test for Difference of Means Χ1 − Χ 2 t= S X1 − X 2
  21. 21. t-Test for Difference of MeansX1 = mean for Group 1X2 = mean for Group 2SX -X = the pooled or combined standard error 1 2 of difference between means.
  22. 22. Pooled Estimate of the Standard Error  ( n1 − 1) S + (n2 − 1) S 2 2 )  1 1 S X1 − X 2 =   1 2  +   n n   n1 + n2 − 2  1 2 
  23. 23. Pooled Estimate of the Standard ErrorS12 = the variance of Group 1S22 = the variance of Group 2n1 = the sample size of Group 1n2 = the sample size of Group 2
  24. 24. Pooled Estimate of the Standard Error t-test for the Difference of Means  ( n1 − 1) S12 + (n2 − 1) S 2 )  1 1  2S X1 − X 2 =    +   n n   n1 + n2 − 2  1 2  S 2 = the variance of Group 1 1 S 2 = the variance of Group 2 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2
  25. 25. Degrees of Freedom• d.f. = n - k• where: – n = n1 + n2 – k = number of groups
  26. 26. t-Test for Difference of Means Example  ( 20)( 2.1) + (13)( 2.6) 2 2  1 1 S X1 − X 2 =    +    21 14   33  = .797
  27. 27. 16.5 − 12.2 4 .3t= = .797 .797 = 5.395
  28. 28. Type of Differences betweenMeasurement two independent groups Nominal Z-test (two proportions)
  29. 29. Comparing Two Groups when Comparing Proportions• Percentage Comparisons• Sample Proportion - P• Population Proportion - Π
  30. 30. Differences Between Two Groups when Comparing ProportionsThe hypothesis is: Ho: Π1 = Π2may be restated as: Ho: Π1 − Π2 = 0
  31. 31. Z-Test for Differences of Proportions Ho : π1 = π 2or Ho : π1 − π 2 = 0
  32. 32. Z-Test for Differences of ProportionsZ= ( p1 − p2 ) − (π1 − π 2 ) S p1 − p2
  33. 33. Z-Test for Differences of Proportionsp1 = sample portion of successes in Group 1p2 = sample portion of successes in Group 2(π1 − π1) = hypothesized population proportion 1 minus hypothesized population proportion 1 minusSp1-p2 = pooled estimate of the standard errors of difference of proportions
  34. 34. Z-Test for Differences of Proportions 1 1S p1 − p2 = pq  −  n n   1 2 
  35. 35. Z-Test for Differences of Proportionspp = pooled estimate of proportion of success in a sample of both groupsq = (1- p or a pooled estimate of proportion ofp p) failures in a sample of both groupsn1= sample size for group 1n2= sample size for group 2
  36. 36. Z-Test for Differences of Proportions n1 p1 + n2 p2p= n1 + n2
  37. 37. Z-Test for Differences of Proportions  1 1 S p1 − p2 = ( .375)( .625)  +   100 100  = .068
  38. 38. A Z-Test for Differences of Proportionsp= (100)( .35) + (100)( .4) 100 + 100 = .375
  39. 39. Differences between Type of three or more Measurement independent groupsInterval or ratio One-way ANOVA
  40. 40. Analysis of VarianceHypothesis when comparing three groupsµ1 = µ2 = µ3
  41. 41. Analysis of Variance F-Ratio Variance − between − groupsF= Variance − within − groups
  42. 42. Analysis of Variance Sum of SquaresSStotal = SSwithin + SSbetween
  43. 43. Analysis of Variance Sum of SquaresTotal n cSStotal = ∑∑ ( X ij − X ) 2 i = 1 j =1
  44. 44. Analysis of Variance Sum of SquaresXiij= individual scores, i.e., the ith observation orp test unit in the jth groupXi = grand meanpn = number of all observations or test units in a groupc = number of jth groups (or columns)
  45. 45. Analysis of Variance Sum of SquaresWithin n cSSwithin = ∑∑ ( X ij − X j ) 2 i = 1 j =1
  46. 46. Analysis of Variance Sum of SquaresWithinXi ij= individual scores, i.e., the ith observation orp test unit in the jth groupXi = grand meanpn = number of all observations or test units in a groupc = number of jth groups (or columns)
  47. 47. Analysis of Variance Sum of Squares Between nSSbetween = ∑ n j ( X j − X ) 2 j =1
  48. 48. Analysis of Variance Sum of squares BetweenX j= individual scores, i.e., the ith observation or test unit in the jth groupX = grand meannj = number of all observations or test units in a group
  49. 49. Analysis of Variance Mean Squares Between SSbetweenMSbetween = c −1
  50. 50. Analysis of Variance Mean Square Within SS withinMSwithin = cn − c
  51. 51. Analysis of Variance F-Ratio MSbetweenF= MS within
  52. 52. A Test Market Experiment on Pricing Sales in Units (thousands) Regular Price Reduced Price Cents-Off Coupon $.99 $.89 Regular PriceTest Market A, B, or C 130 145 153Test Market D, E, or F 118 143 129Test Market G, H, or I 87 120 96Test Market J, K, or L 84 131 99Mean X1=104.75 X2=134.75 X1=119.25Grand Mean X=119.58
  53. 53. ANOVA Summary Table Source of Variation• Between groups• Sum of squares – SSbetween• Degrees of freedom – c-1 where c=number of groups• Mean squared-MSbetween – SSbetween/c-1
  54. 54. ANOVA Summary Table Source of Variation• Within groups• Sum of squares – SSwithin• Degrees of freedom – cn-c where c=number of groups, n= number of observations in a group• Mean squared-MSwithin – SSwithin/cn-c
  55. 55. ANOVA Summary Table Source of Variation• Total• Sum of Squares – SStotal• Degrees of Freedom – cn-1 where c=number of groups, n= number of observations in a group MSBETWEEN F= MSWITHIN

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