ProbStat - Jan 08

8/10/2019 ProbStat - Jan 08

1/11

$,'

NIV+RSITI

TEKNOLOGI

PETRONAS

COURSE

DATE

TIME

SMB3O23EAB2123 PROBABILITY

STATISTICS

16 MAY2008

FRIDAY)

', /

3.00PM 6.00PM

3

hours)

INSTRUCTIONS

O CANDIDATES

1.

2.

3.

4.

5.

Answer

IVE

5)

outof SIX

6) questions

rom

he

Question

ooklet.

BeginEAGH

nswer

n a new

page

n

he

Answer

Booklet.

lndicate

learly nswers

hat

arecancelled,

f

any.

Where

pplicable,how learly tepsaken n

arriving

t hesolutionsnd

indicate

LL assumptions.

Donot

open his

Question

ooklet ntil nstructed.

l t .

i i i .

There are SIXTEEN

(16)

pages

in this

Question

Booklet

Including

he

cover

page

and appendices.

Graph

paper

will

be

provided.

FormulaeBookletwill

be

provided.

Uni-versi t i

Teknoloqi PETRONAS

Note

'.

:1

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2/11

1.

SMB

3023

EAB

2123

a.

A

certain

machine

makes

electrical

esistors

having

a

mean

resistancef 48

ohms

and

a

standard

eviation

f 2.5

ohms.

Find

he

probability

hat

a

random

ample

f 64

of

these

esistors

ill

have

a

i.

mean

esistance

t

least

45

ohms,

[4

marksJ

ii.

mean

esistance

etween

l.S

and

4g.3

ohms,

and

iii.

total

esistance

ot

more

han

3040

ohms.

[4 marks]

[4

marksJ

b-

A

recent

urvey

f

1000

ndividuals

ndicated

hat

39.6%

f the

people

surveyed

will

vote

or

he

current

overnment

n

the

upcoming

fection

in DreamCountry.

l .

Gonstruct

99o/o

onfidence

nterval

or

the

true

proportion

f

people

hat

will

vote

or

the

current

government

n

the

upcoming

election.

[5

marks]

Use

the

confidence

nterval

n

part

(bxi)

to

test

the

hypothesis

that

he

proportion

f

people

hatvote or thecurrent overnment

is

not

4oo/o.

C

1

nef

1rv1'1'

'

62

[3

marks]

2


3/11

SMB

3023

EAB

2123

2.

a.

Traffic

along

a

certain

city

is

being

studied.

The

daily traffic

evel

s

rated

as either

f low,

medium,

r high,

epending

n

how

many

cars

/+awI

traversqthe

street

n

a day.

The

probability

hat

an

accident

occurred

during

he

day

s

0,05.0.20

and 0.90

or

low,

medlum

or high

raffic

leVels,

espectively.

he

probability

hat

he

traffic

evet

will

be low

on

any

given

day is

twice

he

probability

hat

it

will

be medium.

The

probability

hat

the

traffic

evel

will

be

medium

s

twice

he

probability

that

t

will

be high.

i .

what is the probabilityhat he traffic evers row

on

a randomly

selected

ay?

[4

marks]

what

is

the

probability

hat

accidents

iil

occur

on

a randomty

selected

day?

[4

marks]

) : '

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3


4/11

2.

SMB3023

EAB2123

b.

A tollbridge

erves

hreesuburban

esidential

istricts , B

and

C as

shown n Figure

Q2b.

During

jglgrs

of eachday,

he

Urage

volumes f vehicularraffic rom he threedistricts re 2,3, and4

vehicles

er

minute,

espectively.

ssume,that-the

eak

hour

raffic

from each district

and

the

total

peak

hour raffic

at the

toll

bridge s

a

Poisson

rocess.

\___-/

/l"ll-ttd

FIGURE

Q2b

Find

the

expected

number of vehicles

at

the toll

bridge in

1

minute.

[3

marks]

Determine

he

probability

hat more

han2 vehicleswill cross he

[4

marks]

bridge

n

1

minute

\

Find the

probability

n"t the

time

interval

between

two

consecutivears

hat cross he bridge

s

less han 20

seconds.

[5

marks]

il .

f,

F

i:

.;

I,i


5/11

SMB3023

EAB2123

3. a.

The

article

"Hydrogeochemical

haracteristicsf Groundwater

n

a

. Mid-WesternCoastalAquiferSystem"presentsmeasurements f

various

properties

f shallow

roundwater

n a certainaquifersystem

in

[email protected]

re he measurementsf electrical

onductivity

in

microsiemens

er

centimetre)

or 23 samples.

Construct stem-and-leaf

lot

and

describe

ts eatures.

[6

marks]

By comparinghe mean

and median,which

measure

f central

location s

preferred?

xplain.

[4 marks]

iii. ls a histogram ppropriate

o describe

he above

data?Explain.

[2

marks]

5


6/11

b.

SMB3023

EAB 2123

A

project

is

described

by the simple

activity network shown

in

FIGUREQ3b below,consistingof directedbranches epresenting

activities, nd nodes

representinghe

beginning

r

termination

f

activities.

Activities

and

B

are

independent ctivities

nd are scheduled

o

startsimultaneously

t day 0. Node

"a"

epresents

ompletion

f

both

activities

and B, and node

b"

represents

ompletion

f the

project.

Activity

C

can start only

when both

A and B are completed.

All

A

jurations

(in

days)

equiredor

the

respective

ctivities re normal

'( ,/ '

u , /

random

ariables s definedn eachof the

branches

n FIGURE

Q3b.

---.-

At

node

a",

activity

C

is scheduled

o

start

at day 60; howeverf there

is a de lay, t cannot

tartuntil15 days

beyond ts

scheduled

tarting

date.

, . : :v toe2

i

o^-*ol,*{

Completion

day 150

o"-w6,zs ' )

B

Day

60

DB-N(45, f i \

- l " to

(

+

tsd"ss)

fvu+

t] ,

FIGURE

3b

Find he

probability

hat

r

i. activityC will starton schedule,/i:p.,0 days after he starting

dateof

the

project.

[3

marks]

r

i i. the

project

will be comple ted n time

".Jt50

days

after he

\-/

6

starting ate.

[5

marks]


7/11

4.

SMB3023

EAB

2123

The

average

wall hickness

f two-litre

olycarbonate

ottles s

4.0

mils.

A

quality

ontrol

engineer

akes

7 samples

of two-litre

olycarbonate

otfles

from

two

types

of bottles:

ype

A

and

type

B;

and

measures

heir

wall

thickness

in

rni s).

he

results

re

:

a, oLt:S

It

is desired

o test

the

average

hickness

f type

A

botiles

based

on

these

hypotheses:

Ho:

1t :4.9

against

Hr:

p

*

4.0

shoufd

a

/

-test

be used

o test

Ho?

lf

so,

perform

he

test

at

g5%

level

of

confidence.

f not,

explain.

[7 marks]

Construct

box

plot

or

wall hickness

f the

type

B botfles.

[5

marks]

At

957oevel

of

confidence,

s it reasonabte

o

conclude

hat

he

mean

wallthickness

f

both ypes

of

polycarbonate

ottfes

re

he

same?

Explain

our

esults.

[8

marks]

a.

b.

c.

7


8/11

SMB3023

EAB 2123

The following

ata

were collected

n B lung

cancer

patients

where

r

measureshe number fyears hepatient moke igaretteorany ormof

nicotine

roduct)

nd

y

is he

physician's

ubjective

valuationf

the

extent

of

fung

damage

n a

scale f

0

to 100.

ry

-tf l

3X-

t>

An analysis

of variance

s conducted

sing

a statisticsoftware

package

and

he

output

s displayed

n

TABLE

Q5.

TABLE Q5

The reoressionouation

s

v=21.228+1.230x

Predictor Coef

SE Coef

T

p-

value

Constant 21.228 9.442 2.248 0.066

x 1.230

0.280

4.397

0.005

S

=

8.17169

R

-

souared

R-squared(adi)=0.724

Analvsis f

Variance

Source

Df

SS

MS

F

p-

value

Regression

1

1290.84

1290.84

19.331 0.005

Residual

rror 6

400.66

66.777

Total

7

1691.50

a. Sketch scatter

plot

for

given

pairs

of

data

point

and also

plot

the

estimated

egression

ine.

[4

marks]

x

(years)

25 35

22

l5

48

39 42

3l

y (0-100)

55

60

50

30

t )

70

71

55


9/11

SMB3023

EAB2123

b. Estimate he

predicted

physician

cores,

on the

extent

of lung

damage,or wopatients hohavesmoking abits f 20 and40 years

respectively.

[3

marks]

c.

Obtain 95

%

confidencenterval

or he

rueslope

.

d.

Estimatehe

coefficientf

determinationR2).

Discuss

value btained.

e. Conducta hypothesisest on the

significance

f

levelof significance

=

0.05,

[5

marks]

briefly

n

the

[3 marks]

the

slope

at

[5

marks]


10/11

6.

SMB3023

EAB2123

A two actor

experimentalesign

was conducted

o investigatehe lifetime

of

a component

eingmanufactured,

he two

factors

are

A

(design)

nd

B

(cost

of material).

wo evels

(+)

and

(-))

of each actor

are

considered.

Threecomponenis

re manufactured

ith

eachcombination

Tdesign

nd

material, nd

he total ifetimemeasured

in

hours) s as shown

n

TABLE

Q6.

a.

A two way analysis f

variancewas

performed

o

estimate

he effects

of

design

nd

material xpense

n

the

componentifetime. omplete

the following NOVA

TABLE

QGa

by

answering

he corresponding

quest ions.Useq=0.05.

A

v0

: :

;t

f

rr

1

.l

I

i

ABLE

Q6a

fo,os.t , to

'1 ,1L

,/"

Source

DF

SS

MS

F

P-

value

lD-t

A (Design)

1

341.33 341.33

i : ' )

ty-t-

=z'b

B

(Material)

1

20t,99

21t,"9

7

s,33

{t

lnteraction

1

180 180

6t. l ,2B(

I

Error

l0

28

: .8

0

Total

11

q

o.

6b

/

x

Total

ifetime

components

(in

hours)

118

'

oh

10


11/11

SMB3023

EAB

2123

t .

Determinehe Error

degreeof freedom,

SSs,SSrot"r, Ss

and

MSerror.

,.."

,

V

marksl

ii.

Calculate

he F statistics

or B

(material)

nd nteraction.

[2

marks]

iii

Find

the

P-value

for each sources

A, B and interaction

(you

maywrite

our

answern nterval

orm).

[4

marks]

iv.

Basedon

your

esults

n

part

(a)(iii),

what

conclusionsan

you

draw

rom he actorial xperiment?

lz

-vatvt+

1 d

ro'Ll -

H,

(

*h+

cf tc+

of

a/ /

fqc-a-o. ;g

s3ntpzta+)

[2

marks]

b.

lndicate

hicheffects

re significanto the

ifetime f a component.

A,

8,

Ag

[2

marks]

c. Write he

east

quare

ittedmodelusing

nly he

significant

ources.

[3 marks]

;=

-

END OF PAPER

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