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NATIONAL UNIVERSITY OF SINGAPORE

EXAlVIIN

ATION

ST52 3

EXPERIMENT L DESIGN

(Semester 1 : AY 2011-2012)

Novernber

2011

- Time Allowed: 2 Hours

INSTRUCTIONS

TO C NDID TES

1.

This examination paper contains

FIVE (5)

questions and comprises

FOURTEEN

(14)

printed pages (including the present page).

2.

Candidates must answer ALL questions on this paper. The total mark for this paper

is

60.

3. Non-programmable calculators can be used.

4.

This is an OPEN BOOK examination.

5. Write down your matriculation number and seat number in the space below.

Matriculation No: _

Seat No: _

Question Points scored

Max. points

1

12

2

12

3

12

12

5

12

Total

60

1

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1. (12pts)Anagronomistwants

to

conductafield trialtocomparethe yieldofthreevarieties

oftomatoin

the

following9plots.

1

2

3

4

5

7 8

91

(a) (4pts)Supposethis

is

theonlyinformationwe have,whichdesigndoyousuggest? ow

shouldtheagronomistdotherandomization?

(b) (4pts)Supposewe alsoknow th t theplotsinthesamerow willhavethesameamount

ofwaterirrigation;whilethe plotsin the different rows mayhave differentamountof

waterirrigation. Theyieldoftomatomaybe affected by variabilityinthe amountof

availablemoisture. With this information,whichdesign doyou suggest? owshould

theagronomistdotherandomization?

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(c) (4pts) Inadditionto theinformationin (b),we also know

that

theplotsinthesame

columnwillhavethesameamountofsunshinejwhiletheplotsinthedifferentcolumns

may have different amount of sun shine. The yield of tomato may be affected by

variabilityin the amount of available sun shine. With this extra information,which

designdo yousuggest? Howshould the agronomistdotherandomization?

2.

(12pts) The Kenton Companywished

to

test 4different packagedesigns for a new break

fast cereal. Sixteen stores, with approximately equal sales volumes, were selected as the

experimentalunits. Eachstorewasrandomlyassignedoneof thepackagedesigns,witheach

packagedesign assigned

to

4stores. Sales, innumber ofcases, were observedfor the study

period,

and

theresultsarerecordedinTable

1.

The

data

isanalyzedusingone-waylayout.

Packagedesign

Sales Samplemean yd

E l

y

..

_y 2

I j l 1 1

1 11 17 16

14

14.5

21

2

12

1

15 19

14.0

46

3

23 20 18

17

19.5

21

4

27

33

22

26 27.0

62

3

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(a) (3pts) Fill up the blanks in the NOV table below.

Resource Degrees of freedom Sum of squres Mean of squares

Treatment

Error

(b) (3pts) Conduct an F test to test the null hypothesis

th t

the four package designs have

the same sales. Note:

O 05

,3,12 =

3.49.

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(c) (3pts) Use the Thkey method to perform multiple comparisons of the four package

designs

t

the 0.05 leveL Note

QO.05,4,12=4.20.

(d) (3pts) Suppose the interest

is

to test the null hypothesis

th t

the mean sales

for

design

1

is the same as the average sales of all four designs. Conduct a t test to this null

hypothesis

t a

0.05. Note:

to.0

25

12 2.18.

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3. (12pts)A company conducted a

27

full factorial experiment with factors

A B

C, D

E

F G.

Then, the

data

is processed

by

Yate's algorithm in R and all the computed effects,

including

the estimated grand

mean,

are stored in the txt file "Effects_output.txt" in standard

order. Assume that the computations in Yate's algorithm are all correct. Read the following

R outputs to answer all the sub-questions.

>

#

Load data,

> "Effects_output.txt" contains estimates

of effects

from Yate's algorithm

>

Effects_data =

read.table(IEffects_output.txt", header

= F)

> Effects

=

Effects_data$Vl

>

#

Remove grandmean

>

Effects = Effects[2:length(Effects)]

>

#

Order

the

effects

> Effects a

= abs(Effects)

> Effects_a_order = order(Effects_a)

>

#

Number

of

effects

> I

= length(Effects)

> # Some

possibly

relevant

information

> cbind(Effects_a_order[(I-8):I], Effects_a[Effects_a_order[(I-8):I]])

[,1] [,2]

[1,] 97 2.043181

[2,]

113

2.119931

[3,]

16 2.434177

[4,]

111 2.456506

[5,]

116 2.800000

[6,]

114

3.000000

[7,]

85

12.000000

[8,]

31

15.000000

[9,]

11

20.000000

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>

# Some possibly relevant

computational

results

median Effects_a)

[ ] 0.7903197

median Effects_a[Effects_a

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(c)

3pts)Six students attending ST5203 independently constructed half normal plots based

upon the computed effects. Only one student is doing the correct work, which one is

it? Point

out

at least one mistake for each of the other plots.

0

N

'

,

;::

1 l

::J

J:i

..

'

0

0

N

'

II

-

,

1!l

-

::J

J:i

'

0

Student 1

0

0

0

o

0

I

2

1

0

2

half

normalquantile

Student3

0

0

0

. cs;tPO

00

-

I I I

0.0

O.S

1,0

1.S 2.0 2.S

halfnormalquantile

0

N

Student2

0

1 l

::J

5

III

J:i

IV

'

-

0

-

'

0

..-

0,0

0.5

0

0

~

1,0 1,S 2,0

2.S

halfnormalquantile

0

N

Student4

0

'"

..

"0

III

J:i

IV

';!

;::

'

0

-

I

0.0 0.5

0

0

~

I I

1.0 1.5 2.0 2.5

half

normalquantile

8

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Student

Student 6

0

N

It

5

0

III

5

co

.0

\0

'

a

0

0

0

.

~

0

N

0

0

5

rg

0

CD

a

:::J

5

'

0

'

\0

a

0

~

0.0

0.5

1.0

1.5

2.0

2.5

2

1

0

2

half normal quantile half normal quantile

(d) (3pts)With the correct half-normal plot in (c),

how

many effects should be declared as

significant? Clearly write down the name{s) of your declared significant effects.

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4

(12pts}Assume

th t five

factors

A, B, C, D

and

E),

each

t

two levels, are studied with

runs given by the following treatment combinations (I),

ade

bd abe

ede

ae

bee

and

abed.

(a) (3pts)Write down the design matrix of the experiment.

(b) (3pts}Write down the defining contrast subgroup of the design.

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(c)

{3pts)Are there any clear effects in this design? yes, list all clear effects. no, explain

why

not.

(d) {3pts)Denote the above design as dll consider another 8-run design d

2

by switching the

signs in all columns of d

1

Consider d

3

to the augmented design of d

1

and d

2

i.e., d

3

has 16 rows. Find a set of generators

for

d

3

hat

has been achieved by adding d

2

?

Explain clearly.

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5. (12pts)Replicate2

3

fullfactorialexperimentswithfactorsA

Band

C3times. Eachreplica

tion

is

arrangedin4blocks. Theeffectsfor constructingtheblockingschemes for replicates

aredisplayedasfollows.

Replicate

1: Dl AC

D

ABC;

Replicate2:

I=BC D =ABC;

Replicate3: Dl AB D =

AC.

After collectingthe data from the experiment, Yate's algorithm is applied to each ofthe

replicates.The obtainedeffect estimatesfromeveryreplicatearedisplayedin the following

tablein

st nd rd order

Rep. 1 Rep. 2 Rep. 3

2.76

3.94 0.59

-0.30 3.36 3.34

8.46

4.96 4.93

-4.09 -1.92 -1.04

2.16 -1.65 -2.29

4.71

4.91

6.17

-0.38 0.64 0.48

-0.26 1.41 2.32

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(a) (3pts)Obtain the estimates

for

ll the effects in this experiment.

(b) (3pts)Estimate j2, the variance for each individual observation. hat are the degrees

of freedom of your estimates.

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(c) 3pts)Construct

95

confidence intervals

for

effects A and G Note: to 0

25

5 2.57.

d) 3pts)Construct 95 confidence intervals for effects

AB

and

BG

nd of

Paper