20. Correlation Patterns

27. Cross-Tabulation A technique for organizing data by groups, categories, or classes, thus facilitating comparisons; a joint frequency distribution of observations on two or more sets of variables Contingency table- The results of a cross- tabulation of two variables, such as survey questions

28. Cross-Tabulation Analyze data by groups or categories Compare differences Contingency table Percentage cross-tabulations

29. Elaboration and Refinement Moderator variable –A third variable that, when introduced into an analysis, alters or has a contingent effect on the relationship between an independent variable and a dependent variable. –Spurious relationship An apparent relationship between two variables that is not authentic.

30. Tworatingscales 4 quadrants two-dimensionaltable Importance-PerformanceAnalysis) Quadrant Analysis

32. Calculating Rank Order Ordinal data Brand preferences

33. Correlation Coefficient A statistical measure of the covariation or association between two variables. Are dollar sales associated with advertising dollar expenditures?

34. The Correlation coefficient for two variables, X and Y is.

35. 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

36. Simple Correlation Coefficient

38. = Variance of X = Variance of Y = Covariance of X and Y Simple Correlation Coefficient Alternative Method

39. X Y NO CORRELATION. Correlation Patterns

40. X Y PERFECT NEGATIVE CORRELATION - r= -1.0. Correlation Patterns

41. X Y A HIGH POSITIVE CORRELATION r = +.98. Correlation Patterns

42. Pg 629 Calculation of r

43. Coefficient of Determination

44. 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

45. Correlation Matrix The standard form for reporting correlational results.

46. Correlation Matrix

50. 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

51. 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

52. Type of Measurement Differences between two independent groups Differences among three or more independent groups Nominal Z-test (two proportions) Chi-square test Common Bivariate Tests

53. Type of Measurement Differences between two independent groups Nominal Chi-square test

54. Differences Between Groups Contingency Tables Cross-Tabulation Chi-Square allows testing for significant differences between groups “Goodness of Fit”

55. Chi-Square Test x² = chi-square statistics O i = observed frequency in the i th cell E i = expected frequency on the i th cell

56. R i = total observed frequency in the i th row C j = total observed frequency in the j th column n = sample size Chi-Square Test

57. Degrees of Freedom (R-1)(C-1)=(2-1)(2-1)=1

58. d.f.=(R-1)(C-1) Degrees of Freedom

59. MenWomenTotal Aware 50 10 60 Unaware 15 25 40 65 35 100 Awareness of Tire Manufacturer’s Brand

60. Chi-Square Test: Differences Among Groups Example

62. X 2 =3.84 with 1 d.f.

63. Type of Measurement Differences between two independent groups Interval and ratio t-test or Z-test

64. 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

65. Null Hypothesis About Mean Differences Between Groups

66. t-Test for Difference of Means

67. X 1 = mean for Group 1 X 2 = mean for Group 2 S X 1 -X 2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means

69. X 1 = mean for Group 1 X 2 = mean for Group 2 S X 1 -X 2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means

70. Pooled Estimate of the Standard Error

71. S 1 2 = the variance of Group 1 S 2 2 = the variance of Group 2 n 1 = the sample size of Group 1 n 2 = the sample size of Group 2 Pooled Estimate of the Standard Error

72. Pooled Estimate of the Standard Error t-test for the Difference of Means S 1 2 = the variance of Group 1 S 2 2 = the variance of Group 2 n 1 = the sample size of Group 1 n 2 = the sample size of Group 2

73. Degrees of Freedom d.f. = n - k where: –n = n 1 + n 2 –k = number of groups

74. t-Test for Difference of Means Example

76. Type of Measurement Differences between two independent groups Nominal Z-test (two proportions)

77. Comparing Two Groups when Comparing Proportions Percentage Comparisons Sample Proportion - P Population Proportion -

78. Differences Between Two Groups when Comparing Proportions The hypothesis is: H o :  1   may be restated as: H o :  1   

79. or Z-Test for Differences of Proportions

81. p 1 = sample portion of successes in Group 1 p 2 = sample portion of successes in Group 2  1  1 )  = hypothesized population proportion 1 minus hypothesized population proportion 1 minus S p1-p2 = pooled estimate of the standard errors of difference of proportions Z-Test for Differences of Proportions

83. 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 n  = sample size for group 1 n  = sample size for group 2 Z-Test for Differences of Proportions

86. A Z-Test for Differences of Proportions

87. Testing a Hypothesis about a Distribution Chi-Square test Test for significance in the analysis of frequency distributions Compare observed frequencies with expected frequencies “Goodness of Fit”

88. Chi-Square Test

89. x² = chi-square statistics O i = observed frequency in the i th cell E i = expected frequency on the i th cell Chi-Square Test

90. Chi-Square Test Estimation for Expected Number for Each Cell

91. R i = total observed frequency in the i th row C j = total observed frequency in the j th column n = sample size

92. Hypothesis Test of a Proportion  is the population proportion p is the sample proportion  is estimated with p

93. 5. :H 5. :H 1 0   Hypothesis Test of a Proportion

96. 0115.S p  000133.S p  1200 16. S p  1200 )8)(.2(. S p  n pq S p  20.p  200,1n  Hypothesis Test of a Proportion: Another Example

97. Indeed.001 the beyond t significant is it level..05 the at rejected be should hypothesis null the so 1.96, exceeds value Z The 348.4Z 0115. 05. Z 0115. 15.20. Z S p Z p       Hypothesis Test of a Proportion: Another Example