Statistics for Managers Using Microsoft Excel/SPSS Chapter 11 Hypothesis Testing With Categorical Data

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1 Statistics for Managers Using Microsoft Excel/SPSS Chapter Hypothesis Testing With Categorical Data 999 Prentice-Hall, Inc. Chap. -

2 Chapter Topics Z Test for Differences in Two Proportions (Independent Samples) c Test for Differences in Two Proportions (Independent Samples) c Test for Differences in c Proportions (Independent Samples) c Test of Independence 999 Prentice-Hall, Inc. Chap. -

3 Z Test for Differences in Two Proportions What it is used for: To determine whether there is a difference between population proportions and whether one is larger than the other. Assumptions: Independent Samples Population follows Binomial Distribution Sample Size Large Enough: np 5 and n(-p) 5 for each population 999 Prentice-Hall, Inc. Chap. - 3

4 999 Prentice-Hall, Inc. Chap. - 4 Z Test Statistic n n ) p ( p ) p p ( ) p p ( Z s s n n X X p Where X = Number of Successes in Sample X = Number of Successes in Sample Pooled Estimate of the Population Proportion

5 Stating The Hypothesis for the Z Test Hypothesis No Difference Any Difference Research Questions Prop Prop Prop < Prop Prop Prop Prop > Prop H 0 p - p = 0 p - p 0 p - p 0 H p - p 0 p - p < 0 p - p > Prentice-Hall, Inc. Chap. - 5

6 Z Test for Two Proportions Example As personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method as fair. 49 of 8 rated Method as fair. At the 0.0 level, is there a difference in perceptions? p S = p S = = n p 5 n ( - p) 5 for both pop. =.598 n = 78 n = Prentice-Hall, Inc. Chap. - 6

7 Calculation of The Test Statistic Z ( p s p ( p s p ) ) ( p n p n ) (. ( )( ) 78 ) p X n X n Prentice-Hall, Inc. Chap

8 Z Test for the Difference of Two Proportions: Solution H 0 : p - p = 0 H : p - p 0 a = 0.0 n = 78 n = 8 Critical Value(s): Reject H 0 Reject H Z Test Statistic: Z. 90 Decision: Reject at a = 0.0 Conclusion: There is evidence of a difference in proportions. 999 Prentice-Hall, Inc. Chap. - 8

9 c Test: Basic Idea Compares observed to expected frequencies if null hypothesis is true The closer observed frequencies are to expected frequencies, the more likely the H 0 is true Measured by squared difference relative to expected frequency Sum of relative squared differences is test statistic 999 Prentice-Hall, Inc. Chap. - 9

10 c Test for Proportions Contingency Table Contingency Table for Comparing Fairness of Performance Evaluation Methods Populations Evaluation Method Perception Total Fair Unfair Total Levels of Variable 999 Prentice-Hall, Inc. Chap. - 0

11 c Test for Proportions Expected Frequencies of 60 Total are fair ( p = /60 ) 78 used evaluation method Expect (78 /60) = 54.6 to be fair Evaluation Method Perception Total Fair Unfair Total Prentice-Hall, Inc. Chap. -

12 c Test Statistic c All Cells f 0 f e f e f 0 = Observed Frequency in a cell f e = Theoretical or Expected Frequency 999 Prentice-Hall, Inc. Chap. -

13 Computation of the c Test Statistic f 0 f e (f 0 - f e ) (f 0 - f e ) (f 0 - f e ) / f e Observed Frequencies Expected Frequencies Sum = Prentice-Hall, Inc. Chap. - 3

14 c Test for Two Proportions Finding Critical Value r = (# rows in Contingency Table) Reject c = (# columns) a =.0 a =.0 c Table (Portion) Upper Tail Area DF df = (r - )(c - ) = c Prentice-Hall, Inc. Chap. - 4

15 H 0 : p - p = 0 H : p - p 0 c Test for Two Proportions: Solution Test Statistic = Decision: Reject at a = 0.0 Conclusion: There is evidence of a difference in proportions. 0 Reject a = c Note: Conclusion obtained using c test is the same as using Z Test. 999 Prentice-Hall, Inc. Chap. - 5

16 c Test for c Proportions Extends the c Test to the General Case of c Independent Populations Tests for Equality (=) of Proportions Only: (Two Tail Tests, No One Tail Tests) One Variable with Several Groups or Levels Uses Contingency Table Assumptions: Independent Random samples Large Sample Size All expected Frequencies 999 Prentice-Hall, Inc. Chap. - 6

17 c Test for c Proportions: Procedure. Set Hypotheses: H 0 : p = p =... = p c H : Not All p j Are Equal. Choose a and Set Up Contingency Table 3. Compute the Overall Proportion: 4. Calculate Test Statistic: 5. Determine Degrees of Freedom 6. Compare Test Statistic with Table Value and Make Decision p X X n n... X... n 999 Prentice-Hall, Inc. Chap. - 7 c All Cells f 0 f e f e c c X n

18 c Test for c Proportions: Example The University is thinking of switching to a trimester academic calendar. A random sample of 00 undergraduates, 50 graduate students and 50 faculty members were surveyed. Opinion Under Grad Faculty Favor Oppose Totals Test at the.0 level of significance to determine is there is evidence of a difference in attitude between the groups. 999 Prentice-Hall, Inc. Chap. - 8

19 c Test for c Proportions: Example. Set Hypothesis: H 0 : p = p = p 3 H : Not All p j Are Equal. Contingency Table: 3. Compute Over All Proportion: p X n X n... X... n c c Prentice-Hall, Inc. Chap. - 9 X n All expected frequencies are large. Opinion Under Grad Faculty Totals Favor Oppose Totals

20 c Test for c Proportions: Example 4. Compute Test Statistic: f 0 f e (f 0 - f e ) (f 0 - f e ) (f 0 - f e ) / f e Test Statistic c = Prentice-Hall, Inc. Chap. - 0

21 c Test for c Proportions: Example Solution H 0 : p = p = p 3 H : Not All p j Are Equal Decision: Do Not Reject H 0 df = c - = 3 - = Reject a =.0 Conclusion: There is no evidence of a difference in attitude among the groups. c 999 Prentice-Hall, Inc. Chap. -

22 c Test of Independence Shows if a relationship exists between factors of interest One sample drawn Each factor has or more levels of responses Does Not show nature of relationship Does Not show causality Similar to testing p = p = = p c Used widely in marketing Uses contingency table 999 Prentice-Hall, Inc. Chap. -

23 c Test of Independence: Procedure. Set Hypotheses: H 0 : The categorical variables are independent H : The categorical variables are related. Choose a and Set Up Contingency Table 3. Compute Theoretical Frequencies: f e 4. Calculate Test Statistic: 5. Determine Degrees of Freedom 6. Compare Test Statistic with Table Value and Make Decision 999 Prentice-Hall, Inc. Chap. - 3 c All Cells f 0 f f e e

24 c Test of Independence: Example A Survey was conducted to determine whether there is a relationship between architectural style (Split level or Ranch) and geographical location (Urban or Rural). Given the survey data, test at the a =.0 level to determine whether there is a relationship between location and architectural style. 999 Prentice-Hall, Inc. Chap. - 4

25 c Test of Independence. Set Hypothesis: Example H 0 : The categorical variables (Architectural Style and Location) are independent H : The categorical variables are related. Contingency Table: Levels of Variable House Location House Style Urban Rural Total Split-Level Ranch Total Levels of Variable 999 Prentice-Hall, Inc. Chap. - 5

26 c Test of Independence Expected Frequencies 3. Computing Expected Frequencies Statistical independence : P(A and B) = P(A) P(B) Compute marginal (row & column) probabilities & multiply for joint probability Expected frequency is sample size times joint probability House Location Urban Rural House Style Obs. Exp. Obs. Exp. Total Split-Level Ranch Total Prentice-Hall, Inc. Chap. - 6

27 c Test of Independence Test Statistic 4. Calculate Test Statistic: c All Cells f 0 f f e e f 0 f e (f 0 - f e ) (f 0 - f e ) (f 0 - f e ) / f e c Test Statistic = Prentice-Hall, Inc. Chap. - 7

28 c Test of Independence: Example Solution H 0 : The categorical variables (Architectural Style and Location) are independent H : The categorical variables are related df = (r - )(c - ) = Decision: Reject H 0 at a =.0 Conclusion: There is evidence that the choice of architectural design and location are related. Reject a = Prentice-Hall, Inc. Chap. - 8 c

29 Chapter Summary Performed Z Test for Differences in Two Proportions (Independent Samples) Discussed c Test for Differences in Two Proportions (Independent Samples) Addressed c Test for Differences in c Proportions (Independent Samples) Described c Test of Independence 999 Prentice-Hall, Inc. Chap. - 9