# CHAPTER 13, PART B - Salisbury University

Statistics for Business and Economics (13e) Statistics for Business and Economics (13e) Anderson, Sweeney, Williams, Camm, Cochran 2017 Cengage Learning Slides by John Loucks St. Edwards University 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 1 Statistics for Business and Economics (13e) Chapter 13, Part B Experimental Design and Analysis of Variance Randomized Block Design Factorial Experiment 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 2 Statistics for Business and Economics (13e) Randomized Block Design

Experimental units are the objects of interest in the experiment. A completely randomized design is an experimental design in which the treatments are randomly assigned to the experimental units. If the experimental units are heterogeneous, blocking can be used to form homogeneous groups, resulting in a randomized block design. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 3 Statistics for Business and Economics (13e) Randomized Block Design ANOVA Procedure For a randomized block design the sum of squares total (SST) is partitioned into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error. SST = SSTR + SSBL + SSE The total degrees of freedom, nT - 1, are partitioned such that k - 1 degrees of freedom go to treatments, b - 1 go to blocks, and (k - 1)(b - 1) go to the error term. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 4 Statistics for Business and Economics (13e)

Randomized Block Design ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Treatments SSTR k-1 Blocks SSBL b-1 Error SSE (k 1)(b 1)

Total SST nT - 1 Mean Square F pValue SSTR MSTR MSTR= 1 MSE SSBL MSBL= 1 SSE MSE= ( 1 )( 1 ) 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

5 Statistics for Business and Economics (13e) Randomized Block Design Example: Crescent Oil Co. Crescent Oil has developed three new blends of gasoline and must decide which blend or blends to produce and distribute. A study of the miles per gallon ratings of the three blends is being conducted to determine if the mean ratings are the same for the three blends. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 6 Statistics for Business and Economics (13e) Randomized Block Design Example: Crescent Oil Co. Five automobiles have been tested using each of the three gasoline blends and the miles per gallon ratings are shown on the next slide. Factor . . . Gasoline blend Treatments . . . Blend X, Blend Y, Blend Z Blocks . . . Automobiles Response variable . . . Miles per gallon 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

7 Statistics for Business and Economics (13e) Randomized Block Design Type of Gasoline (Treatment) Automobile (Block) Blend X Blend Y Blend Z Block Means 1 2 3 4 5 31 30 29

33 26 30 29 29 31 25 30 29 28 29 26 30.333 29.333 28.667 31.000 25.667 Treatment Means 29.8 28.8 28.4

2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8 Statistics for Business and Economics (13e) Randomized Block Design Mean Square Due to Treatments The overall sample mean is 29. Thus, SSTR = 5[(29.8 - 29)2 + (28.8 - 29)2 + (28.4 - 29)2] = 5.2 MSTR = 5.2/(3 - 1) = 2.6 Mean Square Due to Blocks SSBL = 3[(30.333 - 29)2 + . . . + (25.667 - 29)2] = 51.33 MSBL = 51.33/(5 - 1) = 12.8 Mean Square Due to Error SSE = 62 - 5.2 - 51.33 = 5.47 MSE = 5.47/[(3 - 1)(5 - 1)] = .68 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 9 Statistics for Business and Economics (13e) Randomized Block Design ANOVA Table

Source of Variation Sum of Squares Degrees of Freedom Mean Square F p-Value 5.20 2 2.60 3.82 .07 Blocks 51.33

4 12.80 Error 5.47 8 .68 Total 62.00 14 Treatments 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 10 Statistics for Business and Economics (13e) Randomized Block Design

Rejection Rule p-Value Approach: Reject H0 if p-value < .05 Critical Value Approach: Reject H0 if F > 4.46 For = .05, F.05 = 4.46 (2 d.f. numerator and 8 d.f. denominator) 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 11 Statistics for Business and Economics (13e) Randomized Block Design Test Statistic F = MSTR/MSE = 2.6/.68 = 3.82 Conclusion The p-value is greater than .05 (where F = 4.46) and less than .10 (where F = 3.11). (Excel provides a p-value of .07). Therefore, we cannot reject H0. There is insufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends.

2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 12 Statistics for Business and Economics (13e) Factorial Experiment In some experiments we want to draw conclusions about more than one variable or factor. Factorial experiments and their corresponding ANOVA computations are valuable designs when simultaneous conclusions about two or more factors are required. The term factorial is used because the experimental conditions include all possible combinations of the factors. For example, for a levels of factor A and b levels of factor B, the experiment will involve collecting data on ab treatment combinations. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 13 Statistics for Business and Economics (13e) Two-Factor Factorial Experiment ANOVA Procedure The ANOVA procedure for the two-factor factorial experiment is similar to the completely randomized experiment and the randomized block experiment.

We again partition the sum of squares total (SST) into its sources. SST = SSA + SSB + SSAB + SSE The total degrees of freedom, nT - 1, are partitioned such that (a 1) d.f go to Factor A, (b 1) d.f go to Factor B, (a 1)(b 1) d.f. go to Interaction, and ab(r 1) go to Error. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 14 Statistics for Business and Economics (13e) Two-Factor Factorial Experiment Source of Variation Sum of Squares Degrees of Freedom Factor A SSA a-1

Factor B SSB b-1 Interaction SSAB Mean Square F SSA 1 SSB MSB= 1 MSA MSE MSB MSE MSA= MSAB

MSE (a1)(b1) Error SSE ab(r 1) Total SST nT - 1 pValue SSE MSE= ( 1 ) 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 15 Statistics for Business and Economics (13e)

Two-Factor Factorial Experiment Step 1 Compute the total sum of squares SST = ( ) 2 =1 =1 =1 Step 2 Compute the sum of squares for factor A SSA = ( . ) 2 =1 Step 3 Compute the sum of squares for factor B 2

SSB= ( . ) =1 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 16 Statistics for Business and Economics (13e) Two-Factor Factorial Experiment Step 4 Compute the sum of squares for interaction 2 SSAB= ( . . + ) =1 =1 Step 5 Compute the sum of squares due to error SSE = SST SSA SSB - SSAB 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

17 Statistics for Business and Economics (13e) Two-Factor Factorial Experiment Example: State of Ohio Wage Survey A survey was conducted of hourly wages for a sample of workers in two industries at three locations in Ohio. Part of the purpose of the survey was to determine if differences exist in both industry type and location. The sample data are shown on the next slide. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 18 Statistics for Business and Economics (13e) Two-Factor Factorial Experiment Example: State of Ohio Wage Survey Industry I I I II II II Cincinnati

\$12.10 11.80 12.10 12.40 12.50 12.00 Cleveland \$11.80 11.20 12.00 12.60 12.00 12.50 Columbus \$12.90 12.70 12.20 13.00 12.10 12.70 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 19 Statistics for Business and Economics (13e)

Two-Factor Factorial Experiment Factors Factor A: Industry Type (2 levels) Factor B: Location (3 levels) Replications Each experimental condition is repeated 3 times 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 20 Statistics for Business and Economics (13e) Two-Factor Factorial Experiment ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square

F p-Value Factor A .50 1 .50 4.19 .06 Factor B Interaction 1.12 .37 2 2 .56 .19 4.69

1.55 .03 .25 Error 1.43 12 .12 Total 3.42 17 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 21 Statistics for Business and Economics (13e) Two-Factor Factorial Experiment Conclusions Using the Critical Value Approach Industries:

F = 4.19 < F = 4.75 Mean wages do not differ by industry type. Locations: F = 4.69 > F = 3.89 Mean wages differ by location. Interaction: F = 1.55 < F = 3.89 Interaction is not significant. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 22 Statistics for Business and Economics (13e) Two-Factor Factorial Experiment Conclusions Using the p-Value Approach Industries: p-value = .06 > = .05 Mean wages do not differ by industry type. Locations:

p-value = .03 < = .05 Mean wages differ by location. Interaction: p-value = .25 > = .05 Interaction is not significant. 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 23 Statistics for Business and Economics (13e) End of Chapter 13, Part B 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 24

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