statistically quality design topics –four perspectives in quality improvement –review doe topics...

Statistically Quality DesignStatistically Quality Design

TopicsTopics– Four perspectives in quality improvementFour perspectives in quality improvement– Review DOE topics and terminologiesReview DOE topics and terminologies– Implementation plan and procedure for Implementation plan and procedure for

experimental designexperimental design– Full factorial design and Yates’ algorithmFull factorial design and Yates’ algorithm– Full factorial design example: improving wave Full factorial design example: improving wave

solder process at TDY companysolder process at TDY company– Concepts and examples for conducting Block and Concepts and examples for conducting Block and

Latin square designsLatin square designs

Four Perspectives in Quality Improvement Four Perspectives in Quality Improvement

1. Downstream Perspective: Customer’s Quality – such as: fuel consumption, noise, failure rates, pollution, etc.

2. Midstream Perspective: Manufacturing Quality (spec.+

drawings)– important for production or trading.

3. Upstream: Quality of Design (robustness of objective function)

– good for design & development after product planning

4. Origin: Quality of Technology (robustness of technology )– good for technology development prior to product planning– functionality of generic function– example: Hook’s Law for spring

MY

QualityQuality Engineering Engineering提供在產品設計及生產過程中對市場可能會發生故障之預測技術 /方法

Characteristics of Technology DevelopmentCharacteristics of Technology Development

1. Technology Readiness ( 先行性 ): 若產品計畫前作技術開發則設計完後僅須作適當調整

2. Flexibility ( 汎用性 ): 針對一系列或下一代產品作品質改善

3. Reproducibility ( 再現性 ): R & D Manufacturing Market

Design of Experiments (Topics)Design of Experiments (Topics)CRDCRD

- completely randomized designcompletely randomized design

Full FactorialFull Factorial- all possible combinations of factors and levelsall possible combinations of factors and levels

Fractional FactorialFractional Factorial- assume some interaction will not occur, a factor is assignedassume some interaction will not occur, a factor is assigned

Latin SquareLatin Square- each level of each factor appears only once with each level of two other factorseach level of each factor appears only once with each level of two other factors

Yates’ Notation & AlgorithmYates’ Notation & Algorithm

On-line V.S. Off-line Quality ControlOn-line V.S. Off-line Quality Control

Design of Experiments (Topics cont.)Design of Experiments (Topics cont.)

System DesignSystem Design– the selection of materials, parts, equipment, and process parametersthe selection of materials, parts, equipment, and process parameters

Parameter DesignParameter Design– study the effect of noise factor in DOEstudy the effect of noise factor in DOE

Tolerance DesignTolerance Design– the specification of appropriate tolerance, product and process parametersthe specification of appropriate tolerance, product and process parameters

Signal to Noise Ratio (S/N Ratio)Signal to Noise Ratio (S/N Ratio)– a comparison of the influence of control factors (signal) to that of a comparison of the influence of control factors (signal) to that of

noise factorsnoise factors

Orthogonal ArrayOrthogonal Array− a design where correlation between factors is zeroa design where correlation between factors is zero

Outer ArrayOuter Array– in parameter design to identify the combination of noise factorsin parameter design to identify the combination of noise factors

Experimental Design TerminologyExperimental Design Terminology

ANOVAANOVA– Analysis of Variance

Experimental UnitExperimental Unit– largest collection of experiment material

TreatmentTreatment– what is done to the experiment materials

Sampling UnitSampling Unit– a part of experimental unit

Experimental Design Terminology (cont.)Experimental Design Terminology (cont.)

Experimental/ Sampling Errors– a measure of variation

Randomization– a system of using random number

Replication– number of times a specific combination of factor level is run during an

experiment

Factor– an input to a process produces an effect. controllable factors vs. noise

factors

Level– a setting or value of a factor

Run– number of trials for each condition of an experiment

Experimental Design Terminology (cont.)Experimental Design Terminology (cont.)

Quality CharacteristicQuality Characteristic– the response variable (output)the response variable (output)

InteractionInteraction– the combination of two factors generates a result that is different the combination of two factors generates a result that is different

from individual factor. from individual factor. – main effect vs. interaction effectmain effect vs. interaction effect

DOFDOF– independent piece of informationindependent piece of information

ResolutionResolution– number of letters in the shortest length in defining relationnumber of letters in the shortest length in defining relation– the lower the number, the more saturated the design isthe lower the number, the more saturated the design is

ANOVA (ANOVA ( 變異數分析變異數分析 ))

The method of analyzing data collected by CRD / RCBDANOVA equation

ANOVA Table

...... iijiij YYYYYY

i j

iiji j

i

i jij YYYYYY

2.

2...

2..

=> SST = SStrt + SSE

source of variationsource of variation d.f.d.f. SSSS MSMS FF

Between Trt.Between Trt. a -1a -1

ErrorError n – an – a

TotalTotal N - 1N - 1 2

..YYij

2

.iij YY

2

... YY i 1 aSSMS trttrt

aNSSEMSE

MSEMStrt

ANOVAANOVA

Example:Example:

一個製造紙袋用紙的工廠想改善其紙張強度，若已知紙張強度與紙漿之一個製造紙袋用紙的工廠想改善其紙張強度，若已知紙張強度與紙漿之濃度有關，今欲調查濃度有關，今欲調查 55 種不同之濃度種不同之濃度 (Hardwood (Hardwood Concentration)Concentration) ，， 5%5% ，， 10%10% ，， 15%15% ，， 20%20% 及及 25%25% ，每一濃度取，每一濃度取 55 個觀個觀察值察值 (obs.(obs. ，， tensile strength)tensile strength) ，其結果如下表：，其結果如下表：

Obs.Obs.

濃度濃度 11 22 33 44 55

5%5% 77 77 1515 1111 99 4949 9.89.8

10%10% 1212 1717 1212 1818 1818 7777 15.415.4

15%15% 1414 1818 1818 1919 1919 8888 17.617.6

20%20% 1919 2525 2222 1919 2323 108108 21.621.6

25%25% 77 1010 1111 1515 1111 5454 10.810.8

.iY .iY

376.. Y

ANOVA (cont.)ANOVA (cont.)

),0(~ 2 NY ijijiijiij

0 1 2 3 4 5 1 2 3 4 5

0

: ( . . 0)

: 0i

H i e

H

至少有一

source of variationsource of variation d.f.d.f. SSSS MSMS FF

Between Trt.Between Trt. 5 -15 -1 475.56475.56 118.89118.89 14.7506214.75062

ErrorError 25 – 525 – 5 161.20161.20 8.068.06

TotalTotal 25 - 125 - 1 636.96636.96

96.6362

..YYSSTO ij

20.1612

. iij YYSSE 56.4752

... YYSS itrt

Fixed Effect vs. Random EffectFixed Effect vs. Random Effect

Fixed Effect– 一個工廠有三部機器，觀察三者間有無顯著差異 (chosen in a

nonrandom manner/ a small hand-selected factor level)

Radom Effect– 某工廠有 30 部同類型機器，由其中隨機取出三台，由此三台機

器觀察工廠內 30 台機器之管理狀態 (study the source of variability/ the variation associated with a factor)

MSEMSF

MSEMSF

MSEMSF

AB

B

A

0

0

0

MSEMSF

MSMSF

MSMSF

AB

ABB

ABA

0

0

0

Fixed EffectFixed Effect Radom EffectRadom Effect

Effect AEffect A

Effect BEffect B

Effect ABEffect AB

Implementation Plan and Procedure for Implementation Plan and Procedure for Experimental DesignExperimental Design

Basically, a twelve-steps (procedure) approachfor conducting any experimental design can be divided into the following three stages:

Stage 1 (( 準備及設計選擇階段準備及設計選擇階段 ))

1.1. Define the problems and state the objective of the experimentDefine the problems and state the objective of the experiment

2.2. Select quality characteristic (response) and input variables Select quality characteristic (response) and input variables (factors)(factors)

3.3. Determine the desired number of runs and replicationsDetermine the desired number of runs and replications

4.4. Consider the randomization of runs during the selection of the best Consider the randomization of runs during the selection of the best design typedesign type

Stage 2 (( 實驗及分析資料階段實驗及分析資料階段 ))

1.1. Conduct the experiment and record the dataConduct the experiment and record the data

2.2. Analyze the data using analyze of mean, analysis of varianceAnalyze the data using analyze of mean, analysis of variance

3.3. Use Yates’ algorithm and normal probability plot to determine Use Yates’ algorithm and normal probability plot to determine the significant main and interaction effectsthe significant main and interaction effects

Stage 3 (( 建立預估模式及確認評估階建立預估模式及確認評估階段段 ))

1.1. Develop a fitted model using regression analysisDevelop a fitted model using regression analysis

2.2. Draw conclusion and make predictionDraw conclusion and make prediction

3.3. Perform confirmatory testsPerform confirmatory tests

4.4. Assess results and make decisionAssess results and make decision

Steps for Experimental DesignSteps for Experimental Design

1.1. Statement of the problemStatement of the problem:_________________________________:_________________________________

(During this step you should estimate your current level of quality by way of (During this step you should estimate your current level of quality by way of Cpk, dpm, or total loss. This estimate will be compared with improvements Cpk, dpm, or total loss. This estimate will be compared with improvements found after Step 7.)found after Step 7.)

2.2. Objective of the experimentObjective of the experiment:_______________________________:_______________________________

3.3. Star DateStar Date:_____________ ; :_____________ ; End DateEnd Date:_____________:_____________

4.4. Select quality characteristicsSelect quality characteristics

(also known as responses, dependent variables, or output variables). These (also known as responses, dependent variables, or output variables). These characteristics should be related to customer needs and expectations.characteristics should be related to customer needs and expectations.

ResponseResponse TypeType Anticipated RangeAnticipated Range How will you measure the How will you measure the response?response?

112233

Steps for Experimental Design (cont.)Steps for Experimental Design (cont.)

5.5. Select factorsSelect factors (also know as parameters or input variables) which are anticipated to have an (also know as parameters or input variables) which are anticipated to have an

effect on response.effect on response.

6.6. Determine the number of resources to be used in the experimentDetermine the number of resources to be used in the experiment (Consider the desired number, the cost per resource, time per experimental (Consider the desired number, the cost per resource, time per experimental

trial, and maximum allowable number of resources.)trial, and maximum allowable number of resources.)

7.7. Which design types and analysis strategies are appropriateWhich design types and analysis strategies are appropriate?? (Discuss advantage and disadvantages of each.)(Discuss advantage and disadvantages of each.)

FactorFactor TypeType Controllable or Controllable or NoiseNoise

Range of Range of InterestInterest LevelsLevels Anticipated Anticipated

Interactions WithInteractions WithHow How

MeasuredMeasured11

22

33

Steps for Experimental Design (cont.)Steps for Experimental Design (cont.)

8.8. Select the best design type and analysis strategy to suit your needsSelect the best design type and analysis strategy to suit your needs

9.9. Can all the runs be randomizedCan all the runs be randomized?__________________________?__________________________

Which factors are most difficult to randomize?________________Which factors are most difficult to randomize?________________

10.10. Conduct the experiment and record the dataConduct the experiment and record the data (Monitor both of these events for accuracy)(Monitor both of these events for accuracy)

11.11. Analyze the data, draw conclusions, mark predictions, and do Analyze the data, draw conclusions, mark predictions, and do confirmatory testsconfirmatory tests

12.12. Assess results and make decisionsAssess results and make decisions (Evaluate your new state of quality and compare with the quality level prior to the (Evaluate your new state of quality and compare with the quality level prior to the

improvement effort. If necessary, conduct more experimentation.)improvement effort. If necessary, conduct more experimentation.)

Full Factorial Experimental DesignFull Factorial Experimental Design

Principles:

Random Number TableRandom Number Table

There are 400 digits in this random number There are 400 digits in this random number table, 3 appears 41 times.table, 3 appears 41 times.

3 3 Factors, 2 LevelsFactors, 2 Levels

Four dimensional visibility with

823 test combinations in a full factorial matrix

Label The CellsLabel The Cells

8 TestCombinations 32

(1)(1) aa

bb abab

c c acac

bcbc abcabc

A- A+

B-

B+

B-

B+

C-

C+

Yates’ NotationYates’ Notation

823 Test Combinations

Cell A B AB C AC BC ABC (1) - - + - + + - a + - - - - + + b - + - - + - + ab + + + - - - - c - - + + - - + ac + - - + + - - bc - + - + - + - abc + + + + + + +

Yates’ NotationYates’ Notation

1624 Test Combinations

Yates’ Work SessionYates’ Work Session

Y = yield strength , PSI

A, B and C are concentrations of 3 separate elements

5858

5656

3636

3939

5151

5353

3434

3232

5353

48 48

5454

5959

4949

4949

5555

6161

A- A+

B-

B+

B-

B+

C-

C+

Determine the size of each contrast using Yates’ algorithm

What combination of elements will give the highest yield strength?

The Yates’ AlgorithmThe Yates’ Algorithm

two variables; A, B number of variables, n = 2 number of columns, n = 2

For top ½ of each column: ndst 21

Yates’ Work SessionYates’ Work Session

Yates’ Worksheet, 3 VariablesYates’ Worksheet, 3 Variables

Cell

(1)a

b

ab

c

ac

bc

abc

y y 1 2 3 RANK

TOTAL

Y 4

Analysis of Variance for a A×B Factorial Analysis of Variance for a A×B Factorial ExperimentExperiment

ANOVA of factorial experiment:The total sum of squares can be partitioned into :Total SS = SS(A) + SS(B) + SS(AB) + SSE

ANOVA Table For AXB Factorial ExperimentANOVA Table For AXB Factorial Experiment

SourceSource d.f.d.f. SSSS MSMS

Factor AFactor A

Factor BFactor B

Interaction ABInteraction AB

ErrorError

((a-1)a-1)

(b-1)(b-1)

(a-1)(b-1)(a-1)(b-1)

(n-ab)(n-ab)

SS(A)SS(A)

SS(B)SS(B)

SS(AB)SS(AB)

SSESSE

SS(A)/(a-1)SS(A)/(a-1)

SS(B)/(b-1)SS(B)/(b-1)

SS(AB)/((a-1)(b-1)SS(AB)/((a-1)(b-1)

SSE/(n-ab)SSE/(n-ab)

TotalTotal ((n-1)n-1) Total SSTotal SS

n = rabr = number of times each factorial treatment combination appears in the experiment

A×B Factorial Experiment (Cont.)A×B Factorial Experiment (Cont.)

)(

)()()(

)(

)(

)(

2

2

2

ABSSSSBSSASSTOTALSSE

BSSASSCFr

ABABSS

CFra

BBSS

CFrb

AASS

Test each null hypothesis:

( ) ( ) ( )MS A MS B MS ABF and F and F

MSE MSE MSE

Example: A × B Factorial ExperimentExample: A × B Factorial Experiment

The evaluation of a flame retardant was conducted at two different laboratories on three different materials with the following results

MaterialsMaterials

LaboratoryLaboratory 11 22 33

11

22

4.1 , 3.94.1 , 3.9

4.34.3

2.7 , 3.12.7 , 3.1

2.62.6

3.1 , 2.83.1 , 2.8

3.33.3

1.9 , 2.21.9 , 2.2

2.32.3

3.5 , 3.23.5 , 3.2

3.63.6

2.7 , 2.32.7 , 2.3

2.52.5

Example: A×B Factorial Experiment (Cont.) Example: A×B Factorial Experiment (Cont.)

Total For Calculating Sums of SquaresTotal For Calculating Sums of Squares

Material (B)Material (B)

LaboratoryLaboratory 11 22 33 Total (A)Total (A)

11

22

12.312.3

8.48.4

9.29.2

6.46.4

10.310.3

7.57.5

31.831.8

22.322.3

Total (B)Total (B) 20.720.7 15.615.6 17.817.8 54.154.1

There are n = rab = (3)(2)(3) =18 observation

6006.16218

)1.54( 2

CF

6000.1344.1811.20139.59294.7

)()()(

1344.1311.20139.56006.16293.169

)()(3

)5.7...2.93.12(

1811.26006.1627817.1646

)8.176.157.20(

0139.56006.1626144.1679

)3.228.31(

9294.76006.16253.170

)5.2...9.31.4(

222

222

22

222

ABSSBSSASSSSTOTAL

BSSASSCF

CF

CF

CFTotal SS

SS(A)

SS(B)

SS(AB)

SSE

Example: A×B Factorial Experiment (Cont.)Example: A×B Factorial Experiment (Cont.)

ANOVA Table

SourceSource d.fd.f..

SSSS MSMS FF

Laboratory (A)Laboratory (A)

Material (B)Material (B)

Interaction (AB)Interaction (AB)

ErrorError

11

22

22

1212

5.01395.0139

2.18112.1811

.1344.1344

.6000.6000

5.01395.0139

1.09061.0906

0.6720.672

0.05000.0500

100.28100.28

21.8121.81

1.341.34

TotalTotal 1717 7.92947.9294

Testing hypothesis to confirm interaction exists or not

34.105.0

0672.0)(

MSE

ABMSF

Since ,89.312,205.0 F the interaction is not significant

The null hypothesis is not rejected

No differences among interaction

81.210500.0

0906.1)(

MSE

BMSF

The laboratorylaboratory and material are important.

the null hypothesis is rejected. Since ,89.312,205.0 F

Main Effect Larger Than InteractionMain Effect Larger Than Interaction

Interaction Larger Than Main EffectInteraction Larger Than Main Effect

Two-way ANOVATwo-way ANOVA

Open the Open the two_way.mtwtwo_way.mtw worksheet worksheet

Stat ANOVA Two-way Analysis of Variance

response

MaterialsMaterials

Enter OK

ANOVA TableANOVA Table

P-value < 0.05

1. The materials and laboratory are significant (important).2. The interaction is not significant.

Main Effects PlotMain Effects Plot

Stat ANOVA Main Effects Plot

321

3.50

3.25

3.00

2.75

2.50

21

Materials

Mea

n

Laboratory

Main Effects Plot for responseData Means

Interactions PlotsInteractions Plots

Stat ANOVA Interactions Plot

21

4.0

3.5

3.0

2.5

2.0

Laboratory

Mea

n

123

Materials

Interaction Plot for responseData Means

Full Factorial Experiment Example: Full Factorial Experiment Example: Improving Wave Solder Process at TeledyneImproving Wave Solder Process at Teledyne• Objective :

– To determine the effect of flux type and lead length on the DFDAU wave soldering (WS) defects

• Planted steps for statistically designed experiment (1) Select output variables, 2 factors, 2 levels and 8 Runs (2) Randomize the sequence of runs and labels 8 DFDAU boards (3) Select two touchup operators to check the WS defects

consistency (4) Iso-plot the major WS defects for top/rear sides to compare one

operator against another (5) Analyze the data using ANOVA table with interactions or using

Yates' algorithm (6) Plot/interpret the results and draw the conclusions

Wave Soldering Process Flow ChartWave Soldering Process Flow Chart

Statistically Designed ExperimentStatistically Designed Experiment

Run No. Flux Type Lead Length LabelY1 New(OA) Trimmed Leads abY2 Old(RMA) Std. Lead Length (1)Y3 Old(RMA) Std. Lead Length (1)Y4 New(OA) Std. Lead Length bY5 Old(RMA) Trimmed Leads a Y6 New(OA) Trimmed Leads abY7 New(OA) Std. Lead Length b Y8 Old(RMA) Trimmed Leads a

New Flux = Alpha # 857Old Flux = RMAStd. Lead Length (IC connect. point not trimmed; IC 接點處之引線未被切平 )Trimmed Leads ( about .045”)

Yates’ AlgorithmYates’ Algorithm

Notations

422

(1)(1) aa

bb abab

Std. Leads Trimmed Leads

A- A+

B-

B+

OldFlux

NewFlux

ANOVA Table

ContrastsCell A B AB(1) - - +

a + - - b - + -

ab + + +

22 = 4 Combinations

Operator #1

1212

10.510.599

99

7.57.566

77

8899

88

7766

Operator #2

1111

9.59.588

99

7755

77

7777

77

4.54.522

Avg. Operator

10.510.5

1010

9.59.5

7.57.5

7.257.25

77

88

7.5 7.5

77

77

5.755.75

4.54.5

Std. Leads Trm. Leads

Old Flux

New Flux

Std. Leads Trimmed Leads

Old Flux

New Flux

Old Flux

New Flux

ANOVA Analysis For Two Factorial ANOVA Analysis For Two Factorial ExperimentExperiment

Two way ANOVA for flux type and lead length

Interaction Plot for Flux Type and Lead Interaction Plot for Flux Type and Lead LengthLength

TrimmedStd

10

9

8

7

6

Lead Length_aver

Mea

n

newold

Type_averFlux

Interaction Plot for response_averData Means

2 2 Factor Full Factorial Experiment Factor Full Factorial Experiment Summary and ConclusionsSummary and Conclusions

Summary of Findings:• ISPLOT reveals that 2 operators

were fairly consistent in calling out VIA defects

• VIA defects consist of 77 % vs. 93 % of total defects, 1 vs. 2

• Only the rear VIA defects are considered for output measures

• Defect level : 2941 ppm (Trimmed leads)

• Defect level : 3959 ppm (Std. lead length)

Conclusions:• The ANOVA/Yates’s Analysis

indicates lead length to be the most significant factor

• Interaction between flux and lead length proven to be the least significant factor

• 26 % improvement can be expected if using the “ Trimmed Leads“

• 23 % improvement can be expected if using the “ OA “ Flux

• Optimal range for the board temperature needs to be further studied

Latin Square (Latin Square ( 拉丁方格拉丁方格 ))

AA BB CC

BB CC AA

CC AA BB

Model rkjiY ijkkjiijk ...,1,,

Operators I, II, IIIProcesses 1, 2, 3Material Source A, B, C

Operators

ProcessesI II III

1

2

3

Greco-Latin SquareGreco-Latin Square

I II III

1

2

3

A

C

B

Operators I, II, IIIProcesses 1, 2, 3Material # 1 Source A, B, CMaterial # 2 Source ,,

B

C

A

C

B

A

Latin Square DesignLatin Square Design

By using a Latin square design, three sources of variation, A, B and C, can be investigated simultaneously, providing there is no interaction between the three factors and also that each of them has the same number of levels r.

For example, suppose each factor has four levels denoted by

.,,,,,,,,,, 432143214321 CCCCandBBBBAAAA If factor A is associated

with the rows of the table and B with the columns of the table then each levels of factor C must appear once in each row and once in each column. In order to achieve this a systematic cyclic pattern, it canbe set down for the C’s as shown in the table. To randomize the design,the allocation of the A’s and B’s to the rows and columns is then carried out at random.

4B 2B 1B 3B

2A

4A

3A

1A

1C

4C

3C

2C

2C

2C

2C4C

4C

4C

1C

1C

1C3C

3C

3C

jBiA

Latin Square ModelsLatin Square Models

Latin Square Model: rkjiY ijkkjiijk ...,1,, The α’s, β’s, γ’s and ε’s are mutually independent.

Analysis of variance for a Latin Square designAnalysis of variance for a Latin Square design

The total sum of squares is divided into four component parts, onefor each source of variation and one for the residual.

SSCSSBSSASSTSSE

FCYr

SSC

FCYr

SSB

FCYr

SSA

FCySST

N

YFCrNyY

k

J

i

ijk

ijk

....1

....1

....1

..

.....,,...

2

2

2

2

22

Here Yi… is the sum of over the r observations in which factor A is at level i, with similar interpretation for Y.j. and Y..k and Y… is the sum of all the r2 observations.

The analysis and test statistics are summarized in the following ANOVA table.

ANOVA Table for Latin Square

Source d.f S.S M.S F

Factor A r-1 SSA

Factor B r-1 SSB

Factor C r-1 SSC

Residual r2 - 3r + 2 SSE

Total r2 - 1 SST

22̂

23̂2

4̂2

1̂

21

22 ˆˆ

21

23 ˆˆ

21

24 ˆˆ

ExampleExample

The following 4 X 4 Latin Square in which the effects of three factors, farm, type of fertilizer applied, and method of application (C1, C2, C3 or C4) on the yield crop are being investigated.

FertilizerFertilizer

FarmFarm

1A

2A

3A

4A

1B 2B 3B 4B

433C

238C 133C

236C133C

332C 432C 237C435C337C

129C

332C432C

333C 133C 235C

To ease the calculations, the data can be coded by subtracting 33 from each observation. Then the row and column totals and the totals for each method of application are calculated. ( 扣除 33, 不致影響 ANOVA 分析 )

FertilizerFertilizer

1 2 3 4 1 2 3 4 Total Total

1 1

Farm 2Farm 2

33

44

22

88

44

-2-2

TotalTotal 4 2 10 -4 124 2 10 -4 12

MethodMethod

TotalTotal -4 14 2 10 12 -4 14 2 10 12

40C 30C 10C 22C

25C 10C 34C 41C

10C 23C 42C 31C

31C 41C24C 14C

1C 2C 3C 4C

2

45954..4]0214)4[(

25934..4])4(1024[

13922..4])2(482[

85994..)4...(2000

0.91612..

1612...

2222

2222

2222

22222

2

2

SSCSSBSSASSTSSE

FCSSC

FCSSB

FCSSA

FCSST

FC

rNY

The calculations necessary for testing the significant of the threefactors are summarized in the following ANOVA table.

Source d.f S.S M.S FFarm 3 13 4.33 13.0Fertilizer 3 25 8.33 25.0Method 3 45 15.00 45.0Residual 6 2 0.333Total 15 85

Since the critical value are F0.99(3, 6) = 9.78 and F0.999(3, 6) = 23.70, the farm effect is significant at 1 % level. The type of fertilizer used and the method of application are both significant at the 0.1 % level.

Open the Open the Latin Square.mtwLatin Square.mtw worksheet worksheet

Latin Square ModelsLatin Square Models

Stat ANOVA General Linear Model

The farm effect, the type of fertilizer used and the method of application are significant at α= 0.05

P-value < 0.05

statistically quality design topics –four perspectives in quality improvement –review doe topics...

Documents

design development

manufacturing quality

customers quality

effect of noise factor

levelsfractional factorial

noise factorslevela

wave solder process

process parameterssignal