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

© 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

© 2000 Prentice-Hall, Inc. Data Types

© 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?

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

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

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

© 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

© 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

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

© 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

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

© 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

© 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

© 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

© 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

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

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

© 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

© 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

© 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

© 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

© 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

© 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

© 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

© 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

© 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

© 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

© 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

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

© 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:

© 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:

© 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:

© 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

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

© 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

© 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

© 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

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

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

© 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

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

© 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

© 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)

© 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

© 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

© 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

© 2000 Prentice-Hall, Inc. Expected Count Example

© 2000 Prentice-Hall, Inc. Expected Count Example

© 2000 Prentice-Hall, Inc. Expected Count Example Marginal probability =

© 2000 Prentice-Hall, Inc. Expected Count Example Marginal probability =

© 2000 Prentice-Hall, Inc. Expected Count Example Marginal probability = Joint probability =

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

© 2000 Prentice-Hall, Inc. Expected Count Calculation

© 2000 Prentice-Hall, Inc. Expected Count Calculation

© 2000 Prentice-Hall, Inc. Expected Count Calculation 112· · · ·78 160

© 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

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

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

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

© 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:

© 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

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

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

© 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

© 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

© 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

© 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

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

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

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

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

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

© 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

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