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

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© 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.

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© 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.

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© 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.

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© 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.

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© 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.

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

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© 2002 Thomson / South-Western Slide 11-8 Completely Randomized Design Machine Operator 123.................. Valve Opening Measurements

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© 2002 Thomson / South-Western Slide 11-9 Example: Number of Foreign Freighters Docking in each Port per Day Long Beach 57425742 Houston 2354623546 New York 846798846798 New Orleans 3534235342

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© 2002 Thomson / South-Western Slide 11-10 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.

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

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© 2002 Thomson / South-Western Slide 11-12 Partitioning Total Sum of Squares of Variation SST (Total Sum of Squares) SSC (Treatment Sum of Squares) SSE (Error Sum of Squares)

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© 2002 Thomson / South-Western Slide 11-13 One-Way ANOVA: Sums of Squares Definitions

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© 2002 Thomson / South-Western Slide 11-14 One-Way ANOVA: Computational Formulas

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© 2002 Thomson / South-Western Slide 11-15 Freighter One-Way ANOVA: Preliminary Calculations Long Beach 5 7 4 2 T 1 = 18 n 1 = 4 Houston 2 3 5 4 6 T 2 = 20 n 2 = 5 New York 8 4 6 7 9 8 T 3 = 42 n 3 = 6 New Orleans 3 5 3 4 2 T 4 = 17 n 4 = 5 T = 97 N = 20

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© 2002 Thomson / South-Western Slide 11-16 Freighter One-Way ANOVA: Sum of Squares Calculations

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© 2002 Thomson / South-Western Slide 11-17 Freighter One-Way ANOVA: Sum of Squares Calculations, continued

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© 2002 Thomson / South-Western Slide 11-18 Freighter One- Way ANOVA: Mean Square and F Calculations

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© 2002 Thomson / South-Western Slide 11-19 Freighter Example: Analysis of Variance Source of VariancedfSSMSF Between Factor342.3514.125.12 Error1644.202.76 Total1986.55

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© 2002 Thomson / South-Western Slide 11-20 A Portion of the F Table for = 0.05 123456789 1161.45199.50215.71224.58230.16233.99236.77238.88240.54... 154.543.683.293.062.902.792.712.642.59 164.493.633.243.012.852.742.662.592.54 174.453.593.202.962.812.702.612.552.49 Denominator Degrees of Freedom Numerator Degrees of Freedom

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© 2002 Thomson / South-Western Slide 11-21 Freighter One-Way ANOVA: Procedural Summary Rejection Region Critical Value Non rejection Region

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© 2002 Thomson / South-Western Slide 11-22 Excel Output for the Freighter Example Anova: Single Factor SUMMARY GroupsCountSumAverageVariance Long Beach4184.54.3333 Houston52042.5 New York64273.2 New Orleans5173.41.3 ANOVA Source of VariationSSdfMSFP-valueF crit Between Groups42.35314.1175.11010.01143.2389 Within Groups44.2162.7625 Total86.5519

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© 2002 Thomson / South-Western Slide 11-23 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.

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© 2002 Thomson / South-Western Slide 11-24 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.

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© 2002 Thomson / South-Western Slide 11-25 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.

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© 2002 Thomson / South-Western Slide 11-26 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)

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© 2002 Thomson / South-Western Slide 11-27 A Randomized Block Design Individual observations........................ Single Independent Variable Blocking Variable.....

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© 2002 Thomson / South-Western Slide 11-28 Randomized Block Design Treatment Effects: Procedural Overview

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© 2002 Thomson / South-Western Slide 11-29 Randomized Block Design: Computational Formulas

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© 2002 Thomson / South-Western Slide 11-30 Tread-Wear Example: Randomized Block Design Supplier 1 2 3 4 SlowMediumFast Block Means ( ) 3.74.53.13.77 3.43.92.83.37 3.54.13.03.53 3.23.52.63.10 5 Treatment Means( ) 3.94.83.44.03 3.544.162.983.56 Speed C = 3 n = 5 N = 15

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© 2002 Thomson / South-Western Slide 11-31 Tread-wear Randomized Block Design: Sum of Squares Calculations (Part 1)

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© 2002 Thomson / South-Western Slide 11-32 Tread-wear Randomized Block Design: Sum of Squares Calculations (Part 2)

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© 2002 Thomson / South-Western Slide 11-33 Tread-wear Randomized Block Design: Mean Square Calculations

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© 2002 Thomson / South-Western Slide 11-34 Analysis of Variance for the Tread-Wear Example Source of VarianceSSdfMSF Treatment3.48421.74296.78 Block1.54940.387 Error0.14380.018 Total5.17614

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© 2002 Thomson / South-Western Slide 11-35 Tread-wear Randomized Block Design Treatment Effects: Procedural Summary

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© 2002 Thomson / South-Western Slide 11-36 Excel Output for Tread-Wear Randomized Block Design Anova: Two-Factor Without Replication SUMMARYCountSumAverageVariance 1311.33.76666670.4933333 2310.13.36666670.3033333 3310.63.53333330.3033333 439.33.10.21 5312.14.03333330.5033333 Slow517.73.540.073 Medium520.84.160.258 Fast514.92.980.092 ANOVA Source of VariationSSdfMSFP-valueF crit Rows1.549333340.387333321.7196260.00023577.0060651 Columns3.48421.74297.6822432.395E-068.6490672 Error0.142666780.0178333 Total5.17614

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© 2002 Thomson / South-Western Slide 11-37 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.

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© 2002 Thomson / South-Western Slide 11-38 Two-Way Factorial Design Cells........................ Column Treatment Row Treatment.....

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© 2002 Thomson / South-Western Slide 11-39 Two-Way ANOVA: Hypotheses

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© 2002 Thomson / South-Western Slide 11-40 Formulas for Computing a Two-Way ANOVA

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© 2002 Thomson / South-Western Slide 11-41 A 2 3 Factorial Design with Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

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© 2002 Thomson / South-Western Slide 11-42 A 2 3 Factorial Design with Some Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

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© 2002 Thomson / South-Western Slide 11-43 A 2 3 Factorial Design with No Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

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© 2002 Thomson / South-Western Slide 11-44 CEO Dividend 2 3 Factorial Design: Data and Measurements N = 24 n = 4 X=2.7083 1.752.753.625 Location Where Company Stock is Traded How Stockholders are Informed of Dividends NYSEAMEXOTC Annual/Quarterly Reports 21212121 23322332 43434343 2.5 Presentations to Analysts 23122312 33243324 44344434 2.9167 XjXj XiXi X 11 =1.5 X 23 =3.75X 22 =3.0X 21 =2.0 X 13 =3.5X 12 =2.5

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© 2002 Thomson / South-Western Slide 11-45 CEO Dividend 2 3 Factorial Design: Calculations (Part 1)

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© 2002 Thomson / South-Western Slide 11-46 CEO Dividend 2 3 Factorial Design: Calculations (Part 2)

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© 2002 Thomson / South-Western Slide 11-47 CEO Dividend 2 3 Factorial Design: Calculations (Part 3)

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© 2002 Thomson / South-Western Slide 11-48 CEO Dividend : Analysis of Variance Source of VarianceSSdfMSF Row1.041811.04182.42 Column14.083327.041716.35 * Interaction0.083320.04170.10 Error7.7500180.4306 Total22.958323 * Denotes significance at =.01.

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© 2002 Thomson / South-Western Slide 11-49 CEO Dividend Excel Output (Part 1) Anova: Two-Factor With Replication SUMMARYNYSEASEOTCTotal Reports Count44412 Sum6101430 Average1.52.53.52.5 Variance0.3333 1 Presentation Count44412 Sum8121535 Average233.752.9167 Variance0.6667 0.250.9924 Total Count888 Sum142229 Average1.752.753.625 Variance0.5 0.2679

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© 2002 Thomson / South-Western Slide 11-50 CEO Dividend Excel Output (Part 2) ANOVA Source of VariationSSdfMSFP-valueF crit Sample1.04171 2.41940.13734.4139 Columns14.08327.041716.3559E-053.5546 Interaction0.083320.04170.09680.90823.5546 Within7.75180.4306 Total22.95823

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© 2002 Thomson / South-Western Slide 11-51 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.

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© 2002 Thomson / South-Western Slide 11-52 2 Goodness-of-Fit Test

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© 2002 Thomson / South-Western Slide 11-53 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

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© 2002 Thomson / South-Western Slide 11-54 Demonstration Problem 11.4: Hypotheses and Decision Rules

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© 2002 Thomson / South-Western Slide 11-55 Demonstration Problem 11.4: Calculations Monthfofo fefe (f o - f e ) 2 /f e January1,5531,537.250.16 February1,5851,537.251.48 March1,6491,537.258.12 April1,5901,537.251.81 May1,4971,537.251.05 June1,4431,537.25 5.78 July1,4101,537.2510.53 August1,4501,537.25 4.95 September1,4951,537.251.16 October1,5641,537.250.47 November1,6021,537.252.73 December1,6091,537.253.35 18,44718,447.0041.59 O bserved Chi-square = 41.59

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© 2002 Thomson / South-Western Slide 11-56 Demonstration Problem 11.4: Conclusion 0.01 df = 11 24.725 Non Rejection region

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© 2002 Thomson / South-Western Slide 11-57 Defects Example: Using a 2 Goodness-of- Fit Test to Test a Population Proportion

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© 2002 Thomson / South-Western Slide 11-58 Defects Example: Calculations 2 2 3316 167184 16184 18.06+ 1.57 1963 oe ff f e = 2 2.. fofo fefe Defects3316 Nondefects167184 200 n =

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© 2002 Thomson / South-Western Slide 11-59 Defects Example: Conclusion 0.05 df = 1 3.841 Non Rejection region

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© 2002 Thomson / South-Western Slide 11-60 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

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© 2002 Thomson / South-Western Slide 11-61 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

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© 2002 Thomson / South-Western Slide 11-62 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

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© 2002 Thomson / South-Western Slide 11-63 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

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© 2002 Thomson / South-Western Slide 11-64 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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