Ken Black QA ch11

8/3/2019 Ken Black QA ch11

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Business Statistics, 5th ed.

by Ken Black

Chapter 11

Analysis ofVariance

& Design ofExperiments

Discrete Distributions

PowerPoint presentations prepared by Lloyd Jaisingh,Morehead State University


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Learning Objectives

Understand the differences between variousexperimental designs and when to use them.

Compute and interpret the results of a one-wayANOVA.

Compute and interpret the results of a randomblock design. Compute and interpret the results of a two-way

ANOVA. Understand and interpret interaction.

Know when and how to use multiple comparisontechniques.


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Introduction to Design

of Experiments

Experimental Design

- a plan and a structure to test hypotheses inwhich the researcher controls or manipulatesone or more variables.


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Introduction to Design of Experiments

Independent Variable Treatment variable is one that the experimenter

controls or modifies in the experiment.

Classification variable is a characteristic of theexperimental subjects that was present prior to theexperiment, and is not a result of theexperimenters manipulations or control.

Levels or Classifications are the subcategories of

the independent variable used by the researcher inthe experimental design.

Independent variables are also referred to asfactors.


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Introduction to Design

of Experiments

Dependent Variable

- the response to the different levels of the

independent variables. Analysis of Variance (ANOVA)a group

of statistical techniques used to analyzeexperimental designs.


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Three Types

of Experimental Designs

Completely Randomized Designsubjects areassigned randomly to treatments; single

independent variable. Randomized Block Designincludes a blocking

variable; single independent variable.

Factorial Experimentstwo or more independent

variables are explored at the same time; everylevel of each factor are studied under every levelof all other factors.


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Completely Randomized Design

Machine Operator

Valve Opening

Measurements

1

.

.

.

2

.

.

.

4

.

.

.

.

.

.

3


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Valve Openings by Operator

1 2 3 4

6.33 6.26 6.44 6.29

6.26 6.36 6.38 6.23

6.31 6.23 6.58 6.19

6.29 6.27 6.54 6.21

6.4 6.19 6.56

6.5 6.34

6.19 6.58

6.22


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Analysis of Variance: Assumptions

Observations are drawn from normallydistributed populations.

Observations represent random samples

from the populations. Variances of the populations are equal.


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One-Way ANOVA: Procedural

OverviewH

H

ok

a

:

:

1 2 3

At least one of the means is different from the others

FMSC

MSE

If F > , reject H .

If F , do not reject H .

co

co

F

F


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One-Way ANOVA:

Sums of Squares Definitions

valueindividual

levelorgrouptreatmentaofmean=

meangrand=X

leveltmentgiven treaainnsobservatioofnumber

levelstreatmentofnumber=

leveltreatmenta=

leveltreatmentaofmemberparticular:

nn

ij

SSE+SSC=SST

squaresofsumbetween+squaresofsumerror=squaresofsumtotal

X

X

n

X

ij

j

j

1 1

2

1

2

1=i 1j=

2 jj

C

j

iwhere

jijjj i

C

j

C

j

C

XXX

XnX


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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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One-Way ANOVA:

Computational Formulas

MSE

MSCF

SSEMSE

SSCMSC

Nn

ijSST

CNn

jijSSE

Cj

SSC

df

df

dfXX

dfXX

dfXXn

E

C

Tj

C

i

Ei

C

j

C

C

jj

j

j

1

1

1 1

2

1 1

2

1

2

where

X

: i = a particular member of a treatment level

j = a treatment level

C = number of treatment levels

= number of observations in a given treatment level

X = grand mean

column mean

= individual value

j

j

ij

n

X


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One-Way ANOVA:

Preliminary Calculations

1 2 3 4

6.33 6.26 6.44 6.29

6.26 6.36 6.38 6.23

6.31 6.23 6.58 6.19

6.29 6.27 6.54 6.21

6.4 6.19 6.56

6.5 6.34

6.19 6.58

6.22

Tj T1 = 31.59 T2 = 50.22 T3 = 45.42 T4 = 24.92 T = 152.15

nj n1 = 5 n2 = 8 n3 = 7 n4 = 4 N = 24

Mean 6.318000 6.277500 6.488571 6.230000 6.339583


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15492.0

)230.619.6()230.622.6()2775.636.6()2775.626.6()318.64.6(

)318.629.6()318.631.6()318.626.6()318.633.6(

23658.0

)339583.623.6()339583.6488571.6(

)339583.62775.6()339583.6318.6(

22

222

2222

1 1

2

22

22

1

2

47

85[

n

jijSSE

jSSC

j

i

C

j

C

jj

XX

XXn

One-Way ANOVA:

Sum of Squares Calculations


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39150.0

)339583.619.6(

)339583.622.6()339583.631.6(

)339583.626.6()339583.633.6(

2

22

22

1 1

2

n

ijSST

j

i

C

j

XX

One-Way ANOVA:

Sum of Squares Calculations


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One-Way

ANOVA: Mean

Square

and F Calculations

18.10007746.078860.

007746.20

15492.

078860.3

23658.

231241

20424

3141

MSEMSCF

SSEMSE

SSC

MSC

N

CN

C

df

df

df

df

df

E

C

T

E

C


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Analysis of Variance

for Valve Openings

Source of Variance df SS MS F

Between 3 0.23658 0.078860 10.18

Error 20 0.15492 0.007746

Total 23 0.39150


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F 20,3,05.df1

df2

A Portion of the F Table for = 0.05

1 2 3 4 5 6 7 8 9

1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54

18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.4619 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42

20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39

21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37

df2


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One-Way ANOVA:

Procedural Summary

.Hrejectdo,10.3F

.Hreject,10.3>F

oc

oc

F

F

If

If

Rejection Region

Critical Value10.3

11,9,05.F

Non rejection

Region

20

3

2

1

othersthefromdifferentis

meanstheofoneleastAt:H

:H

a

4321o

.Hreject,10.3>10.18=FSince ocF


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Excel Output

for the Valve Opening Example

Anova: Single Factor

SUMMARY

Groups Count Sum Average Variance

Operator 1 5 31.59 6.318 0.00277

Operator 2 8 50.22 6.2775 0.0110786

Operator 3 7 45.42 6.488571429 0.0101143

Operator 4 4 24.92 6.23 0.0018667

ANOVA

Source of Variation SS df MS F P-value F crit

Between Groups 0.236580119 3 0.07886004 10.181025 0.00028 3.09839

Within Groups 0.154915714 20 0.007745786

Total 0.391495833 23


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MINITAB Output

for the Valve Opening Example


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Multiple Comparison Tests

An analysis of variance (ANOVA) test is anoverall test of differences among groups.

Multiple Comparison techniques are used toidentify which pairs of means are

significantly differentgiven that theANOVA test reveals overall significance. Tukeys honestly significant difference

(HSD) test requires equal sample sizes

Tukey-Kramer Procedure is used whensample sizes are unequal.


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Tukeys Honestly Significant

Difference (HSD) Test

HSDMSE

n

,C,N-C

,C,N-C

q

q

where: MSE = mean square error

n = sample size

= critical value of the studentized range distribution from Table A.10


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q Values for = .01

Degrees ofFreedom

1

2

3

4

.

11

12

2 3 4 5

90 135 164 186

14 19 22.3 24.7

8.26 10.6 12.2 13.3

6.51 8.12 9.17 9.96

4.39 5.14 5.62 5.97

4.32 5.04 5.50 5.84

.

...

Number of Populations

. , ,.

01 3 12504q


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Tukeys HSD Test

for the Employee Age Data

HSDMSE

nC N Cq

X

X

X

, ,.

..

. . .

. . .

. . .

504163

52 88

28 2 32 0 38

28 2 24 8 34

32 0 24 8 7 2

2

3

3

1

1

2

X

X

X


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Tukeys HSD Test for the Employee

Age Data using MINITAB

Intervals

do not

contain 0,

so significant

differences

between the

means.


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Tukey-Kramer Procedure:

The Case of Unequal Sample Sizes

HSDMSE

r sn n

,C,N-C

r

th

s

th

,C,N-C

q

n r

n s

q

where: MSE = mean square error

= sample size for sample

= sample size for sample

= critical value of the studentized range distribution from Table A.10

2

1 1( )


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Freighter Example: Means and

Sample Sizes for the Four Operators

Operator Sample Size Mean

1 5 6.31802 8 6.2775

3 7 6.4886

4 4 6.2300


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Tukey-Kramer Results

for the Four Operators

Pair

Critical

Difference

|Actual

Differences|

1 and 2 .1405 .0405

1 and 3 .1443 .1706*

1 and 4 .1653 .0880

2 and 3 .1275 .2111*

2 and 4 .1509 .0475

3 and 4 .1545 .2586*

*denotes significant at .05


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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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A Randomized Block Design

Individual

observations

.

.

.

.

.

.

.

.

.

.

.

.

Single Independent Variable

Blocking

Variable

.

.

.

.

.


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Randomized Block Design Treatment

Effects: Procedural Overview

othersthefromdifferentismeanstheofoneleastAt:H

:H

a

321o

k

FMSC

MSE

If F > , reject H .If F , do not reject H .

c o

co

FF


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Randomized Block Design:

Computational Formulas

SSC n j C

SSR C i

n

SSE ij i iC n N n C

SST ij N

MSCSSC

C

MSRSSR

n

MSESSE

N n C

MSC

MSE

MSR

MSE

X X df

X X df

X X X X df

X X df

F

F

j

C

C

i

n

R

i

n

j

n

E

i

n

j

n

E

treatments

blocks

2

1

2

1

2

11

2

11

1

1

1 1 1

1

1

1

1

( )

( )

( )

( )where: i = block group (row)

j = a treatment level (column)

C = number of treatment levels (columns)

n = number of observations in each treatment level (number of blocks - rows)

individual observation

treatment (column) mean

block (row) mean

X = grand mean

N = total number of observations

ij

j

i

X

X

X

SSC sum of squares columns (treatment)

SSR = sum of squares rows (blocking)

SSE = sum of squares error

SST = sum of squares total


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Randomized Block Design:Tread-Wear Example

Supplier

1

2

3

4

Slow Medium FastBlockMeans( )

3.7 4.5 3.1 3.77

3.4 3.9 2.8 3.37

3.5 4.1 3.0 3.53

3.2 3.5 2.6 3.10

5Treatment

Means( )

3.9 4.8 3.4 4.03

3.54 4.16 2.98 3.56

Speed

jX

iX

X

C = 3

n = 5

N = 15


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SSC n j

SSR C i

X X

X X

j

C

i

n

2

1

2 2 2

2

1

2 2 2 2 2

5

3

54 356 16 356 98 3563484

77 356 37 356 53 356 10 356 03 356

1549

( )

(3. . ) (4. . ) (2. . ).

( )

(3. . ) (3. . ) (3. . ) (3. . ) (4. . )

.

[

[ ]


Sum of Squares Calculations (Part 1)


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Sum of Squares Calculations (Part 2)

176.5

)56.34.3()56.36.2()56.34.3()56.37.3(

)(

143.0

)56.303.498.24.3()56.310.398.26.2(

)56.337.354.34.3()56.377.354.37.3(

)(

2222

1 1

2

22

22

1 1

2

n

i

C

j

n

i

C

j

XX

XXXX

ijSST

ijijSSE


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Mean Square Calculations

MSCSSC

C

MSRSSR

n

MSESSE

N n C

FMSC

MSE

1

3484

2

1742

1

1549

40 387

1

0143

80 018

1742

0 01896 78

..

..

..

.

..


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for the Tread-Wear Example

Source of VarianceSS df MS F

Treatment 3.484 2 1.742 96.78Block 1.549 4 0.387 21.50

Error 0.143 8 0.018

Total 5.176 14


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Randomized Block Design Treatment

Effects: Procedural SummaryH

H

o

a

:

:

1 2 3

At least one of the means is different from the others

78.96018.0

742.1

MSE

MSCF

F = 96.78 > = 8.65, reject H ..01,2,8

oF


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Randomized Block Design Blocking

Effects: Procedural Overview

H

H

o

a

:

:

1 2 3 4 5

At least one of the blocking means is different from the others

5.21018.

387.

MSE

MSRF

F = 21.5 > = 7.01, reject H .F o. , ,01 4 8


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Excel Output for Tread-Wear

Example: Randomized Block DesignAnova: Two-Factor Without Replication

SUMMARY Count Sum Average Variance

Supplier 1 3 11.3 3.7666667 0.4933333

Supplier 2 3 10.1 3.3666667 0.3033333

Supplier 3 3 10.6 3.5333333 0.3033333

Supplier 4 3 9.3 3.1 0.21

Supplier 5 3 12.1 4.0333333 0.5033333

Slow 5 17.7 3.54 0.073

Medium 5 20.8 4.16 0.258

Fast 5 14.9 2.98 0.092

ANOVASource of Variation SS df MS F P-value F critRows 1.5493333 4 0.3873333 21.719626 0.0002357 7.0060651

Columns 3.484 2 1.742 97.682243 2.395E-06 8.6490672

Error 0.1426667 8 0.0178333

Total 5.176 14


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MINITAB Output for Tread-Wear

Example: Randomized Block Design

Blocking variable

Suppliers


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Two-Way Factorial Design

Cells

.

.

.

.

.

.

.

.

.

.

.

.

Column Treatment

Row

Treatment

.

.

.

.

.


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Formulas for Computing

a Two-Way ANOVA

SSR nC i

R

SSC nR j C

SSI n ij i j R C

SSE ijk ij RC n

SST ijk N

MSRSSR

R

MSR

MSE

MSC

X X df

X X df

X X X X df

X X df

X X df

F

i

R

R

j

C

C

j

C

i

R

I

k

n

j

C

i

R

E

a

n

r

R

c

C

T

R

2

1

2

1

2

11

2

111

2

111

1

1

1 1

1

1

1

( )

( )

( )

( )

( )

SSC

C

MSC

MSE

MSISSI

R C

MSI

MSE

MSESSE

RC n

where

C

I

FF

1

1 1

1

:

n = number of observations per cell

C = number of column treatments

R = number of row treatments

i = row treatment level

j = column treatment level

k = cell member

= individual observation

= cell mean

= row mean

= column mean

X = grand mean

ijk

ij

i

j

X

X

XX


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A 2 3 Factorial Designwith Interaction

Cell

Means

C1 C2 C3

Row effects

R1

R2

Column


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A 2 3 Factorial Designwith Some Interaction

Cell

Means

C1 C2 C3

Row effects

R1

R2

Column

A 2 3 i i


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A 2 3 Factorial Designwith No Interaction

Cell

Means

C1 C2 C3

Row effects

R1

R2

Column

A 2 3 Factorial Design: Data and


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A 2 3 Factorial Design: Data andMeasurements for CEO Dividend Example

N = 24

n = 4

X=2.7083

1.75 2.75 3.625

Location Where CompanyStock is Traded

How Stockholdersare Informed of

DividendsNYSE AMEX OTC

Annual/QuarterlyReports

2

121

2

332

4

343

2.5

Presentations to

Analysts

23

12

33

24

44

34 2.9167

Xj

Xi

X11=1.5

X23=3.75X22=3.0X21=2.0

X13=3.5X12=2.5


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A 2 3 Factorial Design: Calculationsfor the CEO Dividend Example (Part 1)

SSR X X

SSCX

X

SSI X X X X

nCi

nR

j

n ij i j

i

R

j

C

j

C

i

R

2

1

2 2

2

1

2 2 2

2

11

2

4 3 2 5 2 7083 2 9167 2 7083

4 2 175 2 7083 2 75 2 7083 3625 2 7083

4 15 2 5 175 2 7083

10418

140833

( )

.

( )

.

( )

( )( )[( . . ) ( . . ) ]

( )( )[( . . ) ( . . ) ( . . ) ]

[( . . . . ) ( . . . . )

( . . . . ) ( . . . . )

( . . . . ) ( . . . . ) ]

.

2 5 2 5 2 75 2 7083

35 2 5 3625 2 7083 2 0 2 9167 175 2 7083

30 2 9167 2 75 2 7083 375 2 9167 3625 2 7083

2

2 2

2 2

00833


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SSE X X

SST X X

ijk ij

ijk

k

n

j

C

i

R

a

n

r

R

c

C

2

1112 2 2 2

2

111

2 2 2 2

2 15 1 15 3 375 4 375

77500

2 27083 1 27083 3 27083 4 27083

229583

( )

( . ) ( . ) ( . ) ( . )

.

( )

( . ) ( . ) ( . ) ( . )

.


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MSRSSR

R

MSR

MSE

MSCSSC

C

MSC

MSE

MSISSI

R C

MSI

MSE

MSESSE

RC n

R

C

I

F

F

F

1

10418

110418

10418

043062 42

1

14 0833

2

7 04177 0417

04306

1635

1 1

00833

200417

00417

04306010

1

7 7500

1804306

..

.

..

..

.

.

.

..

.

..

..

A l i f V i


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for the CEO Dividend Problem

Source of VarianceSS df MS F

Row 1.0418 1 1.0418 2.42

Column 14.0833 2 7.0417 16.35*

Interaction 0.0833 2 0.0417 0.10

Error 7.7500 18 0.4306

Total 22.9583 23

*Denotes significance at = .01.


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Excel

Outputfor the

CEO

DividendExample

(Part 1)

Anova: Two-Factor With Replication

SUMMARY NYSE ASE OTC Total

AQReportCount 4 4 4 12

Sum 6 10 14 30

Average 1.5 2.5 3.5 2.5

Variance 0.3333 0.3333 0.3333 1

Presentation

Count 4 4 4 12Sum 8 12 15 35

Average 2 3 3.75 2.9167

Variance 0.6667 0.6667 0.25 0.9924

Total

Count 8 8 8

Sum 14 22 29Average 1.75 2.75 3.625

Variance 0.5 0.5 0.2679

E l O t t f th


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Excel Output for the

CEO Dividend Example (Part 2)

ANOVA

Source of Variation SS df MS F P-value F critSample 1.0417 1 1.0417 2.4194 0.1373 4.4139

Columns 14.083 2 7.0417 16.355 9E-05 3.5546Interaction 0.0833 2 0.0417 0.0968 0.9082 3.5546

Within 7.75 18 0.4306

Total 22.958 23


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MINITAB Output for the

Demonstration Problem 11.4:



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Demonstration Problem 11.4:

Interaction Plots

321

4

3

2

1

4321

4

3

2

1

Warehouses

Length

1

2

34

Warehouses

1

2

3

Length

Interaction Plot (data means) for DaysAbsent


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Ken Black QA ch11

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Transcript of Ken Black QA ch11