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# Business Research Methods, 9th ed.Chapter 22

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

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### Transcript of "Business Research Methods, 9th ed.Chapter 22"

1. 1. Business Research Methods William G. Zikmund Chapter 22: Bivariate Analysis - Tests of Differences
2. 2. Type of Measurement Differences between two independent groups Differences among three or more independent groups Interval and ratio Independent groups: t-test or Z-test One-way ANOVA Common Bivariate Tests
3. 3. Type of Measurement Differences between two independent groups Differences among three or more independent groups Ordinal Mann-Whitney U-test Wilcoxon test Kruskal-Wallis test Common Bivariate Tests
4. 4. Type of Measurement Differences between two independent groups Differences among three or more independent groups Nominal Z-test (two proportions) Chi-square test Chi-square test Common Bivariate Tests
5. 5. Type of Measurement Differences between 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 ∑ − = i ii )²( ² E EO x x² = chi-square statistics Oi = observed frequency in the ith cell Ei = expected frequency on the ith cell
8. 8. n CR E ji ij = Ri = total observed frequency in the ith row Cj = total observed frequency in the jth column n = sample size Chi-Square Test
9. 9. Degrees of Freedom (R-1)(C-1)=(2-1)(2-1)=1
10. 10. d.f.=(R-1)(C-1) Degrees of Freedom
11. 11. Men Women Total Aware 50 10 60 Unaware 15 25 40 65 35 100 Awareness of Tire Manufacturer’s Brand
12. 12. Chi-Square Test: Differences Among Groups Example 21 )2110( 39 )3950( 22 2 − + − =X 14 )1425( 26 )2615( 22 − + − +
13. 13. 161.22 643.8654.4762.5102.3 2 2 = =+++= χ χ
14. 14. 1)12)(12(.. )1)(1(.. =−−= −−= fd CRfd X2 =3.84 with 1 d.f.
15. 15. Type of Measurement Differences between two independent groups Interval and ratio t-test or 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. 021 21 =− − µµ µµ OR Null Hypothesis About Mean Differences Between Groups
18. 18. meansrandomofyVariabilit 2mean-1mean t = t-Test for Difference of Means
19. 19. 21 21 XX S t − Χ−Χ = X1 = mean for Group 1 X2 = mean for Group 2 SX1-X2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means
20. 20. 21 21 XX S t − Χ−Χ = t-Test for Difference of Means
21. 21. X1 = mean for Group 1 X2 = mean for Group 2 SX1-X2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means
22. 22. Pooled Estimate of the Standard Error ( )       +      −+ −+− =− 2121 2 22 2 11 11 2 ))1(1 21 nnnn SnSn S XX
23. 23. S1 2 = the variance of Group 1 S2 2 = the variance of Group 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2 Pooled Estimate of the Standard Error
24. 24. Pooled Estimate of the Standard Error t-test for the Difference of Means ( )       +      −+ −+− =− 2121 2 22 2 11 11 2 ))1(1 21 nnnn SnSn S XX S1 2 = the variance of Group 1 S2 2 = the variance of Group 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. ( )( ) ( )( )       +        + =− 14 1 21 1 33 6.2131.220 22 21 XX S 797.= t-Test for Difference of Means Example
27. 27. 797. 2.125.16 − =t 797. 3.4 = 395.5=
28. 28. Type of Measurement Differences between two independent groups Nominal Z-test (two proportions)
29. 29. Π Comparing Two Groups when Comparing Proportions • Percentage Comparisons • Sample Proportion - P • Population Proportion -
30. 30. Differences Between Two Groups when Comparing Proportions The hypothesis is: Ho: Π1 = Π2 may be restated as: Ho: Π1 − Π2 = 0
31. 31. 21: ππ =oH or 0: 21 =− ππoH Z-Test for Differences of Proportions
32. 32. Z-Test for Differences of Proportions ( ) ( ) 21 2121 ppS pp Z − −−− = ππ
33. 33. p1 = sample portion of successes in Group 1 p2 = sample portion of successes in Group 2 (π1 − π1) = hypothesized population proportion 1 minus hypothesized population proportion 1 minus Sp1-p2 = pooled estimate of the standard errors of difference of proportions Z-Test for Differences of Proportions
34. 34. Z-Test for Differences of Proportions       −=− 21 11 21 nn qpS pp
35. 35. p = pooled estimate of proportion of success in a sample of both groups p = (1- p) or a pooled estimate of proportion of failures in a sample of both groups n1= sample size for group 1 n2= sample size for group 2 p q p Z-Test for Differences of Proportions
36. 36. Z-Test for Differences of Proportions 21 2211 nn pnpn p + + =
37. 37. ( )( )       +=− 100 1 100 1 625.375.21 ppS 068.= Z-Test for Differences of Proportions
38. 38. ( )( ) ( )( ) 100100 4.10035.100 + + =p 375.= A Z-Test for Differences of Proportions
39. 39. Type of Measurement Differences between three or more independent groups Interval or ratio One-way ANOVA
40. 40. Analysis of Variance Hypothesis when comparing three groups µ1 = µ2 = µ3
41. 41. groupswithinVariance groupsbetweenVariance F −− −− = Analysis of Variance F-Ratio
42. 42. Analysis of Variance Sum of Squares betweenwithintotal SSSSSS +=
43. 43. ∑∑= = −= n i c j1 1 2 total )(SS XXij Analysis of Variance Sum of SquaresTotal
44. 44. Analysis of Variance Sum of Squares pi = individual scores, i.e., the ith observation or test unit in the jth group pi = grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) ijX X
45. 45. ∑∑= = −= n i c j j 1 1 2 within )(SS XXij Analysis of Variance Sum of SquaresWithin
46. 46. Analysis of Variance Sum of SquaresWithin pi = individual scores, i.e., the ith observation or test unit in the jth group pi = grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) ijX X
47. 47. ∑= −= n j jjn 1 2 between )(SS XX Analysis of Variance Sum of Squares Between
48. 48. Analysis of Variance Sum of squares Between = individual scores, i.e., the ith observation or test unit in the jth group = grand mean nj = number of all observations or test units in a group jX X
49. 49. 1− = c SS MS between between Analysis of Variance Mean Squares Between
50. 50. ccn SS MS within within − = Analysis of Variance Mean Square Within
51. 51. within between MS MS F = Analysis of Variance F-Ratio
52. 52. Sales in Units (thousands) Regular Price \$.99 130 118 87 84 X1=104.75 X=119.58 Reduced Price \$.89 145 143 120 131 X2=134.75 Cents-Off Coupon Regular Price 153 129 96 99 X1=119.25 Test Market A, B, or C Test Market D, E, or F Test Market G, H, or I Test Market J, K, or L Mean Grand Mean A Test Market Experiment on Pricing
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. WITHIN BETWEEN MS MS F = 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