Quantitative Methods for Decision Making-1

8/10/2019 Quantitative Methods for Decision Making-1

1/61

Quantitative Methods

for Decision MakingA Practical and Philosophical approach

By,

Yaseen Ahmed MeenaiFaculty, FCS-IBA

[email protected]

In the Name of ALLAH, the ene!cent, the Merciful,

O Allah, send your salutations upon uhammad !"B#$% &on the Family o' uhammad !"B#$% as you sent yoursalutations upon Ibrahim & on the Family o' Ibrahim (erily

you are ost "raise)orthy & *lorious+
mailto:[email protected]:[email protected]


2/61

hat is Statistics !A science or an

art%

An acti(ity o' obtainin/ data and then0

Compilin/, summari1in/, presentin/,analy1in/, interpretin/ and+.

2ra)in/ conclusions, is called "tatistics.In short it is0

Data Process Information#$onclusions

Statistics is sort o' a mi3ture o' science andart, till process it is a SCI45C4 and dra)in/conclusions is an indi(idual6s A78.


3/61

hat is 2A8A !A )ord or a

9ey)ord%

2A8A is a /roup o' ra) 'act and :/ures)hich may ;A7< 'rom0

"erson to "erson, Ob=ect to Ob=ect,2istance to 2istance and 8ime to

8ime+.

Only the absence o' ;A7IA8IO5 can

cause a CO5S8A58 and it doesn6te3ists in our physical )orld. Onlyspiritualism can de:ne a CO5S8A58.


4/61

Data v#s %ariale

;ariable is the stora/e o' data, its bein/ represented by letters X,Y,Zetc.&here are t'o t(pes of variales)

Qualitative %ariale) It deals 'ith the data 'hich ma( var( ( itkind, 'hich provides laels, or names, for categories of like

items, i*e* a set of oservations 'here an( single oservation isa )ord or code that represents a class or categor(*

Gender, Complexion, Weather, Type are some examples

Quantitative %ariale) It deals 'ith the numeric data, 'hichmeasures either ho' much or ho' man( of something, i*e* a set

of oservations 'here an( single oservation is a number thatrepresents an amount or a count*

Age, Height, number, price are some examples of uantitati!e !ariable"

#ource$ http$%%&&&"microbiologybytes"com%maths%'('')'*"html
http://www.microbiologybytes.com/maths/1011-17.htmlhttp://www.microbiologybytes.com/maths/1011-17.html


5/61

Inactivit( reaker +

-ect) Allocate a blank pa/e 'rom your )ritin/ material and di(ide thatpa/e into t)o columns in the 'ollo)in/ manner>

8ry to )rite atleast ? (ariables in each column by obser(in/ se(eral :elds

like mana/ement, a/riculture, medical, en/ineerin/, /eolo/y etc. Submit thesame sheet by )ritin/ your 'ull name on the top.

Qualitative %ariales Quantitative%ariales

- *ender - A/e

?- Comple3ion ?- $ei/ht

- uali:cation - ei/ht

D- eather D- "rice

?. ?.


6/61

Data "ources

8here are three ma=or sources o' data>

. "urve(#$ensus) An oEcial, usually periodicenumeration o' a population, o'ten includin/ thecollection o' related demo/raphic in'ormation,is called census. "urve( means to inspect anddetermine the conditions o' interest.

.* /0periment) Any acti(ity, )hich is usuallybein/ conducted )ithin an isolated atmosphere,and produces results, is called e0periment.

1*"imulation) An arti:cial )ay o' data collection.


7/61

Question of the Da(+*

hat do you think aboutuality o' the 'ollo)in/ in IBA

- 8eachin/ ,?,,D,?- Administration,?,,D,

- Structure,?,,D,

here -;ery "oor -43cellent


8/61

Data$ollection#compilation

8eachin/ 7anks )here ')+ery oor, -).xcellent

D. .G D. . ?.GD.G .H D. .D D.

.H ?.G D. .D .?

.G . .H .H .G.J . D.? D. D.?

D. . D. ..G D.H .? D.? D.

D.? . ?.

2ata collectionKcompilation is needed 'or /ettin/actual beha(ior o' the (ariable.

Note$ The abo!e data is simulated !ersion of the actual"


9/61

Data &aulation !*roupin/43ercise%

"tep 2 34) 5inding the range

7an/e L a3. M in L .-?.G L?.

"tep 2 3.) 5inding the numer of classes

5o. o' classes L N . lo/!n% L N. lo/!G% L J.G

"tep 2 31) 5inding the 'idth or height 6h7

h L 7an/eK5o. o' classesL ?.KJ.G L .GG .D

$lass Interval) One o' the inter(als into )hich the ran/e o' a(ariable o' a distribution is di(ided, esp. one o' thedi(isions o' the base line o' a bar chart or histo/ram.

A'ter 'ormin/ the structure o' Class-Inter(als and 'reuencies byusin/ methods o' tally-marks, )e can obser(e the actual beha(ior.


10/61

Data Process Information

RanksFreque

ncy

?.G

.

.D .H

D.?

D.J

8he abo(e mentioned 'reuency distribution table and the$isto/ram are re(ealin/ the shape o' thou/hts /enerated 'romthe minds o' students. I' )e disco(er a subseuent athematicalodel, it )ill called a "robability distribution.

?DJH

?

Histogram

8anks

5re9uenc(


11/61

Data Process Information

8he abo(e mentioned 'reuency distribution table and the$isto/ram are re(ealin/ the shape o' thou/hts /enerated 'romthe minds o' students. I' )e disco(er a subseuent athematicalodel, it )ill called a "robability distribution.

@

?

D

J

H

A@

A?

Histogram

8anks

5re9uenc(

RanksFreque

ncy

?.G G

.

.D .H J

D.?

D.J


12/61

:rouping the data6M"/;$/L7

2ata Analysis option is located in the Data menu, incase i' it is not present there )e can acti(ate it byrunnin/ theAdd)/ns present in /0cel ptions


13/61

:rouping the data 6M"/;$/L7cont+

=in numers 8hese numbers represent the inter(als thatyou )ant the $isto/ram tool to use 'or measurin/ the inputdata in the data analysis.

A'ter pro(idin/data>rangeoutput

options, )e can:nd thehisto/rameither in the

ne) )orksheetor in thespeci:c place o'the e3istin/sheet.


14/61

"tatistical Measures 6Anintroduction7

&he phrase ?descriptive statistics< is used genericall(in place of statistical measures*

&hese statistic6s7 descrie or summari@e the 9ualitiesof data*

Another name is ?summar( statistics


15/61

"tatistical Measures 6An/0ample7

Class

Inter(als

Freuenc

y

7elati(e

Freuency

!7.F.%

Cumulati(e

7elati(e Freuency

!C.7.F%

.BC ? ?K? L .H .HCB K? L .? .?H

BE K? L .J .JD

EB43 G GK? L .?H .?

43B4. ? ?K? L .H .

'L? 7.F.L

Consider the 'ollo)in/ /roup data>

&he aove data sho'ing Income in 4333Fs of 8upees ofsome individuals in late 4GE3Fs

Class

Inter(als

Freuenc

y

7elati(e

Freuency

!7.F.%

Cumulati(e

7elati(e Freuency

!C.7.F%

.BC ? ?K? L .H .HCB K? L .? .?H

BE K? L .J .JD

EB43 G GK? L .?H .?

43B4. ? ?K? L .H .

'L? 7.F.L


16/61

"tatistical Measures6Quartiles7

8hese are (alues respecti(elyrepresented by , ? and and di(ides

the data into D eual parts.

4ach part contains ?Q obser(ations uartiles #sually hi/hli/ht D diRerent

classes i.e. o)er class, o)er iddle,#pper iddle and #pper class..

Lo'er$lass

.Lo'erMiddle

.JpperMiddle

.Jpper$lass

Q

4

Q

.

Q

1

Min

Ma0


17/61

Computin/ uartiles

Class

Inter(als

Freuenc

y

Cumulati(e

Freuency!C.F.%

.BC ? ?CB G

BE JEB43 G ?43B4. ? ?

'L?

In order to computer uartile ;alues, )eneed to consider the same 'reuencydistribution in addition to the column o'Cumulati(e Freuency.


18/61

$omputing Quartiles6Procedure7

For any /roup-data, uartiles can be computed by'ollo)in/ t)o simple steps>

Step-> Findin/ the location o' ithuartile> !)herei0',1 and 23

Step-?> Findin/ the (alue o' ithuartile>

Where l = lower limit of captured class, h=class-width, f=class

frequency, C.F.=previous class C.F.


19/61

Computin/ uartiles !2emo%

ClassInter(als

Freuency

Cumulati(eFreuency

!C.F.%.BC ? ?

CB GBE JEB43 G ?43B4. ? ?

'L?

"tep>4 65or Q47) 64 0 .7 # C K *.

"tep>.) Q4KC.# 6*. > .7 K *Note) $lass 'idthKh=2

4st

Quartile

$lass


20/61

Quartiles 6Income$lasses7

.Lo'er$lass

.Lo'erMiddle

.JpperMiddle

.Jpper$lass

Q

4

Q

.

Q

1

Mi

n

Ma

0 ? G G??? HGHJ

?

uartiles can be computed usin/ S4TC4,un/roup 'orm o' data is needed there, thesynta3 is /i(en belo)>LQJA8&IL/6Data 8ange,i7 )here i0',1,2sho)in/ uartile numbers.


21/61

Quartiles, Deciles andPercentiles

Quartiles)8o di(ide thedata into Deual parts.Quartiles are

three (alues, ?and

"tep>4)

i=1,2,3

"tep>.)

Deciles)8o di(ide thedata into eual parts.Deciles are

5ine (alues 2,

2?, 2+ 2.

"tep>4)

i=1,2,3,,

"tep>.)

Percentiles)8o di(ide thedata into eual parts.Percentiles are

5inty nine (alues", "?,+. "

"tep>4)

i=1,2,3,

"tep>.)


22/61

Practice Questions

Q* hat should e the interval of income'hich covers middle 3 individualsO

Ans* G to HGHJ

Q* hat should e the interval of income'hich covers middle C3 individualsO

Q* hat should e the interval of income'hich covers middle 13 individualsO

Mi

n

Ma

0

433

C3 1313

D1 D


23/61

/0plorator( Data Anal(sis 6/DA7( "ir ohn ilder &uke(

8here are t)o types o' studies>

$ypothetical Study

43ploratory Study

In 43ploratory study, )e can per'orm ouranalysis by a(oidin/ con(entionalmethodolo/ies. In 42A, )e can obser(e

the trend o' data by applyin/ diRerentprocesses on the data.

8he Bo3-plot is a (ery use'ul part o' 42A.


24/61

&he =o0>Plot

543

Teachin Ranks

Boxplot of Teaching

Min Q4 Q. Q1

Ma0

Inter>9uartile8angeKQ1>Q4


25/61

Processing Data using=o0>Plots

leAge

maleA

45

35

25

Boxplots of Female Ages - Male Ages(means are inicate !" soli circles#

ales are


26/61

/0plorator( Anal(sis for Qualit(ranks from Aventis 5ield Managers

t%r

Amin

hing

5

4

3

2

Boxplots of Teaching& Aministration ' $tr%ct%re

(means are inicate !" soli circles#


27/61

"tatistical Measures 6$entral &endenc(7

6Mean, Median and Mode7

&he main prolem associated 'ith themean value of some data is that it issensitive to outliers*

&he median is simpl( the middle valueamong some scores of a variale* ItFsthe .ndQuartile 6Q.7 of an( data*

&he most fre9uent response or valuefor a variale* Multiple modes arepossile) imodal or multimodal*


28/61

Mean, Median and Mode

8he Mode is based on the principal o'democracy, )hile median 6Q.7'ollo)s the rule o'

moderation. Mean took its place a'ter bein/inUuenced by the hi/her (alues o'measurements. 8he abo(e mentioned distributionis N(ely ske)ed.

Measurements are on x-axis andfrequencies are on y-axis


29/61

Mean and Mode6$omputations7

$lass

Intervals

5re9uenc(

fi

Mid>Points

xi f i xi

.BC ? !?ND%K? L ?

CB f'L !DNJ%K? L

BE fmL !JNH%K? LG G

EB43 f1LG !HN%K? L G

43B4. ? !N?%K?

L?

fiK. f i

xiK4G

Modal

$lass

!ode= ".333 = "333#-

!a$ority%s &ncome

$lass

Intervals

5re9uenc(

fi

Mid>Points

xi f i xi

.BC ? !?ND%K? L ?CB f'L !DNJ%K? L BE fmL !JNH%K? LG G

EB43 f1LG !HN%K? L G

43B4. ? !N?%K?L

?

fiK. f i

xiK4G

= "1'(#- is the

)vera*e &ncome


30/61

/mpirical relationship #'Mean, Median and Mode

Follo)in/ are the (alues 'or ean,edian and ode obtained 'rom theIncome data>

)(

333.72

222.7

160.725

179

21

1

2

s+ewedvelysli*htlyisdatathe,hus!ode!edian!ean

hfff

ff

l!ode

-!edian

f

.f

!ean

m

m

i

ii


31/61

Arithmetic Mean, :eometricMean and Harmonic Mean

For any un/roup data, 8he Arithmetic eanis>

For any un/roup data, 8he *eometric eanis>

For any un/roup data, 8he $armonic ean is>

Where xiare the obser!ations

and n is the sample si4e


32/61

Arithmetic Mean, :eometricMean and Harmonic Mean

Consider the Follo)in/ un/roup data andcompute A.. , *.. and $..>

X/ $ ',1,2,5,- n0-

A"6" 0 7'8182858-3%-

0 '-%- = 3.0

G"6" 0 7'x1x2x5x-3

'%-

0 7'1(3 '%-= 2.6052

H"6" 0 - % 7'%'8'%18'%2898'%-3

-%1":222 = 2.!"!


33/61

&heorems related to AM, :M HM

#m$irica%%y $ro&e the fo%%o'in( )heorems*

)heorem No. *

A6;G6;H6

3.0 + 2.6052 + 2.!"!

)heorem No. 2*

A6 x H6 G61

2"( x 1"':


34/61

Arithmetic Mean, :eometric Meanand Harmonic Mean for :roup Data

For any *roup data, 8he Arithmetic ean is>

For any *roup data, 8he *eometric ean is>

For any *roup data, 8he $armonic ean is>

Where xi are the 6id)oints

and fiare class fre>uencies"


35/61

AM, :M HM6$omputations7

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

.BC ?

CB

BE G

EB43 G 43B4. ?

fiK.

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

.BC ?

CB

BE G

EB43 G 43B4. ?

fiK.

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

.BC ? 23CB

BE G

EB43 G 43B4. ?

fiK.

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

.BC ? 23

CB 55

BE G )*

EB43 G *)43B4. ? 2++

fiK.

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

.BC ? 23

CB 55

BE G )*

EB43 G *)43B4. ? 2++

fiK. fi

xiK4G

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

ifi

.BC ? 23

CB 55

BE G )*

EB43 G *)43B4. ? 2++

fiK. fi

xiK4G

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

ifi

.BC ? 23 3 2

CB 55

BE G )*

EB43 G *)43B4. ? 2++

fiK. fi

xiK4G

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

ifi

.BC ? 23 3 2

CB 55

BE G )*

EB43 G *)43B4. ? 2++

fiK. fi

xiK4G

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

ifi

.BC ? 23 3 2

CB 55 5 5

BE G )* * )

EB43 G *) ) *43B4. ? 2++ ++ 2

fiK. fi

xiK4G

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

ifi

.BC ? 23 3 2

CB 55 5 5

BE G )* * )

EB43 G *) ) *43B4. ? 2++ ++ 2

fiK. fi

xiK4G

xifi

5or

A*M*

5or

:*M*

5or

H*M*

$lass

Intervals

5re9uenc

(fi

Mid>

Pointsxi

fi

/ i

ifi fi# i

.BC ? 23 3 2 ?KCB 55 5 5 KBE G )* * ) KG

EB43 G *) ) * GK43B4. ? 2++ ++ 2 ?K

fiK. fi

xiK4G

xifi fi/ xi


36/61

Mean, Median and Mode

S4TC4 synta3es 'or :ndin/ threemeasures o' central tendency are0

LA(era/e!2ata 7an/e% For ean

Luartile!2ata 7an/e,?% Foredian

Lode!2ata 7an/e% For ode


37/61

"tatistical Measures6Dispersion7

hat is 2IS"47SIO5A dart-/ame can help us in this+

=ased on the &isua%

oser&ationR 'e can declarePla(er>A as a 'innerecause)/%ayer is1ore consistentKess;ariableK$omo/enousKess2ispersed

A n d/%ayer is1ess ConsistentKore;ariableK$etero/eneousKore

dispersed

=ut'e

still

donFt

kno

',ho'

MJ$H

dispers

edthep

la(er

=

isOOOAnd

Ho'

much

nsist

entthep

la(erA

isOOOOSSSS


38/61

Measures of Dispersion

"ome Important Measures ofDispersion are)

7an/eLa3-in

;ariance

Standard 2e(iation

ean 2e(iation Inter-uartile 7an/e

CoeEcient o' ;ariation !C.;.%

i i


39/61

Dispersion Measures6$ont+7

( )n

..011ariance i ==2

)(

Variance of the following

ungroup data:

: 1!2!3!"!5Mean=3

2)( ==0#tandard $e%iation&&1."1" '''


40/61


41/61

%ariance "tandard deviation6group>data7

$lass

Intervals

5re9uenc

(

fi

Mid>

Points

xi

fix

i

fi

6xi

>mean7.

.BC ? !?ND%K?L ?

CB !DNJ%K?L

BE !JNH%K?LG G

EB43 G !HN%K?L

G

43B4. ? !N?%K?

L?

fiK. f i xiK4G

( )"5."

25

3".111)(

2

==

==

i

ii

f

..f011ariance

$lass

Intervals

5re9uenc

(

fi

Mid>

Points

xi

fi

xi

fi

6xi

>mean7.

.BC ? !?ND%K?L ? ?! -

G.J%?LD.JCB !DNJ%K?L ! -

G.J%?L?.

BE !JNH%K?LG

G !G -G.J%?L.?

EB43 G !HN%K?L

G G! -

G.J%?L?.J43B4. ? !N?%K?

L? ?! -

G.J%?L?.D

fiK. f i xiK4G K444*1C


42/61

%ariale $omparison 6Propert(of $*%*7

CoeEcient o' ;ariation 'or ,?,,D, !n 0 -% is,

And 'or the Income-data ! fi 0 1- %0 it is,

So technically, Income data is more consistentthan the :rst :(e natural numbers.

1."71003

"1".1100

)(.. ===

0

0C

".29100

16.7

111.2100

)(.. ===

0

0C

d !l l i


43/61

Hand>Pro!le Anal(sis6An e0plorator( approach7

&hum 6;47

incms

;.;1;C

;

"pan

6;7

Length 6;7

"*No*

Measurements 6;7

T

? T?

T

D TD

T

J TJ

G TGDetermine the

Mean, "tandarddeviation and$oecient of%ariation*

$omputing Mean and "tandard


44/61

$omputing Mean and "tandardDeviation Jsing "cienti!c

$alculatorsNe' Models 6/""eries7"ress MD/Select "&A&Select 4>%ar/nter the Data inappeared data column+5or 5inding Mean and"tandard Deviation)"ress Shi't and then press

Select ;A7Select 'or meanSelectn'or Standard2e(iation

Prev* Models 6M""eries7"ress MD/Select "D/ntering the Data)s4 Ms. Ms1 Mdo it for all remaining dataoser&ations*

5or 5inding Mean and"tand* Dev*"ress Shi't and "ress ?Select 'or meanSelectn'or Standard

2e(iation


45/61

h( =ell>"haped "(mmetricalDistriutionOO

8here are se(eral Symmetrical2istributions


46/61

h( =ell>"haped "(mmetricalDistriutionOO

In a Bell-shaped distribution, e3treme(alues come )ith less 'reuency.

a=ority 'alls )ithin one standard

de(iation. It6s 5ature6s 2istribution. *od created

almost all natural measures )ith a bell-shaped distribution.


47/61

/mpirical Proof for the Appro0*$on!dence Intervals

Brin/ One 5eem ea' and measure its len/thin cms.

Obtain ean and Standard 2e(iation 4mpirically pro(e the 'ollo)in/ theorems>

% )ill co(er appro3imately JHQ obser(ations

?% ?)ill co(er appro3imately Q obser(ations

% )ill co(er appro3imately .HQ obser(ations

6:roup the data and prove that its=ell>shaped s(mmetric in nature7


48/61

&he Normal 6:aussian7 Distriution!2istribution o' a continuous random (ariable%

Bell-shaped distribution or cur(e

"er'ectly symmetrical about the mean.

ean L median L mode

8ails are asymptotic> closer and closer tohori1ontal a3is but ne(er reach it.Appro3imate domain 'ormula is -TN


49/61

&he Normal Proailit( Densit(5unction

8he "2F is )ritten as>

here W6 and W6 are t)o parameters)hich are ean and Standard 2e(iation,respecti(ely.

Simpli'y the Wf7X3?i' L and L Simpli:ed 'orm is said to be the #tandard@ormal istribution.

N l d


50/61

Normal curves andproailit(


51/61

5inding Area Jnder the"tandard Normal $urve+

Standard 5ormal 8able comprises allpossible Areas under the Standard5ormal Cur(e.

8hese Areas are to the le't o' XL1 i.e.,

8his can be )itten as "!XY .H% L .HH

i di d h


52/61


2etermine the 'ollo)in/ AreasKprobabilitiesusin/ the Standard 5ormal 8able>

- "!X.?% L

?- "!XV -.% L- "!XL -.% L

D- "!XN.% L

Solution,

"!XN.%L M "!XV N.%L M .HD L .HG

8heorem> "!XN.% L "!X -.%

i di d h


53/61


2etermine the 'ollo)in/ AreasKprobabilities usin/the Standard 5ormal 8able>

- "!-. X N.% L

Solution,

"!-. X N.% L "!X N.% M "!X V-.%

&heorem)

"!a X b% L "!X b% M "!X V a%

J- "!-?.? X -.% L


54/61

serving Quantiles 6Inverseconsideration of "tandard Normal &ale7

2etermine the 'ollo)in/uantilesK"ercenta/e "ointsKX-scoresusin/ the Standard 5ormal 8able>

G- "!X a% L .?

8here'ore, the ans)er )ill be aL -.J

T 3*3G + 3*3 + 3*33

-.

..

-. .?

+


55/61

serving Quantiles 6Inverseconsideration of "tandard Normal &ale7

H- "!X b% L .

8here'ore, b L -Z.J N !.DN.%K?[L

-.JD

/lse'here 'e can also consider thenearest value*

T 3*3G 3*3 3*3C + 3*33

-.

..-.J .D .

+


56/61

Normal Distriution 6$ases7

"oft>drink Anal(sis from UJ canteens

Amount o' so't-drink )ithin a /lass 'ollo)s a

5ormal 2istribution )ith L?? ml. and L ml.

I' a student purchases one /lass o' so't-drink

then determine the probability that he )ill /etless than ? ml )ithin his /lass>

"!TV?% L

e must use the 1-trans'ormation> X L !T-%K, so>

"Z!T-%K V !?-??%K[ L

"! X V - . % L .HG


57/61

Normal Distriution 6$ases7

"oft>drink Anal(sis from UJ canteens

"!TV?% L .HGQ

- 8here is a JQ chance that he )ill /et lessthan ?ml )ithin his /lass.

?- e are JQ con:dent that he )ill /et lessthan ? ml. )ithin his /lass.

- I' students purchasin/ /lasses o' so't-

drink then appro3. 3 .HG H o' them )ill

be ha(in/ less than ? ml. )ithin their/lasses+

Find> "! ? T ?? % L "!T ??% M "!TV?%

N l P iliti J i M"


58/61

Normal Proailities Jsing M">/;$/L

For any 5ormal distribution )ith L?

and L, )e can obtain the "!TV?D%

usin/ the 'ollo)in/ synta3>

LNormdist60,,,cumulative7

KNormdist6.C,.3,,47

And 'or "!T\?%

K4 > Normdist6.,.3,,47e can apply the same scenario on a so't-drink case

study.


59/61

Inde0 Numers

Inde3 5umbers are 74A8I;4 measures.

Inde3 5umbers Could be "rice 7elati(es oruantity 7elati(es.

Inde3 5umbers are ha(in/ t)o ma=or types>% Simple Inde3 ?% Composite Inde3

# Simple Inde3 5umber can be obtained

usin/ this 'ormula> InKPn#P3433)here,

n is the current year 7time3 and o is the Base year

7time3

"imple Inde0 6/0ample7


60/61

"imple Inde0 6/0ample7 Consider the 'ollo)in/ table comprisin/ prices o'

a commodity in diRerent years>

I' )e )ant to use a 5i0ed ase method by :3in/the base year as ?J then the possible Indices

)ill be computed by di(idin/ all "rice (alues )ithD.

In $hain ase method0 the precedin/ year price)ill be used as base.

Years

Price68s#>7

?J D?G J

?H JG

5i0ed =ase

InKPn#C 433

LDKD L

.Q

LJKD L

.Q

LJGKD L

?D.Q

$hain =ase

InKPn#Pn433

LDKD L

.Q

LJKD L

.Q

LJGKJ L

.GQ

$omposite Inde0 6/0ample7


61/61

$omposite Inde0 6/0ample7

Consider the 'ollo)in/ table comprisin/ prices o'

a commodity in diRerent years 'or three diRerentcities>

Be'ore computin/ the :3ed base or chain based

inde3 numbers, )e ha(e to obtain a sum 'or allprices in the ne3t column.

Finally )e can compute both Fi3ed base and chainbase indices 'or the " column usin/ the same

Years

Price$it(4

Price$it(.

Price$it(1

?J

D ?

?G

J J J?

?

H

JG J JH

5i0ed =ase

InKPn#4

433

Q

.Q

?H.?Q

$hain =ase

InKPn#P3

433

Q

.Q

J.Q

"um

P

J

HG

?

Date post:	02-Jun-2018
Category:	Documents
Upload:	mazhar-ali
View:	223 times
Download:	0 times

Quantitative Methods for Decision Making-1

Documents