1. 13 - 1 © 2000 Prentice-Hall, Inc. Statistics The Chi-Square Test & The Analysis of Contingency Tables Chapter 13

2. 13 - 2 © 2000 Prentice-Hall, Inc. Learning Objectives 1.Explain  2 Test for Proportions 2.Explain  2 Test of Independence 3.Solve Hypothesis Testing Problems Two or More Population Proportions Two or More Population Proportions Independence Independence

3. 13 - 3 © 2000 Prentice-Hall, Inc. Data Types

4. 13 - 4 © 2000 Prentice-Hall, Inc. Qualitative Data 1.Qualitative Random Variables Yield Responses That Classify Example: Gender (Male, Female) Example: Gender (Male, Female) 2.Measurement Reflects # in Category 3.Nominal or Ordinal Scale 4.Examples Do You Own Savings Bonds? Do You Own Savings Bonds? Do You Live On-Campus or Off-Campus? Do You Live On-Campus or Off-Campus?

5. 13 - 5 © 2000 Prentice-Hall, Inc. Hypothesis Tests Qualitative Data

6. 13 - 6 © 2000 Prentice-Hall, Inc. Chi-Square (  2 ) Test for k Proportions

7. 13 - 7 © 2000 Prentice-Hall, Inc. Hypothesis Tests Qualitative Data

8. 13 - 8 © 2000 Prentice-Hall, Inc. Chi-Square (  2 ) Test for k Proportions 1.Tests Equality (=) of Proportions Only Example: p 1 =.2, p 2 =.3, p 3 =.5 Example: p 1 =.2, p 2 =.3, p 3 =.5 2.One Variable With Several Levels 3.Assumptions Multinomial Experiment Multinomial Experiment Large Sample Size Large Sample Size All Expected Counts  5 All Expected Counts  5 4.Uses One-Way Contingency Table

9. 13 - 9 © 2000 Prentice-Hall, Inc. Multinomial Experiment 1.n Identical Trial 2.k Outcomes to Each Trial 3.Constant Outcome Probability, p k 4.Independent Trials 5.Random Variable is Count, n k 6.Example: Ask 100 People (n) Which of 3 Candidates (k) They Will Vote For

10. 13 - 10 © 2000 Prentice-Hall, Inc. One-Way Contingency Table 1.Shows # Observations in k Independent Groups (Outcomes or Variable Levels)

11. 13 - 11 © 2000 Prentice-Hall, Inc. One-Way Contingency Table 1.Shows # Observations in k Independent Groups (Outcomes or Variable Levels) Outcomes (k = 3) Number of responses

12. 13 - 12 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Hypotheses & Statistic

13. 13 - 13 © 2000 Prentice-Hall, Inc. 1.Hypotheses H 0 : p 1 = p 1,0, p 2 = p 2,0,..., p k = p k,0 H 0 : p 1 = p 1,0, p 2 = p 2,0,..., p k = p k,0 H a : Not all p i are equal H a : Not all p i are equal  2 Test for k Proportions Hypotheses & Statistic Hypothesized probability

14. 13 - 14 © 2000 Prentice-Hall, Inc. 1.Hypotheses H 0 : p 1 = p 1,0, p 2 = p 2,0,..., p k = p k,0 H 0 : p 1 = p 1,0, p 2 = p 2,0,..., p k = p k,0 H a : Not all p i are equal H a : Not all p i are equal 2.Test Statistic  2 Test for k Proportions Hypotheses & Statistic Observed count Expected count Hypothesized probability

15. 13 - 15 © 2000 Prentice-Hall, Inc. 1.Hypotheses H 0 : p 1 = p 1,0, p 2 = p 2,0,..., p k = p k,0 H 0 : p 1 = p 1,0, p 2 = p 2,0,..., p k = p k,0 H a : Not all p i are equal H a : Not all p i are equal 2.Test Statistic 3.Degrees of Freedom: k - 1  2 Test for k Proportions Hypotheses & Statistic Observed count Expected count Number of outcomes Hypothesized probability

16. 13 - 16 © 2000 Prentice-Hall, Inc.  2 Test Basic Idea 1.Compares Observed Count to Expected Count If Null Hypothesis Is True 2.Closer Observed Count to Expected Count, the More Likely the H 0 Is True Measured by Squared Difference Relative to Expected Count Measured by Squared Difference Relative to Expected Count Reject Large Values Reject Large Values

17. 13 - 17 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?

18. 13 - 18 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  2 Table (Portion)

19. 13 - 19 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  2 Table (Portion) If n i = E(n i ),  2 = 0. Do not reject H 0

20. 13 - 20 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) If n i = E(n i ),  2 = 0. Do not reject H 0

21. 13 - 21 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) If n i = E(n i ),  2 = 0. Do not reject H 0

22. 13 - 22 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) If n i = E(n i ),  2 = 0. Do not reject H 0

23. 13 - 23 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) If n i = E(n i ),  2 = 0. Do not reject H 0

24. 13 - 24 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) df= k - 1 = 2 If n i = E(n i ),  2 = 0. Do not reject H 0

25. 13 - 25 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) df= k - 1 = 2 If n i = E(n i ),  2 = 0. Do not reject H 0

26. 13 - 26 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) df= k - 1 = 2 If n i = E(n i ),  2 = 0. Do not reject H 0

27. 13 - 27 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) df= k - 1 = 2 If n i = E(n i ),  2 = 0. Do not reject H 0

28. 13 - 28 © 2000 Prentice-Hall, Inc. Finding Critical Value Example What is the critical  2 value if k = 3, &  =.05?  =.05  2 Table (Portion) df= k - 1 = 2 If n i = E(n i ),  2 = 0. Do not reject H 0

29. 13 - 29 © 2000 Prentice-Hall, Inc. As personnel director, you want to test the perception of fairness of three methods of performance evaluation. Of 180 employees, 63 rated Method 1 as fair. 45 rated Method 2 as fair. 72 rated Method 3 as fair. At the.05 level, is there a difference in perceptions?  2 Test for k Proportions Example

30. 13 - 30 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution

31. 13 - 31 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution H 0 : H a :  = n 1 = n 2 = n 3 = Critical Value(s): Test Statistic: Decision:Conclusion:

32. 13 - 32 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution H 0 : p 1 = p 2 = p 3 = 1/3 H a : At least 1 is different  = n 1 = n 2 = n 3 = Critical Value(s): Test Statistic: Decision:Conclusion:

33. 13 - 33 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution H 0 : p 1 = p 2 = p 3 = 1/3 H a : At least 1 is different  =.05 n 1 = 63 n 2 = 45 n 3 = 72 Critical Value(s): Test Statistic: Decision:Conclusion:

34. 13 - 34 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution H 0 : p 1 = p 2 = p 3 = 1/3 H a : At least 1 is different  =.05 n 1 = 63 n 2 = 45 n 3 = 72 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05

35. 13 - 35 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution

36. 13 - 36 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution H 0 : p 1 = p 2 = p 3 = 1/3 H a : At least 1 is different  =.05 n 1 = 63 n 2 = 45 n 3 = 72 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05  2 = 6.3

37. 13 - 37 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution H 0 : p 1 = p 2 = p 3 = 1/3 H a : At least 1 is different  =.05 n 1 = 63 n 2 = 45 n 3 = 72 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05  2 = 6.3 Reject at  =.05

38. 13 - 38 © 2000 Prentice-Hall, Inc.  2 Test for k Proportions Solution H 0 : p 1 = p 2 = p 3 = 1/3 H a : At least 1 is different  =.05 n 1 = 63 n 2 = 45 n 3 = 72 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence of a difference in proportions  =.05  2 = 6.3

39. 13 - 39 © 2000 Prentice-Hall, Inc.  2 Test of Independence

40. 13 - 40 © 2000 Prentice-Hall, Inc. Hypothesis Tests Qualitative Data

41. 13 - 41 © 2000 Prentice-Hall, Inc.  2 Test of Independence 1.Shows If a Relationship Exists Between 2 Qualitative Variables One Sample Is Drawn One Sample Is Drawn Does Not Show Causality Does Not Show Causality 2.Assumptions Multinomial Experiment Multinomial Experiment All Expected Counts  5 All Expected Counts  5 3.Uses Two-Way Contingency Table

42. 13 - 42 © 2000 Prentice-Hall, Inc.  2 Test of Independence Contingency Table 1.Shows # Observations From 1 Sample Jointly in 2 Qualitative Variables

43. 13 - 43 © 2000 Prentice-Hall, Inc.  2 Test of Independence Contingency Table 1.Shows # Observations From 1 Sample Jointly in 2 Qualitative Variables Levels of variable 2 Levels of variable 1

44. 13 - 44 © 2000 Prentice-Hall, Inc.  2 Test of Independence Hypotheses & Statistic 1.Hypotheses H 0 : Variables Are Independent H 0 : Variables Are Independent H a : Variables Are Related (Dependent) H a : Variables Are Related (Dependent)

45. 13 - 45 © 2000 Prentice-Hall, Inc.  2 Test of Independence Hypotheses & Statistic 1.Hypotheses H 0 : Variables Are Independent H 0 : Variables Are Independent H a : Variables Are Related (Dependent) H a : Variables Are Related (Dependent) 2.Test Statistic Observed count Expected count

46. 13 - 46 © 2000 Prentice-Hall, Inc.  2 Test of Independence Hypotheses & Statistic 1.Hypotheses H 0 : Variables Are Independent H 0 : Variables Are Independent H a : Variables Are Related (Dependent) H a : Variables Are Related (Dependent) 2.Test Statistic Degrees of Freedom: (r - 1)(c - 1) Rows Columns Observed count Expected count

47. 13 - 47 © 2000 Prentice-Hall, Inc.  2 Test of Independence Expected Counts 1.Statistical Independence Means Joint Probability Equals Product of Marginal Probabilities 2.Compute Marginal Probabilities & Multiply for Joint Probability 3.Expected Count Is Sample Size Times Joint Probability

48. 13 - 48 © 2000 Prentice-Hall, Inc. Expected Count Example

49. 13 - 49 © 2000 Prentice-Hall, Inc. Expected Count Example

50. 13 - 50 © 2000 Prentice-Hall, Inc. Expected Count Example 112 160 Marginal probability =

51. 13 - 51 © 2000 Prentice-Hall, Inc. Expected Count Example 112 160 78 160 Marginal probability =

52. 13 - 52 © 2000 Prentice-Hall, Inc. Expected Count Example 112 160 78 160 Marginal probability = Joint probability = 112 160 78 160

53. 13 - 53 © 2000 Prentice-Hall, Inc. Expected Count Example 112 160 78 160 Marginal probability = Joint probability = 112 160 78 160 Expected count = 160· 112 160 78 160 = 54.6

54. 13 - 54 © 2000 Prentice-Hall, Inc. Expected Count Calculation

55. 13 - 55 © 2000 Prentice-Hall, Inc. Expected Count Calculation

56. 13 - 56 © 2000 Prentice-Hall, Inc. Expected Count Calculation 112·82 160 48·78 160 48·82 160 112·78 160

57. 13 - 57 © 2000 Prentice-Hall, Inc. You’re a marketing research analyst. You ask a random sample of 286 consumers if they purchase Diet Pepsi or Diet Coke. At the.05 level, is there evidence of a relationship?  2 Test of Independence Example

58. 13 - 58 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution

59. 13 - 59 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

60. 13 - 60 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution H 0 : No Relationship H a : Relationship  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

61. 13 - 61 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution H 0 : No Relationship H a : Relationship  =.05 df = (2 - 1)(2 - 1) = 1 Critical Value(s): Test Statistic: Decision:Conclusion:

62. 13 - 62 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution H 0 : No Relationship H a : Relationship  =.05 df = (2 - 1)(2 - 1) = 1 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05

63. 13 - 63 © 2000 Prentice-Hall, Inc. E(n ij )  5 in all cells 170·132 286 170·154 286 116·132 286 154·132 286  2 Test of Independence Solution

64. 13 - 64 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution

65. 13 - 65 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution H 0 : No Relationship H a : Relationship  =.05 df = (2 - 1)(2 - 1) = 1 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05  2 = 54.29

66. 13 - 66 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution H 0 : No Relationship H a : Relationship  =.05 df = (2 - 1)(2 - 1) = 1 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05  =.05  2 = 54.29

67. 13 - 67 © 2000 Prentice-Hall, Inc.  2 Test of Independence Solution H 0 : No Relationship H a : Relationship  =.05 df = (2 - 1)(2 - 1) = 1 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence of a relationship  =.05  2 = 54.29

68. 13 - 68 © 2000 Prentice-Hall, Inc. OK. There is a statistically significant relationship between purchasing Diet Coke & Diet Pepsi. So what do you think the relationship is? Aren’t they competitors?  2 Test of Independence Thinking Challenge

69. 13 - 69 © 2000 Prentice-Hall, Inc. You Re-Analyze the Data

70. 13 - 70 © 2000 Prentice-Hall, Inc. You Re-Analyze the Data High Income

71. 13 - 71 © 2000 Prentice-Hall, Inc. You Re-Analyze the Data Low Income High Income

72. 13 - 72 © 2000 Prentice-Hall, Inc. True Relationships* Apparent relation Underlying causal relation Control or intervening variable (true cause) Diet Coke Diet Pepsi

73. 13 - 73 © 2000 Prentice-Hall, Inc. Moral of the Story* Numbers don’t think - People do! © 1984-1994 T/Maker Co.

74. 13 - 74 © 2000 Prentice-Hall, Inc. Conclusion 1.Explained  2 Test for Proportions 2.Explained  2 Test of Independence 3.Solved Hypothesis Testing Problems Two or More Population Proportions Two or More Population Proportions Independence Independence

75. End of Chapter Any blank slides that follow are blank intentionally.