O

Lshmatíon Throny
Sec
uused a
ass
ohouse Value
Stnishc Ca lleel an
Pon4 sme H Pova mele.
estimetov
(OR) a
Pavicular
PAya metes
An eshmehe th
Of the esimator
dbtaine y1 SamplJ
Dumen n l Va lue

I huo
yp
()Poit esimehier

() Tnerial estimertion

Sec
Pont Estimahon
Parameler
Popuah on
eshmae of a
an
Singe Valu, then he eshmale
ven by Payamee r .
Ca lled a Point eshme of
the

L a Poim estima
F Sample mean

Popution maan

POint
eStimae of
VaYIa nce
Sample
Popubhion Variance -

Point esti males

Metds nding Value of
the
Tvn obtained u suS acheve the
Statish c denved fro Suavey data to
the Populai on'S Coses pondin g
oes+ approxation af
e sti mato S CAn be
Unknowwh parameler. Thu Point
as is
methodS, each
Cakuh ed uSin yaty
Own Pope
co 1
Momevts
Method f about he
with mown facts
T that
beginsaffeck S a Sample
Paramee r

Populetion poPulti on
h+ Connec
O Coeale
PTamokers.
moments unknaun
 Sam ple from which
Select the popula-tion
P o p u l a t h io m momen+s
esti me th
momeNS
The Sample of he Population
3The
sed Solve the u a h onms

e S t i m c r h on
acumae
Tis helds mas +

unKmaw pOpulathon Paame esS.

Example denSt of
the
paAmelerSo,, K
the Opuatio

L3 fo find the Pns of Sample

Our In lerest P a a meB
,, ,8k
momenS hat w es male h

densi fn the popuahm,

OLo+ f(ol , O2, 9 ) > dens- dsaw0.
Sigeen Samples
fem cahich (1y n)
then Tel2-K.
fu;o,0,-, 8.)dz,
be fhe
In Genral wl

pAYAmeers

we P , 8ar.. Oc
Solving

4Honda,

,(mm-,)

eanlation
obtainnd 6y ment t Smpe
whese m
t O rdor Mo

Noe
Parameess, we
eStimaR the
we wish to
about the ongn
Compule k mome ntS
Should

Metwod oment
estimatos S Aye
uSuall
ess efficien than MLE's
laximum ltolhood ochmator Poi n t
kethocd Osthmrtos
pAramoers
tool
hat tie s ftndunknos h
dale Sets
Jtalhod funchom T Corrpares
th nitASe he
he best t fo datn bH usin g
and ds
model
Poven
Values

Dein
.kelhood

maximu m

Sahhc
&(x ,1)
A
e hma h o
XA) u
Samle , ,1., A , ,
Onch
T fo hea+ maxi miys
PArRm ets
Va h Ro L (0/x 2/X»)
Ielhoat funchion

maximum
ikelih aod Hhe
Methed 0 f esti mato Y

amethod, a good
hw
T he Lkel.hoo d M a k muM

mxim
s Callel Hh
whch
One f(.,o). It u
f(1,,0),. ,

L f , O); e stimatos (MLE). Populti m

MLE Sos t
Litelhood the
t
Sehsie
Pavomet o

ecivalen
L +ve, o
COT)

Psa chca
convehient
=o a
to as ikelihood
uSunly T*ffered
omd itF

euah or

Nole u sualy
Peram e e
Tw MLLE of
denond
MaK im um
ke lihoo d OshmaosS
Propomes ef
COnsisent
MLE a
oose fficiontt
2) MLE
 fSufficiend SeShimntor5 4
MLEs e Sufficiens,

unbiased
hot necossaily
MLE S C
nve he nyaianco PDpe*
MLEs
u t MLE sb , hen

So) will be a MLE fos

nos mau
MLE emds to
t h the dismbuthon
forv Nexge Amples

Chacesi cs Estimtion
estimatos should
Point
A Food
following Coilena
SaHsfy
C Unbiasedness

( Consisknc
3) EF: oen
C4) Sfficien
S ) RobuSt ness

() Unbiased ness hn
am
eshmqtur
Le+ be Said to be
(A12,.A)
th e Shmatos rameter
unbia sed estimae of

I Eo) - o to
be Poshvely
I E0)>8, then &aid

iased oe
E (0)< 0, hen T Said hatively
IP
rased amount
ef a S
Caled
EC)-o}

denoted 4
.bo) EC)-
om
Uased
E), he Somple ean an

esma a popuhhon mean

w heve ,- 7
unbiased s t mak

Bu Ae

U brse d Sn

Also, S nS
n-

Cons+
e
obleid from
Stetshc be
Consi ten+
Said to
Smpie Si , I f i+ corme S close
Premees
Ww hen n be con9
th ree ,
amd Claser
Jage nd avg
I ezo
To-o|
n
s e] =

a Consis ent if it ConNer

esti motor 0
to P, S t Sample Siy

) E fcitnc
two more Cons Sent
ge+
eimato1S th Same PaYamees
fo
2 Consisen eshmtto1S exist fo Same
PaTamekr 9, then tw Stohsh c w th th Smalley

Vanane Called an
efficien4 estimtor fO

wule otw Stah sh c Called an mefi ciênt

esh metoí of
fov tw o Consi sen e shato 5 and

fos we have V(D, )< V(O) then

n fhient esi n atov

c o n si Sk nt astimator Od a a r mekr 0,
th One wth he Smalles VAnana Called
mosh efAiie eshator best Sh mtor e
 3

S
i
The median is a obust meaSu
Ccensal

cndeny
TaK Some dlalaset f2,3,5,6,9 1. - | o00
daapbin wth Value
T ue add anol
+1000
hen the median wil cma sligy
t med an
the
wll Shll e Simla
bu dsta
the o gnml

and i n e aarhle
3 hu median bsolue deviation
S tatishal
en 2 ae vobu St+ meSus s af
oeviaton and
disprSiom, whele t Stendrd
a ng ee e

S t i mators
win Sonsed
and
Timmed stimator S
me thadd to
make StotiStics meve obst

3enere
cass of Simple Stetish cs,
L- eshmaorS
Y
neral
obs wmle M-estmatos S
Stah sthcs and Ye
Y obust
Class Can be involved
tha
Preeed Son though
calcue

APplhcan yeressiom problems,

also exist fo
I+ u para me ler dismbuh n
modelsS
naluggd inar
d i s m b u t i on s
Vanouss
fo R

Pera mc hoc StahsFcs So that

esthmato
imfluea funchr b
designing
0)5y
0)0 behaviour
P e . Seleckd

achieved. th
e ophn

eplain 1 eshmato s smbuth on

Gi B omT)

Under
ascumphon
that
w th OR oistmbuti ons de)
deaved foT, other
ateas+ freedom.
t-dishmb h m
with ow degsa
MSing tw
OR
 With mixture tup moTe isabuhons
PobemS-
Sude d o liong
PAsamek iS
Nocati on
Est mtin
Scale

Tessi Coe-fficic ntS

Thedels
n
Esh mti on model- Staes
for whhch
in Stae-Spele fovm,
pressed
the Standad metmod eauivalent

Kalman fhes
PaoblemS esti mator
an
unbiase d
mean
3T the Sample
Populthon mean
for the

Soln
TP: E(Z)» .
andomn Sample
the
Let Sie n

E3E1-E(Za) E

t Ean
E,
(H+H)

estimrto
anblased
an

r a m e hrs s
with
binomi a Varine
ProPoton
observed POpatawom of
n and P hen ST n tho p a T a m e k r
UnbiaSe d estimato ef
Succe SSeS an

Soln bivoial Vaia

th ParameksS

a
GYen
and P, hen Mean (X)- np
 TT E(): P
E(x)
Le E() %
An
Unbiased esimcoT fhe
Parameler p

Show t h a t

3 Tf 3 m Unbiased estimae
a biad eStimato7
is

Sol u an
Unbiased
esti me
Guven
>ElO) 0.
E () # 0*
T.P
Vea()ElO) -¬c6)
Naw
Naw
(E(6) )*
ElO)- Vaa(6) +
Van (0) + 0

ElO) 2
biased estimatoT 0
a

faom a Novmal
a
Tando m Sample
4)I N(), then,
PoPul ati m
2 an Unbiasedd esti mator
ST

Soln
Given Vanlxi) - I
amd
E (Ai)-
El) +
EËJ
Et
ECt)E
Let
et,

.'. ECt) +
 A) Si n
5P.T,for andom Sample
drawn om a ven ange Populaan
CP), a3)
An Unbiased estimato f h Parame k1
S ( x - ) u an unbiase d esti marto
n-

Soln

(-5) 2 [-) - (-PJ

a(-T-H)
- + -N)
-

A
i

+n-H)-2G-
2
2 (i-)
(-)

2T-) (7-u
2 H) +n(7-H)-

nli-4)
-

O
Ci-H)-

Given -7)
T

- n-)J from
s (-

E - - nE (-
El
E(i-p)
w.kT E(;-) -

ElS') xha -*

unbi ood

n
estimatos

ConSider,
-
an
unbiaSed estmor f*.
6) ST `- 5 (x,-5) Un bia se esimortor
n- ef
tHhe PaYame les

Soln

2( -3) -H+H- <)

2,- - (%-p)]

-H) + (Í-)J
- a (4i-H)

+n-H)
-)-a-H) (o;-H)

2(-¥)n(-4) +n (i-4)
-p) -

(o-H) - nlz-H)

Given 5(-a)
- p ) - o (i-H]
n-i

E S ) E - - n EC=-P)J
n-

na
(na- J

Unbiased esimator
an
Hence, 8 z%- th PaTAmeR1
n-
 12
are Sample Vaiables based o n
YandomSamples ef Siyes . . , Nr respe ctvely
Populatim with Vasiance
drawn from a lavge O

Omd T S; 3 = an Unbiased estmak ef

find h Value 8

Soln
T an Unbiased estimshe
2
ECT).

E( 4)-
2

EC) -

i)
( i - ) =*

7 n; 6,+n,t... + n) -1

Te X,72,
andom Sample from NC)
an Unbiase d
omd T 1-HI, hen TVi
esti mak

S ln

T -H

ECT El%-
deviaho about h
E-H the meann

oms di sm bi om
mean e iven
wich

ECT)
 unbiased
E (T) an

estimo

N (0,)
trom
Tf , n Tan dom Sample
Unbiased
e stime
ond 2,
if T: then T an
a n

on

ECT) E (4)
i

N(o, s)
Elx)» Van(I;)
+
E ()

ECT)L.pr* ea
obtai nad fem
the Valuns
oBelow dou are
iven taken from
obserVatim
Tandom Sample
Popul ahon .
infin e 34
35
32 34 an
for . Ts

find Point estimato

? Explain.
un bi a se d est mre
Awo an
the Point eshmtos for Ts
Fnd
unbiaSed esti maR o Explain
Point+ e stimatos for
Rind

the
Nhat Can be Said a bout
Sampin
disibehn Be Suse to aiscuss
expeched Valu Standard deviati on, and

th Shapee Samping disibuhsn

Solr
eSim atoT
A= 32t 34t35+31
@Point

35.
an CAnbiased e stimato s d P.

(from problem 1)
 Point eshmaor 2 ( - T *
. s(d-3
n-

(32-35) +(3 4-3 5) + (35-35)+ (39-35)

4

9+1+ 0 +6 = 8.66L*
3
eshimato
T s an un biased
(from problem 3)

e Shimator Vs 6644
Point
= 2.94 39

ual to
dishibti on
mas m ean

The Samplinq
mean. Tie Sampling dish bcti
m
has
he POpulation divided
he Population SD f
a S D equal
of he Sampk
by The SqaAY
oot Si
Sam phng dismbethi m
he
T Shape
nbma Cuwe
4 drawn
, a,x4) of Si Consider
1A Tandom Sample C unKmo wn
mean

populatien
from aa noma eshimae
esti m afors

he followin

i) ta= It 24 tA3 where

3 Find . Ave

an Unbiased estimto
9t Hhe
estimator
Teas o n s
unbia sed ? Stae
ti,t2 and t3
bes+ am ong
a m on
which

Sol ECx)-H,
Van (i) -a

We have
0, i#j =l,2,n.
Cov(i, X)
 ECE) E + t+ g+1

EC . F 3
3

unbiased estimator
an

C3+AJ-
3 t3

+ Vlt,)+V(z
V) . [Va,)+ VOa)

+vla) + V)
V)= [v(z)+Via,)
+*
%( ) +

(a1) + v()
v) V
5
Sense

i the best estime

least, t
Vt)
leas4 Vaaianc
andom Sample of
amd 4 amd Vanance o
12) Le+ o4, man
Value
PoPulaion

from a
esimatdo S used +o estimae
e
Let 2,T
Value whese
 T +232 +33 -5ol4

g: K +2 +3 +4
estimator s2
Unbiased
Tand a r e
(i ind
whethe Unbiased estimator
s
01) nd he Value ef k
k
3

for
ConSisknt

(t) With t w

estimao"?
best eshmato
Cw) Ahat s the

Poma
Soln an
Tan dom Sample
Sine , 2 , g, 4 VaianCe
and
Populati m with mean
n)
V (x)
= Co (a1,3)-o, Ci4j=l,3..
E () H,
E CT) =
Mtp+H - 2=B
C Ts a Unbised
esti matox

ECT)= H+2t 3u -5.

=
. an
un b i a s e

B alSo
wich implies
that T,
esh mator af

Gi) We ase ven ECT) -F

F + H+M+H -

3M+K =
4
K

-.
Cif) wth K,
G: (x,+ +3 +4) Populahm
Con st6 Hent e sh m a t t o r
a
men
numberS, T3 is a
Sanpl Jarg
the heetlaw
by esti natoT -
Consi s e n t
 4V(4) 1
oV Var T Vx V()+V(4) +

Vav(T) t 4010- a5
3

Va (T). t o n a ) - 4 e s i martos
he hest
Van(T,) u mi mimum,
Sens I
mi nimurm Vanante.
in

Tandom ob 8evati o w and O

Values
3)Tf , 2 which assumes

Bemou lhs
Vai ak md (1-)respechively. ST
Pobab1hhes
with
estimto1
unbiase d
T(n-T) an

n (n-1)

T: X4+*t. t An
Where a
a y the
nepladng
neplacing y thu
- p
=II-p , =:
p.
sol Convestiona
(
(ta
-O )
-) =

md P
: F
PP(-)
( x - ) =F ean nP
T +3l2t +In d i s h b h a n
wIth

amol vananu
mp
a biomI
fnow S ECT) -ECT))
Van (T) =

amd
ET) nP =np
2

E CT)rp9tnp

[ n ECT) E CT*J
nn-J nin-l
No E -npy -)

n(n-1)

-n p4)
(np
n (n-1)
14Tf X,a,nj s a ran dom Sample Sie
d i s m b u h on, the p.mf
dqwn Aom eomehC
wnch iven Pla-1): y , T:1,2.. P.T
COnsis Hnt+ eshmrtor
h mean Sample
mean
Popuaiom

So Romei c
iven
Mean Vaa d
PoPulhon

Fl): E (24) E)
-

Van (x)
Van()
Vavn (oti)
and Van ()=0
E)
e StimatD e
C&a Con S S lent
Populatiom man

most effiet esti matos and t2

5 Tf t PAsameer)
Unbi a Se d .esh matos (of Some Populohom

, nd if the Coselati on coficient P

wth e f Cien

behwan ti and t u f S.T P Ve.

Son
a n d t2
Variance
e te
the
Let and V
Yes pech vely Then

e- V -
V2
h +q ,
where p+
le Van a L f
md ort V3 4e the

Then V Va pt,+Vb)
V +9V. +2pq Cov(t,,)
 Cov (ty,t2

CovCt) -f V
V( 2 )V,-
Ve

dess
e simatoT
the
1

Cie v , 2 (p:9.V
Ve
PA & fom

e)
e

ie) whn
when
rolds
o lds Jood,
the eauality
In ,

2(f -)

P-ve
efficient
esti moavtos s
both mos
Eamd t are
f t, i the ave sage
Vand
16) Tf Vanana

V 1+ P),
wkere f
with eual
that Van (t3)
nd t, Tmore
Corelah on Aoetwen t ond t2
Coepp cien

SoIn
Let
Van (al, 1 bl,) a van(t) 4h van lt,)
2ab Cov (t,, 6,)=O

Vanlta). an (Kt Kt) (by O.

+

VA(th) Van(t,) y * y Cov (tit)

yV+PV VCP)
Sec2
Method ef moments
w
wth
h
o in the Populatian
) Find the of
estim atov
metvoa

density funchonfa)=0 o 4X <1; O>o, by

moments.

Soln ovdes moment (about ovin)

The fis

the Populatiom

Ca, 2 , A n )

th Sample
Order momen+ ef
th fis+

alout

m o m e n t S

method f
b
o+)

a(-) -

-3

the Unifon
Sample frem
ayandom
_ a<a<b.
2) Tf (A,3 (X,a, b) =
denSr f b-a
Cwth
t methoel
he
Populah m

e s h matoTS y
Fnd
momen} s

S
Soln b-a.
d b-4
d EI
2
2 (b-a)

dx
3 (b-)

a (b +ab+a
3 Cb-

- btabr4

m, o= 2

and m L53
momendS
hfrst ond Second oder
mehad
are
He orgin, hen hen
ther
about
the Sample
momets ve

b= a-a
atb 2
abtb 38
3sby
ata (2- a) t (22-a)=
)a2ta
+ (4 i - 3 5 ) - o
4(43)
V4*-
-
2 2

a t a()
ain
fern

have
a= - V 3 ( z - )
we
a cb,
amd

fx,)-
fa,p) - 3Cy.
3C -
a-p)
P 23
Pm.f

3For he
he
 obtain w estimtoi
if the
8 Pdoy the method of (2
feanenGe at
X:12, 3 are
momenB,
18 res pectively 22, 20 ond

Soln
flp) (3p)
-Ci-P)*
moment
a bout the Oiin
OTde
fvst iven distibuhom b
he mean
he

.3

the mean o h Ob Served Sam le n by

X 0) (22) +(2) (20) + 3 (18) 21

5
22 +20 +18

momentS,
the method f

e) 3P
3p-3p+p3

29p-81p + 42 =b

P 871 5123

58
0.605
2 315 1 )

2-3150, b o.6o 5

andom Sample from

fom
be a

4) Let (X,Xi Xn) with p4f $ X

dishibi m
exponential

fC,6) 0e

e Ise wher
e.
=
e,
fmonme t s
EStimaf si
method
Son
24)
0 to
One pAramekr
Heve theve Only
e shmaBe
So E (x) X
expontnial
di shi buio
tor h
ECx)

TeSuHs
in
ECx) X

estimah on oef
momert

So O he

follous
Component

el echonic units
Tha hme failuse an
R r a meles
an
exponenhal dismbutm with he folloong
o
md hestd Tesuting
ae andomly Selece d
falure time Cin hours).
23.42
03, 6.07, 68.
4L,I7.11 32.54, 8.77,12.4
13
he m Omen estmae A
ind

Solo
k 1 3 - 0 3 t 6.07+G&49 P17.II t32 554 +8.77
He se
+12.14 t 23.42)

C181 52
22.6

The momet esthmae

22.49= o.0440t

Tancdom Sample fem

G) SupP ase #hat X,X Xn a

homa dishbuhon N(Ho). Find the moment

a

estmoxs C m a2.

Soln
 Soln 29
Fo tho Nomal disbrtion
2

E()-M md E(x*) =uto

eauating E(x)to X ve
md ECx) to X;

M X, M+a*
w momemt
Solving use eans ves
estiators

and
X

1x-x)

Valuas ,0,1,2
,2 with
w th
takes he
A) A Tan dom
Tespective
Vanable
PHobabiles
X

2
- t2(1-0
amd

the Parameers
Omd
+C2n-) o , wAhese
e Populati o n
dsauwn
d s a w nn from
fom
15

Tfa a Sample of Si ectve

Tespecive
ruencies
with
Valuus O,),2 estimaoTS V«araod
8ielded the estmotors

Pnd the
27, 3 8 , o ectively
mment S.
metho d f

Salo - ( - o ) +(1D+2
(-doj
H-E
+ (2-1o }
1- +2%

2- +
(Lo-2) &
Sample
The m m q fu obseved
m' 3P(1) +lo2) 58
75
15
 about tho on7n s
Second Ordes moment
Fiven b
2

hemehhod
he omentS

omd M, 3

2
O
Omd
2-2+ (6«-2) 0 =1e
8 olving O
amd 0 - 1
33
So
Method af aximu Likelihood Esthmotor

ConSides characenishc hat OCCUS in

e Tandopm Sample PSie

POpulhon Let X,X2,Xn
n sO
I - p omd
P[X;-J- > fo I=
PLXo] ikelihood
OEbe. obdain the Maximum
Hhese
eshmatos
(OR) for he
likelihood estmatox
rd he maximum WhurTe
binomial dish bukm B(NP),
Pasameke» P of
on thu basis fSantle
but finite,
N Vesy age
Vananco
S i x n. Also findS

Soln he
binomial dishibuim
he P.m.f
o -0,2 N
P(x- «)- p(x3NPD . NGP (1-P)
Sample
random
he ikelihood dn qPhe
C2 n)
n

P)- Tncy p ( I - P " - 2

L(
ToP (1-P)

n (m-be ) ie Cr-7
o s ( n ) +nxagP +
n

og L

ikdiwod ean

i n n(n-X)
-P

-P) P(n- 7)= o

-> -PA -Po + P= 6
- Pns o

P
 Kao Cramer's for mwla
(29)

-E 1 -
-P

-E - h(n-x)

C1-p

-P)
+ (n-K})
-P)

PI-PUJ PA

P6
'.Vanl P) -
2

n om
andom Sample 3i
2 Let X,,2Xn b e a
th Poisson dis buti an
f()A"e oe
stmator ef
maxMu m Mkal ihood
obtain

Soln ea
P(x.)
Th P.d f

LL,, ,) - T e

og L Log ( A
 M M M
EM M
M

N
 inerShak, he
he numbes f
4) Ln On area
alon Call follo u s
Con necti ber
dsoppe wireless hone
PP. Fem 4 calls, mumber dvopped
kali hood
Rnd h m a Xinmum
Connechim 2,0, 3,,1.
estmsk of

Soln
- 2 +0 +3+ 1.5
4

a oma Populhon
fron
5For Yndom Samplin estimators
the maximum ikdiho od
N p , , find

,When nown
i When Kno un

dii The SornuH aneo esimtim ana

Sol t
omaldisibuion N (H, ) ,
fo
Aikelihood nib n

L= Slok, o), f(x,,0), ... f(

o N 27

ag- Loj 7 -

, S_ (x;-
LO
D i - ) (n -np)
The MLE H tr

Cog )o
 1,
H MLE of
TbSean ha+

both Sdto O w.1to we go

1 2 (,-")

the
he MLE of i n b

epesthimatio b
Ctri The imutan eou

Oo - o (eg L) o

Cie A

USing , we a

-7.

Note
k E) E(S) /*.
El) El):p
=

the MLE ned ot hecsanly be unbiased

PoCesS making takes biomass

6) on reon soline
Suco se omd ComvestS it into 9selina
in the fovm of
unng catalyhe Techions A+ Onu >lep i apolepillo
Plat PrOce SS, he Output includes Cabon chaina
engt 3. Rfhean UnS wrth Same
artalyst Poduaad
 (33)
elds (gal)
5 57 5 76 4 18 4.6 4 T02 6.62 6.33 7.24
5.53
24 6.43 5.59 5.31
Treatng he ields a andom
noml
P opulthon Sample from a
@ cbtain the Yax imum
mean eld and
fhe Va ziancee
kelihood e Sh maes

ob tain the maXi mum

kelth0od esimate op the
Coqfcien Vasiahon
Soln
= -55t 5.16
+.+5.1
5
IS
Recall that he MAX
keluhood eshmsk
Vasanauses divi Sor O, moO+ h-
5

S-3) n -C16.263) 1.0S

Tre Cfcrent fVasiehm
and a So
fancthn of R
itS a x i mum likeli hood estimate n
that Same of and o 2

-034 O43

0btain maxi mum likelihoo d esimatoYS

amd b in ksmS Sample obsevations
2 , An taken fe exponenttal POpulah o
wth
densty functiom
Fa,a,b) =
-b (-a)
2a,b>o
Soln
To Snd Unknbwn Conshan K we howe
 I
33)
fl,a,b) do =

eb (x-a)

ke ed=

b ao
keab
ke

Cos) kb
(-a)

f(xa,b) = be a,b>0

-))
Now, L (ot,/a n,a, b) =b"e
be-bnx -na)
O.
for the 3imultaneous
Treiklihood ms
a omd b are
estimatm

(da L) -o
-
from O, we
nb (x -a)
Log L: n og b -

>nb -0, which means b o, coich

u gven to e
abSard s

> -n Cz-a)-o

Un u e Mot
>b
detniely 5own
Agan fom O, we note that Lu maximum
3
for a
given tve Valu f , when e nb(a)maximum

e ) N hen (-a) u mmimum

de When ( - a ) + (x,-a) t + ( t ,- ) ] mnimum

e) When aSmallest among 1, ,n, Say

Valu o

eshmtor f in
Rnd he maximum likelhood
he Populahim oith dens t lat, CHO)
bas ed On Tandom Samplee
Suficient
Sigen Test whethes hi eshmato
eshmator .
s
32 TnesVal Estimahm

an
eShmre a POpulatim P s Y a m A e r

betw ean cwhich

dishnc4 umbtrS
gn by 2
Considered to Iie, then
the Parameter e
Ca lled tewal estimsk
S t t m ae
the PaY qm etr
an Upper Condence
I haVing oe
Con fi d e w
imit and
limit omod lwes
condains
Pobabily ha hi mewal
assi t
Populah m Pasameey
thw dnknouwn
table Valuas
two tail one tail

Con fdee level C-) Zlov) t

1.282
q07 64
196 1.645
95
1.960
233
98
2 515 2.326

without any refeseno to 3.00 2.576

Con fidevce Aevel

te POpulahon Parameler
Cnenval e Sti mses

Mean
- Z <+
Cor)

e ( a t Z, S.E ())
Or

s)
 1
Tf 95 Confdene hon

I 99/ Confi deno Hhen

H ( t 3 58

2 Piopohon P

Pe Ipt Z SE(
3) Diffkrende means (MI-H2)

Hi-a L(7-3) SE (%,-7)

4Difeyene Popoh ono CP-P):
P-P eLCh- F) Zy sE (p,-+,2J

3.2
MaXi mum Feror of estmate E

3 2 (a)
Maximum a ap estm ae (fw drae
Samples)
E Z ( r mown)
Poblems

Sample Si
Si
oohas S.D oS
A Tand om
abou+ he maximum eTror
Wha+ Can Jou Say
with 95 Confide e

Soln n (oO, Ta 5 .
C ven
esDY
maX mum

9sZ7 19%
Where Zy or
 aximum
( =14

= 94 0.98

mlan
inends to u s e th
2) An indushial engincos eshmat

Si n-15o
andom Sample
m i a ured by
do as
mechanical p
avesage in e w oakeis
In a laa
Cesfan rs) aSse mby
ncu, thu
the evginer
egir
expene
epenen a,

indu sh4 IP on the ba SiS

data, What
ndushy fos Such
assume hat =6 2 h
Con about
wh
PMobabi lity o99
asseF
Can he
OF eroT
maximum Siy

SoIn
ne15, 6.2
Given

the MaXMum
99/= 2.575
at

E Z
E- 2-515. 6.2

Probrbility
= I30 with
asse
Can
Hence engince 30.
moSt
be at
o.99 that esoos

aTandDm
Sample
ha+ 20.o,
ho la ge that h
o.95
3) A SSumi ^ with Probability
be taken o
to asset the true mecn
will ot disfes from
meah
Sample 3.0 poinds
han
To findn.

Sol =96,
E:3,
tven, : 20, Zy
W k.T

EZ
3- C19 20
 vn 46) (20) 3
3

e)n13.O64
o. 72 I).

Sample s 20 amd
Stand dvd doviatiom
Te t
wrth
9 9 confideme 72
the aximum
Semple migh e
how lag

Soln
E 2, Z 2.58
iven, 20,

EZ
(re) 12 2 58 20

n 2-58 x 20 30
2

th Taured Sample Si

Can exCep+
one
s the MAxi mum mea

5 whet he
with Pobabi l ty 0.9, when using estim<e
ma ke
64
o
Tandom Sample Sie
Populatin with o2.5647
th mean of

Sel
Given, n= 64

Max eoT E

64 5 (90/ confi deea)

SD 2-54 = 16

.. E -C1-64 5) ( ) -o 32
e bor=0 329
Maximum
6 A Yesearch woker wants fo deemi ne
h ave sag

timit Hakes a mech ani c to Totak the tyses

ASses wth
to be able +o
Cex md he a n t s
mean eP ww Sample
that
5 onfidence Lf he Can
Car resuma
atmostos mnuk s. 6 minus,how a 2
b y by
ex pene nce 4ha+
tem Past
4o take ?
will ave
Sample

Soln
Civen E 0.5, T : 6 minuk s
96

n Sample Sig?

E Z
o.5 C96) 6
o.5 3 134

Vn 3.13 -212
o.5
D39-33 39
the Sample
thes0ised Siee
39

use he mean
College WantS amon+

avesa
dean
)Tha Sample
estimae
one
class to the
rand om from +o aSSet wth
of a to ge
Studets take to e
able
Hme watS O.25
and She atmosSt

next the expenence

f e m e x p e n e n e

292Confidence
be presumed
mimues. If it
Can
an
will She
a Sample
how asge
min,
tha 4
take?
ave

2 51S
So E -0.25
Gaven

o.25: 515)
 0.25 36oS

3 6o5
o.2 S

n 2o1.93 208

mean mumber
to esimak he
8 T+ is desiyed until a Cerain
use
ef houYS of Cohmuous TepairS.
Tf + Can be

Compuk*x wil st Sample

a
asSumed hat 48 hour g how lage
to asset
be neede d So ha+ one e able
mea n

wth o onfidena that the Sample

at most lo housS.

Soln
E lo, :48 Zy-645

E Z,
Ci64 5) ( )
1o

= 78.9

Vn 79 6 7.894
lo

> n - 62 34 62
3.2 (b) Max mum
Eo of eshmaie for Smal
42
S amles)

E-t lUnknown)
In Six de emminations af mei Point

alumnlum alloy, a chemist Obtained 532.26

deges. Celsius wth a Standard deviotion ed - 4 dgre

L he uses th meann Ho esimak the actual

the aloy, wh Can the chemiS+

me Hing POin
wrth 98 Confidenco about the MaX imum
aSset

Soln
Given n -6, 3 II4 md t.o 3.365 (Sr n-i- 5 d.of)

E= 3 365. 4 1.566

Sset with 8 Confrdenc

Hecethe chemist+ Can
h aluminium
Hing P
that hn fos me o
gu1e
MoS 1.566 degreas.
Off
2 the Population mean for
Confide o ineswa for
u
Sample (o
s Kmown)

Cor

taken Pem a

A nndom Sample & na

mean
Given that th Sample
Populaion wrth o-5.1
Inteswa)
for

2 1 6 , Conshuc a 5 Confdance
the Population man

Soln 116
n 10D, 21-6, T- 5, y
ven
- < +
36-|96.5 H < 216 +1-46
Vio Vioo
imlewal frem
either
(o) 20.-6 << 22-6. of CourSe, ort do@s
th populatiom
mean M,
20.6 t bntain 5
22. means that the method
a 95 Confdent
ot
Yot but we
obtaintd "woks"" 95/
was
by wich the ieva
9P the hme.
a
Sumptim for
Con
eleciCy
2) h owemge monthy
elechi
t250
unit s. Assumins
Sam ple Uofamilieo familie nSo
elech c Con sump tion all estimk
SD 5/ Confidence
intewAl

Con 9huC+
UntS, actual Ran eleci c oomSum ptin.
te

Soln 6
Gwen X125o, So,

1250 1-96 g =
125o t 29 4o unis
T
%
h Populahon
Thus fos 9 5 leve f Confi denCa
fall be tw-earn P 2 e 12206o UnitB
YYan M
n 214-40 Units
i e 1220 60 u<12440
 4 inboduto Some i mcenive for waer balanca
3 T ordes
aCcoun S, a andom
Snmple Si 64
Savings bxana was
Stud e
at a n k's
Savngs accounls
monthly Balana in Savin
estimae e avea found o
Dwese
bn adount S Tho man
m R 200o Yespechvely
e and (3) 94
(9) 24/
()95/
Fnd (0) 90 m e an
the populath on
mlvalS
Consdeme
(lub-) 7 for
Confdena
evel
i mitS wth
Confde a cCounts ave ven
balan in Savings
Av onthly
9 0 / Confderca lm
645
850 t 64 5/290 8 5o0 t 320

85oo t 41.25 =(8090,8910)

q5 confdena JimtS

16 85o t 146 (20D0

64
= S0o i 3 9 2

= 8 5oot496 =(8010,8990)

ImS
Ciri 4 / Confdena
2-515
(2000
85o t 2.515(
85 t515o
8 5o 644 =(7856.25,9143.75) limts
meswa l ox
T+may be hoted Hhat he
desieJevel S COnfidece
gers wider a

increa Sed.
 3 20) Confiden
45
tval for the Populahan mean fos
lar amples (wun a Unhmawn)

dnte n:50,
305 58n m, amd
manollas Aeigh
-b366.86 (ene B 36.14 m),
Conshuc a
Z Conf dence inkewal for h
Populto Ymean all neno pllas.

Solo
GIven n5o 305.58,8 36.91 and

ZooS 2.515, we

305.58 -2.515|36 <p< 305.55t 5)

01) 21212<H<319.04
we e 99 Confi dent hat the inteaval fPom
212.12 nm 3 9.04 nm Contains the mean

nanopi lIaY uigh+

mean
inkewal Aov he Populatiom
32(e) Confdeneo
Smali Samples (o u UnKmew
fo1

that SiIk fbres are vey tough bu

We Kno
Shok Supply Engnars are mating bseak thughs
to Crea Sy nthetic SiIk bses that Can im pove
Car Sumper S to ullet- Proofs
every Th em
Vests or to mAke amfcialblod vessels One

h Stati sh cs
reseaT ch Troup epoS SummM
n 18, 226, S.
 Prewd f h e
for Th ughiness (MJ/) oy
inkwal for t mLan toughnas
Con shuct a 15 Confidenco nosmal.
th PopulatHon
hese fbre ASsumne th

Soln
I1 dlagree qf
Given far n-
n
02s s * 2.1ho
.o2
faed om
fomula for
The 95 Confidento
becomes

- 2 H1o) 15.1<H<(22-4 t2410)

Cor)
<30.41 MT/m
1411 < the frem
inewal
15/ Conf dene that
are
we mean toughnay fah
Contains t
(419 to 36.4 MI/m the Cumet proca
fioes Creatd
To SSible ahfcial data bt
oi nal
da met ive
Thu ancle aSSumpt m b
homal
moderael thu
n
Outier exisls

Cntia unlegs
mo+ Use
n

knded fo
Cam
shafts 0 2 9nd
ec centr City
2)A Sample e breakd

engi n
heve o data may
asoline
ASSuminj Popuhion
O.444 noma
aS.D frem the actue
random
Sample i mewal
a
a
15/ Confsde
delemine
w Cam Shaft )
eccenrichy of
meah

=o2
Soln m-0 h Sma
226 for 9df

O.314

i nkwa for t actual

a5 Gonf denc

an
eccenrcity
o2 0.312 (o.1023, 1334)
3Ten b bean made by a contain pocasym ve
mean diamee f o.5o6o Cm wrth Sd f
A Sumi n data L ee Ken
O O o y o Cm

ndom Snmple favm nosmal populrti en

AS
q5/ onfidena nlewa for he acual

Consmc he bl heri ng
diamekkr

Sol
h 10 oSo 60, 8 0 boo
Small Sample

fo 9 5 confidene
T emwy f PSh mak

Evm
(n-) df
at P5 with
en t
2.262
1 df
CIC) o0S
o.025
Emax 2.262 xO.00

0 . 0022S 6
limfs r e
inlewal

Now the 95 Confdene

O S o lo o.00286

= (o.5031, o.5o$ F)

two populath on
th differencon beiweon
COngideme inerva for
meanS for lag Samples u mown)

, -

Paoblem two nolependrnt Processrs

there 410
In Ces +an facoy weght Im
Samu idem. The avera
manufactring One pocez
250 1He mS Pooduad hom
Sampl the SD
12 Ozs wlile
found toe 20 Ozs , wit
400 ens fo m
Comespondiug figures Sampl
 tu

ctHh TYOce ises ave 24 Ozs and 14 Ozs F md

n th +
/ Confdexe lim ytt dferent
of Hems P.odutd Paocesses es pretiely

CTiven
n 25b, 120, S12
4 8 , 124, S l4
Zy 5 1e
(HI-PD
fir
11 onf.denee mi

( Z 2 ( -t Z

120-2lyt5 12
250 45D

4t (25T) (103 24)

4t24635

(H) (134, 6.66)

between two
f the differenne
3-20q) Confdene inkwal
Sma Samples
Populatim meanSS

(-)-ay <-t<(a7
Con
Valu
twit, n+n2-2
tha tebulakd
Whee
at a level Significah.
reodorn
deg

Pnoblem
ai ven two oups Studmb he
maks obtaine 4 as followS

Tar 18 20 3 S0 41 3l 34 4 41

21 28 26 35 30 44 44
 90% &
ConShutta 95/ Oonfide me newa t mean
manks Secured by Shudents of akove 2 Too up 5

Fer the diffesena

q4/ Confidena inewal
Consmc+ two Kinds ef i'gt bulbs,
meann Tifein
berwon bulbs ef
Yandom Sample 40 g+
ven Mat
418 hours
thefrS+ Kine 9ased he ave
Second
amd 50 1gh+ ulb s th
Contiuow Use
ContimuOU
kind aShd
on the
av 402 rS
KnDwn e o:26
use ThR poPulat cn S.D ave

O 22

Soln from+able, 0.03 - Ss.

FoT 0.06, we fnd
Confidene
inewal for P-H
4
418-402) - 18 922
H-H < (41F -y o2) tl88
 5)
6.3,-H<25.7
He ne w
4 4/Confident hat heinewal
betwee
G.3 25. rs Contains he achual Aifferena
bulbs
men e t i me w hwo Kinds of lig
Sugysis
Theac %oth tve
Confidence linds e
hat
on he stkind ght ulbs
Second kind.
Supei

3.2(h) Intewal esthmathm fVans nu.

Value
Variance f
nomn opuhen
ra ndon Smple Si n fom
hen
(o-) 4
n-1) 82
(n-)

inkNAl
a (-d) IG0 / Confrdente

PHobl em
an
% rst ns tu asolino ConSumh
T ConstmmctF

expermnentm) engine
had
S.b of
a 22aallons.
Ch m e a Sure

992 Confidene newal or h

Consumbhmf
tu tre Vanability T asolin

engine
Soln:
iven n= l4, 8 12, 3 1 . o I . and
O D05, 15

0.915, 15 4 o 1 .

become S

2
Is (2-2 S(2.2)(22)
3 2 So 4-6o

e 2.2) <o*< 15.18

 t u o Vaia mo
3.2 Tnewal
T a i o
eshmaion
Variaa
a Vqlus th
Te and Sines n and na frr
Samble of
independet TAndom

hen
nomal
populhon

fa, n, -i, ar
n,-1,-
inewal t G*
l00Confidentu
a(-)

Problems Hhe hi cotina

mode +o Compese
Stuy has be eigaretes. D ote
A
two brands
Content s mlramS
Contnt 3.1
nicoHne
Drand A and
an
oN cigraks a
Brand B
had

with o5 while 8
01
nicoin Conknt 21 "g° wi th S.P
an
Ar av
daa I'ndependent
myAsSumin tha+ he 2 Sets are

homal populahi on wth eana

Bandom Sample rem fer h a
95 nowa
VAnane, (a)aonShruct
A onfidente the
ehween he man
nicohne conke nts
nlewal
Aiffesen
(b) Find a 18/ confidende
2 brandh Cigaretes

Sole
 the ntevwal
Obtained mre includes Hh

ther e .U
ako
Possibility that Hhe
Teal evidence h assum FHon
gainst
eqnal Populatiom Vanace

Problems
Exh
Sle v o i CO was
So
D A andom Sample Sle nvoicm.
populatm of
taken Som a |age Rs. 20oo wth
Cwas found +o be
The owa Value
0 Confide ncz
S.D R 540. Fnd Value al he
mean
th tauo
intewa) for
SAles

Soln X 3 540,
u 20oO, -

The
mfomati m
given
l0/

G1.5o
54 0
Sx 64
= 6 4 (Srom nosma
and Za t bhe)

populatm
inewAl
RMI e d Confidena
the
m ean

C61-50) =2060
tlo.10
+ 64
200 o
invoic fo
Sale
he Rs. 188936
mean
he between
tena to fall
Populahon
ikely
t whol
2110.10.

omd Rs I 8 8 9 . 3 o H < 2110.10

Cie

Ioo
Obsev aw tields
andom Sample »f Vanance 400.
2 A Samble
X- I50 and
Sam ple mean
nlewal
i newal for thu
Confidence
and 99/
95
Compuk
man
Topulatim
Soh
Given S4oo
n 00 X =(50,
>S 20

S = 2 2
S.E
Z y 1-94
Cof ovel
at 5 = 2.se
t

(1) at 95
X 96 5.Fx

x2 =
So t 312
I 5 o - 9 6

os
146.o8
= I 5 3 . 9 2

2) at 91
X t -59 S.F
=I5o t 2-58x2

16 C6T) t44. 84
=
I55.