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© 2002 Thomson / South-Western Slide 11-1 Chapter 11 Analysis of Variance and Chi-Square Applications.

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Presentation on theme: "© 2002 Thomson / South-Western Slide 11-1 Chapter 11 Analysis of Variance and Chi-Square Applications."— Presentation transcript:

1 © 2002 Thomson / South-Western Slide 11-1 Chapter 11 Analysis of Variance and Chi-Square Applications

2 © 2002 Thomson / South-Western Slide 11-2 Learning Objectives Understand the differences between various experimental designs and when to use them. Compute and interpret the results of a one-way ANOVA. Compute and interpret the results of a random block design.

3 © 2002 Thomson / South-Western Slide 11-3 Learning Objectives, continued Compute and interpret the results of a two-way ANOVA. Understand and interpret interaction. Understand the chi-square goodness- of-fit test and how to use it. Analyze data by using the chi-square test of independence.

4 © 2002 Thomson / South-Western Slide 11-4 Introduction to Design of Experiments An Experimental Design is a plan and a structure to test hypotheses in which the business analyst controls or manipulates one or more variables. It contains independent and dependent variables. Factors is another name for the independent variables of an experimental design.

5 © 2002 Thomson / South-Western Slide 11-5 Design of Experiments, continued Treatment variable is the independent variable that the experimenter either controls or modifies. Classification variable is the independent variable that was present prior to the experiment, and is not a result of the experimenter’s manipulations or control.

6 © 2002 Thomson / South-Western Slide 11-6 Design of Experiments, continued Levels or Classifications are the subcategories of the independent variable used by the business analyst in the experimental design. The Dependent Variable is the response to the different levels of the independent variables.

7 © 2002 Thomson / South-Western Slide 11-7 Three Types of Experimental Designs Completely Randomized Design Randomized Block Design Factorial Experiments

8 © 2002 Thomson / South-Western Slide 11-8 Completely Randomized Design Machine Operator Valve Opening Measurements

9 © 2002 Thomson / South-Western Slide 11-9 Example: Number of Foreign Freighters Docking in each Port per Day Long Beach Houston New York New Orleans

10 © 2002 Thomson / South-Western Slide Analysis of Variance (ANOVA): Assumptions Observations are drawn from normally distributed populations. Observations represent random samples from the populations. Variances of the populations are equal.

11 © 2002 Thomson / South-Western Slide One-Way ANOVA: Procedural Overview

12 © 2002 Thomson / South-Western Slide Partitioning Total Sum of Squares of Variation SST (Total Sum of Squares) SSC (Treatment Sum of Squares) SSE (Error Sum of Squares)

13 © 2002 Thomson / South-Western Slide One-Way ANOVA: Sums of Squares Definitions

14 © 2002 Thomson / South-Western Slide One-Way ANOVA: Computational Formulas

15 © 2002 Thomson / South-Western Slide Freighter One-Way ANOVA: Preliminary Calculations Long Beach T 1 = 18 n 1 = 4 Houston T 2 = 20 n 2 = 5 New York T 3 = 42 n 3 = 6 New Orleans T 4 = 17 n 4 = 5 T = 97 N = 20

16 © 2002 Thomson / South-Western Slide Freighter One-Way ANOVA: Sum of Squares Calculations

17 © 2002 Thomson / South-Western Slide Freighter One-Way ANOVA: Sum of Squares Calculations, continued

18 © 2002 Thomson / South-Western Slide Freighter One- Way ANOVA: Mean Square and F Calculations

19 © 2002 Thomson / South-Western Slide Freighter Example: Analysis of Variance Source of VariancedfSSMSF Between Factor Error Total

20 © 2002 Thomson / South-Western Slide A Portion of the F Table for  = Denominator Degrees of Freedom Numerator Degrees of Freedom

21 © 2002 Thomson / South-Western Slide Freighter One-Way ANOVA: Procedural Summary Rejection Region  Critical Value Non rejection Region

22 © 2002 Thomson / South-Western Slide Excel Output for the Freighter Example Anova: Single Factor SUMMARY GroupsCountSumAverageVariance Long Beach Houston New York New Orleans ANOVA Source of VariationSSdfMSFP-valueF crit Between Groups Within Groups Total

23 © 2002 Thomson / South-Western Slide Multiple Comparison Tests An analysis of variance (ANOVA) test is an overall test of differences among groups. Multiple Comparison techniques are used to identify which pairs of means are significantly different given that the ANOVA test reveals overall significance.

24 © 2002 Thomson / South-Western Slide Randomized Block Design An experimental design in which there is one independent variable, and a second variable known as a blocking variable, that is used to control for confounding or concomitant variables. Confounding or concomitant variable are not being controlled by the business analyst but can have an effect on the outcome of the treatment being studied.

25 © 2002 Thomson / South-Western Slide Randomized Block Design, continued Blocking variable is a variable that the business analyst wants to control but is not the treatment variable of interest. Repeated measures design is a randomized block design in which each block level is an individual item or person, and that person or item is measured across all treatments.

26 © 2002 Thomson / South-Western Slide Partitioning the Total Sum of Squares in the Randomized Block Design SST (total sum of squares) SSC (treatment sum of squares) SSE (error sum of squares) SSR (sum of squares blocks) SSE’ (sum of squares error)

27 © 2002 Thomson / South-Western Slide A Randomized Block Design Individual observations Single Independent Variable Blocking Variable.....

28 © 2002 Thomson / South-Western Slide Randomized Block Design Treatment Effects: Procedural Overview

29 © 2002 Thomson / South-Western Slide Randomized Block Design: Computational Formulas

30 © 2002 Thomson / South-Western Slide Tread-Wear Example: Randomized Block Design Supplier SlowMediumFast Block Means ( ) Treatment Means( ) Speed C = 3 n = 5 N = 15

31 © 2002 Thomson / South-Western Slide Tread-wear Randomized Block Design: Sum of Squares Calculations (Part 1)

32 © 2002 Thomson / South-Western Slide Tread-wear Randomized Block Design: Sum of Squares Calculations (Part 2)

33 © 2002 Thomson / South-Western Slide Tread-wear Randomized Block Design: Mean Square Calculations

34 © 2002 Thomson / South-Western Slide Analysis of Variance for the Tread-Wear Example Source of VarianceSSdfMSF Treatment Block Error Total

35 © 2002 Thomson / South-Western Slide Tread-wear Randomized Block Design Treatment Effects: Procedural Summary

36 © 2002 Thomson / South-Western Slide Excel Output for Tread-Wear Randomized Block Design Anova: Two-Factor Without Replication SUMMARYCountSumAverageVariance Slow Medium Fast ANOVA Source of VariationSSdfMSFP-valueF crit Rows Columns E Error Total

37 © 2002 Thomson / South-Western Slide Two-Way Factorial Design An experimental design in which two ot more independent variables are studied simultaneously and every level of treatment is studied under the conditions of every level of all other treatments. Also called a factorial experiment.

38 © 2002 Thomson / South-Western Slide Two-Way Factorial Design Cells Column Treatment Row Treatment.....

39 © 2002 Thomson / South-Western Slide Two-Way ANOVA: Hypotheses

40 © 2002 Thomson / South-Western Slide Formulas for Computing a Two-Way ANOVA

41 © 2002 Thomson / South-Western Slide A 2  3 Factorial Design with Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

42 © 2002 Thomson / South-Western Slide A 2  3 Factorial Design with Some Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

43 © 2002 Thomson / South-Western Slide A 2  3 Factorial Design with No Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

44 © 2002 Thomson / South-Western Slide CEO Dividend 2  3 Factorial Design: Data and Measurements N = 24 n = 4 X= Location Where Company Stock is Traded How Stockholders are Informed of Dividends NYSEAMEXOTC Annual/Quarterly Reports Presentations to Analysts XjXj XiXi X 11 =1.5 X 23 =3.75X 22 =3.0X 21 =2.0 X 13 =3.5X 12 =2.5

45 © 2002 Thomson / South-Western Slide CEO Dividend 2  3 Factorial Design: Calculations (Part 1)

46 © 2002 Thomson / South-Western Slide CEO Dividend 2  3 Factorial Design: Calculations (Part 2)

47 © 2002 Thomson / South-Western Slide CEO Dividend 2  3 Factorial Design: Calculations (Part 3)

48 © 2002 Thomson / South-Western Slide CEO Dividend : Analysis of Variance Source of VarianceSSdfMSF Row Column * Interaction Error Total * Denotes significance at  =.01.

49 © 2002 Thomson / South-Western Slide CEO Dividend Excel Output (Part 1) Anova: Two-Factor With Replication SUMMARYNYSEASEOTCTotal Reports Count44412 Sum Average Variance Presentation Count44412 Sum Average Variance Total Count888 Sum Average Variance

50 © 2002 Thomson / South-Western Slide CEO Dividend Excel Output (Part 2) ANOVA Source of VariationSSdfMSFP-valueF crit Sample Columns E Interaction Within Total

51 © 2002 Thomson / South-Western Slide  2 Goodness-of-Fit Test The  2 goodness-of-fit test compares expected (theoretical) frequencies of categories from a population distribution to the observed (actual) frequencies from a distribution to determine whether there is a difference between what was expected and what was observed.

52 © 2002 Thomson / South-Western Slide  2 Goodness-of-Fit Test

53 © 2002 Thomson / South-Western Slide Month Gallons January1,553 February1,585 March1,649 April1,590 May1,497 June1,443 July1,410 August1,450 September1,495 October1,564 November1,602 December1,609 18,447 Milk Sales Data for Demonstration Problem 11.4

54 © 2002 Thomson / South-Western Slide Demonstration Problem 11.4: Hypotheses and Decision Rules

55 © 2002 Thomson / South-Western Slide Demonstration Problem 11.4: Calculations Monthfofo fefe (f o - f e ) 2 /f e January1,5531, February1,5851, March1,6491, April1,5901, May1,4971, June1,4431, July1,4101, August1,4501, September1,4951, October1,5641, November1,6021, December1,6091, ,44718, O bserved Chi-square = 41.59

56 © 2002 Thomson / South-Western Slide Demonstration Problem 11.4: Conclusion 0.01 df = Non Rejection region

57 © 2002 Thomson / South-Western Slide Defects Example: Using a  2 Goodness-of- Fit Test to Test a Population Proportion

58 © 2002 Thomson / South-Western Slide Defects Example: Calculations              oe ff f e = fofo fefe Defects3316 Nondefects n =

59 © 2002 Thomson / South-Western Slide Defects Example: Conclusion 0.05 df = Non Rejection region

60 © 2002 Thomson / South-Western Slide Contingency Analysis:  2 Test of Independence A statistical test used to analyze the frequencies of two variables with multiple categories to determine whether the two variables are independent. Qualitative Variables Nominal Data

61 © 2002 Thomson / South-Western Slide Investment Example   2 Test of Independence In which region of the country do you reside? A. Northeast B. Midwest C. South D. West Which type of financial investment are you most likely to make today? E. Stocks F. Bonds G. Treasury bills

62 © 2002 Thomson / South-Western Slide Investment Example   2 Test of Independence Type of financial Investment EFG AO 13 nAnA Geographic BnBnB Region CnCnC DnDnD nEnE nFnF nGnG N Contingency Table

63 © 2002 Thomson / South-Western Slide Investment Example   2 Test of Independence Type of Financial Investment EFG Ae 12 nAnA Geographic BnBnB Region CnCnC DnDnD nEnE nFnF nGnG N Contingency Table

64 © 2002 Thomson / South-Western Slide  2 Test of Independence: Formulas   ij ij e nn N where    : i= the row j= the column the total of row i the total of column j N= the total of all frequencies i j n n  2 2     oe where ff f e : df= (r- 1)(c- 1) r= the number of rows c= the number of columns Expected Frequencies Calculated   (Observed   )


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