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Dr. Jerrell T. Stracener

EMIS 7370 STAT 5340

Probability and Statistics for Scientists and Engineers

Department of Engineering Management, Information and Systems

SMU BOBBY B. LYLESCHOOL OF ENGINEERING

EMIS - SYSTEMS ENGINEERING PROGRAM

Design of Experiments& One Factor Experiments

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Provides a systematic approach to planning whatdata is required and the analysis to be performed

Designing an experiment - Planning an experimentso that information will be collected whichrelevant to the problem under investigation.

The design of an experiment is the completesequence of steps taken ahead of time to insurethat the appropriate data will be obtained in a waywhich permits an objective analysis leading tovalid inferences with respect to the stated problem.

Method of Experimental Design

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1. There must be a clearly defined objective.2. The effects of the factors should not be obscuredby other variables.3. The results should not be influenced by conscious or unconscious bias in the experiment or on the part of the experimenter.4. The experiment should provide some measureof precision.5. The experiment must have sufficient precisionto accomplish its purpose.

Some requisites of a good experiment:

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That unit to which a single treatment, which maybe a combination of many factors, is applied inone replication of the basic experiment.

A factor refers to an independent variable.

Experimental Units

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Implies the particular set of experimental conditions which will be imposed on an experimental condition which will be imposed on an experimental unit within the confines of the chosen design.

Treatment and Treatment Combinations

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The allocation of the experimental units to blocksin such a manner that the units within a block arerelatively homogenous while the greater part of the predictable variation among units has beenconfounded with the effect of blocks.

Blocking

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The placing of a set of homogenous experimentalunits into groups in order that the different groups may be subjected to different treatments

Grouping

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Obtaining the experimental units, the grouping,the blocking and the assignment of the treatmentsto the experimental units in such a way that abalanced configuration exists.

Balancing

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The results of experiments are affected not only by the action of the treatments, but also by extraneous variations which tend to mask theeffects of the treatments. The term ‘experimentalerrors’ is often applied to these variations, wherethe word errors is not synonymous with ‘mistakes’,but includes all types of extraneous variation.

Two main sources of experimental errors are:1. Inherent variability in the experimental material2. Lack of uniformity in the physical conduct of the experiment, or failure to standardize theexperimental technique.

Experimental Error

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1. Replication

2. Randomization

3. Local Control

Basic Principles of Experimental Design

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Repetition of the basic experiment. In order to evaluate the effects of factors, a measure of precision must be available. In situations where the measurement of precision must be obtainedfrom the experiment itself, replication provides themeasure. It also provides an opportunity for theeffects of uncontrolled factors to balance out, andthus aids randomization as a bias-decreasing tool.Replication will also help to spot gross errors inmeasurement.

Replication makes a test of significance possible.

Replication

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By insisting on a random assignment of treatments to the experimental units, we can proceed as though the assumption: ‘Observations are independently distributed’, is true.

Randomization makes the test valid by making itappropriate to analyze the data as though theassumption of independent error is true.

Randomization

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The amount of balancing, blocking, and groupingof the experimental units that is employed in theadopted statistical design. The function of localcontrol is to make the experimental design moreefficient. That is, local control makes any test ofsignificance more sensitive. This increase in efficiency (or sensitivity) results because a properuse of local control will reduce the magnitude ofthe estimate of experimental error.

Local Control

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A statistically designed experiment consists of thefollowing steps:1. Statement of the problem.2. Formulation of hypothesis.3. Devising of experimental technique and design.4. Examination of possible outcomes and referenceback to the reasons for the inquiry to be sure theexperiment provides the required information toan adequate extent.5. Consideration of the possible results from thepoint of view of the statistical procedures whichwill be applied to them, to ensure that the conditions necessary for these procedures to bevalid are satisfied.

Steps in Designing an Experiment

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6. Performance of experiment.7. Application of statistical techniques to theexperimental results. 8. Drawing conclusions with measures of thereliability of estimates of any quantities that areevaluated, careful consideration being given to the validity of the conclusions for the populationof objects or events to which they are to apply.9. Evaluation of the whole investigation, particularly with other investigations on the sameor similar problem.

Note: Frequently, there is a formidable barrierto communications which must be overcome.

Steps in Designing an Experiment continue

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A. Obtain a clear statement of the problem.B. Collect available background information.C. Design a test program

1. Hold a conference of all parties concerned2. Design the program in preliminary form3. Review the design with all concerned

D. Plan and carry out the experimental work.E. Analyze the dataF. Interpret the resultsG. Prepare the report

Check List for Planning Test Programs

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One Factor Designs

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• A design in which the treatments are assignedcompletely at random to the experimental units, or vice versa. It imposes no restrictions, such asblocking, or the allocation of the treatments tothe experimental units.

• Used because of its simplicity

• Restricted to cases in which homogenous experimental units are available

Completely Randomized Designs

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The basic assumption for a completely randomizeddesign with one observation per experimental unitis that the observations may be represented mathematically by the linear statistical model

Yij = + i + ij , i = 1, 2, …, kj = 1, 2, …, n

whereYij is the observation associated with the ith treatment and jth experimental unit, is the true mean effect (constant)i is the true effect of the ith treatment

and ij is the experimental error

Model

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The results of a completely random experiment with one observation per experiment unit maybe exhibited as follows:

Treatment Total1 2 . . .

Y11 Y21 . . . Y

Y12 Y22 . . . .

. . . . . .

Y1n Y2n . . . Yn

Totals Y1· Y2· . . . Y· Y··

Number of Obs.n n . . . n n

Means Y1 Y2 . . . Y Y

Statistical Layout

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In partitioning the total variation of the observations into the variation attributable tomean, treatments, and random error, the Sum-of-Squares is used:

Total Sum of Squares = Treatment Sum of Squares + Error Sum of Squares

SST = SSA + SSE

k

i

n

jiij

k

ii

k

i

n

jij yyyynyy

1 1

2.

1

2...

1 1

2..

Analysis of Variance

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where

and

k

i

n

jij yy

1 1

2..SST

k

ii yyn

1

2...SSA

k

i

n

jiij yy

1 1

2.SSE

Analysis of Variance

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•

•

• SSE = SST - SSA

k

i

n

jij nk

YY

1

2..

1

2SST

k

i

i

nk

Y

n

Y

1

2..

2.SSA

Analysis of Variance Computational Formulas

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Sources of Degrees of Sum of Mean F- CriticalVariation Freedom Squares Square Ratio Value of F

Treatments SSA

Error (n-1) SSE

Total n-1 SST

1)-K(n

SSEs2

1K

SSAs 2

1

2

21

s

s

Analysis of Variance Table (AOV Table or ANOVA Table)

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Suppose that an appliance manufacturer is interested indetermining whether the brand of laundry detergent usedaffects the amount of dirt removed from standard householdlaundry loads. In particular, the manufacturer wants tocompare four different brands of detergent (labeled A, B, C,and D). Suppose that, after a random assignment of ten loadsto each brand, the amount of dirt removed (measured inmilligrams) was determined, with the results summarized below.

A B C D11 12 18 1113 14 16 1217 17 18 1617 19 20 1515 21 22 1416 18 15 1714 19 17 1310 18 21 1612 16 16 1714 18 20 18

AMOUNT OF DIRT (MG) REMOVED BY FOUR DETERGENT BRANDSBrand

Example

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The statistical layout is:

Treatment(Brand) TotalA B C D

11 12 18 11

13 14 16 12

… … … …

14 18 20 18

Totals 139 172 183 149 643

Number of Obs. 10 10 10 10 40

Means 13.9 17.2 18.3 14.9 16.075

Example - Solution

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Plot of Means

13.9

17.2 18.3

14.9

0

5

10

15

20

A B C D

Detergent Brand

Ave

rag

e

overall average=16.075

Example - Solution

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The sum of squares are calculated as follows:

The mean squares are:

The calculated F-ratio is:

123.275)(SSA 24

1

i

i YYn

500.213)(SSE 24

1

10

1

i j

iij YY

775.336)(SST4

1

10

1

2 i j

ij YY

93.6

9306.5339

0917.4114

MSE

MSAF

SSEMSE

SSAMSA

Example - Solution

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Analysis of Variance Table

Sources of Degrees of Sum of Mean of F- CriticalVariation Freedom Squares Squares Ratio Value of F

Treatments 3 123.275 41.09 6.92 2.866

Error 36 213.5 5.93

Total 39 336.775

Example - Solution

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Since the probability of obtaining an F statistic of6.93 or larger when the null hypothesis is true isapproximately 0.001 and less than the specified of 0.05, the null hypotheses is rejected.orsince the calculated F-ratio of 6.93 is greater thanthe critical value of 2.87, the null hypotheses isrejected.

Therefore conclude that the different brands ofdetergent are not equally effective.

Example - Conclusion