bel
Unit - \V ae
Estimation 2 Tests_of Hypothesis
fanametesys : The population measurmonds oP observations
Ge cabled parameters. Tt is doroted by OC.
Eat a, oN
Statistics + The statistiza) measurements op observations
ina Sample ane Gllled Statstics. Tt is Aonotes by eB.
x. £, Sy
Estimate : H BA Staterout moadets Bad Unknown
Population parametey using sample. Stat sks .
estimator: The Rotedure of Velo to dotermine Linke
Populotinny paramates. Using gample Statiaticg, is Called
an estimator.
eu cee) =
2) €@) =o
1%, rane Estimates ,
Bro ap estimations :
fone ane 2 tape OF estimations
1) Point Estimation)
9) Tukevva) éstimakivn
) Point Estimation: Tp an estimate apa Population is
en by a Single Valua Hon it that estimate i Called
Penk estimetion of Reramatey.
x: The mean hide of a Stadout th a college x 165cm. Unbiased Ectimaton :
A Samp Statatic © 5 Said to be Unbiaredl estimator
af population ganametcy ©. Tp E[E)=O
Boe) CCX)
ea ce
Biased estimator :
A sample siataic Bs kaid tole biered estimator
of Population pananeter ©, IP e(6) +e.
Theorem +
Boo that cample inca) H i, UW
Populativn cath bean co.
irc mor eae Nace a Random Brmple
7 Rom a prpullaten toda moan 40
een x,
n
Vv
e@ = &(4 2%)
re arise eae Xa Wi
nbiared estimotoY oF
ECT) 2 ke Lor vat ree -----t¥)
2% [EG)+e Go) +-- - + EG)
aa [ut Up = == ut |
et fou]
= Je
6 Gomeaen “Teovem : a
Shoo that S$” is an Unbiased eytimaboy af the
Parametey o-* whe i _t Ge yi
o.. ai Wl Tey /
VrooP + Considoy
ee (tea by
ee se = [oy oH) aCe wi ee cee a
ee
ae ree 22 G41) Csi) He (ent)
i a ; 1) 9 C5) B61) +h (Z— 00
2; 07-0 esp [Accor - _ afl
ee EO, cui) 2 Ot-—4) |¢ fone) + (%)-0) 4-4
4 (Ta)
Aire ihe Ts
s pHY — 20% if) 14444 44,04)
=
+n i)
Ty Cl eee Act ne ey
y
me = MAM 4IG+ ety
» oe Q
= o OF UY — 9 (
w
Houevey ee Per 4
Chaka ea
came)
Be rg
ew) ane [no?ne2|
_ 4 7 @)
ge
YL
ade
eset’ ae Most efficient Estimator ¢ iiss
2, 6), 8, ave tivo unbiared estenatray op dua Same
Populatin pancioter 6". Tp W6)2V (6) then
6, iS most efficient estimator ae ©' than
(or)
All possible unbiaked ostinatosy ap pananuter ©, the
one with the Sialllest Vaniante (4 Called most egeictert
estimatoyY @.
Gee Esti «
A Good eatimatd a the one cohich is as clore to the
Aus Value op parametey as possible.
(or)
Ropestes of Good slater:
a
D censistingy + estmaby ©, conergs 0 6, #30
then é 8 @ consistant .
2) Unbiagednes + The estimator & {3 callct unbiareet
i Efe)zo
2) EAfiCheuty ; A Stectistic 6 43 Gaid to be Amore afficient
“hhiaxed eatmetor ofthe pararctoy © thanthe 3¥
stataic @, if
@ & w B are both unbiared estimahs oF ©
®) vc)
whee PS LS Cyr)”
Narkinun) ec of extunake )
re rth orate “Glimaks & with Chot\ probobitik,
for astimating population mean wig f =
= an
4
Gre op Large Kmples .
Sinibrly € = on {2 Pov astinaking population Proportion
‘Tn Gre OF Small Bmpls , EC hy, Se Ay estivekirns, da
Bample Size Ne (Men y”
The Cnfidente Limit ana
G2 4R) - (rae, G Fre )
The Confidence = Co) 100
—5 Tp p & kaw unknown then sample’ Sige
“ate aes ect)
5 Je Pu population Proakion fran
fample Sac m= (S)" (PA) PF
TF Confidence 1390 % then Confidente lawib ase H+41.46Y ‘]
tas, then TH1-46
qa7. then ZEDS8
99.734. then HED.
Problems »
SS
D what is dhe Sige of He Small’ Bamph sequiied to catinaté
an unknown propastion to Within G& maim ery? Of
0666 with athart GS, Conkidence .
go Marimum Enoy & =6.06
coniauncoP th = 4s, = Ui-d)¥90=95%
[he 6.95 = b= 6.05
S =0.015
wet
Confidon(e ts 457. thon Bay = 1.46 Pa 16)
net 4 0 2:
4 e987". 26618 £261
nN = 267 ‘
2) Tr we Gn asest With aS % that the marimam
JF qc l-@so8, By, =, 96 Chor 95 4.)
Mantimam EVN E = eu), - (ee)
0,05 a1 GG | O2%O%
‘ oo
B.2 0.8 K CAG)
(6.05)
MV
So my
Y= 246
mae i eee Len
3 |Suming that o-=20,0 , hoa lange a random Sarnple ke a"
“taken te conert weit protelab ty ots that the Sarple moan
tall nat differ fon the tua maan by more than 3.0 pola ?
i) F
won.
SL GNer watinum Eyvoy B= 3.0, gr = 20.05 Za, ae lt
n= (#4, ne aay (061)
= (146 <22) = ety
nem te Mahe om
4) tohat Js the marcimam error one Can expeck
Probability ¢.40 when Using dhe mean OP @ nando gemple
Biae ney to estimate fra mean oF ppulation with o7 2256.
gl nay, The preobabtlily -0.40
om
oanse BD oz (28 ahé
Grbidena Linck =0% Buty = 1.64UF [aor 64)
Maximum ornon € = ey, q
2 eg x bebe = 0327
ey
5) The mean 9 the $.0 aa Popdation an Il,195 # 14osy
sepectisely 1p n=50, find 957. Gofidance intevual fer the
mean,
Sa_ Mean af population 4 = N195-
3.) of Population o = hesy Gaus
n=50
Morin enag= Ey «=
Ex for 95% Confidene = 196
21.96 x HOSE joata4
A \G
eonfidente wshowia) = (Bh, K+ a)
= (ISH agqq, mas +3847)
Se
= “
Pay” fF 6) Find 97. Gonfitlence Limits fem dhe mean ap @ normality
dibibuted Population Prom thir ho Pollaoing Semple Gas
taken 15, 17,10, 18, 16,9, 7) 1113 1Y
Bh og Sar tOt HIGHT A THIF _ Ig
= aS
aye ae
nA
oh [Gs-2% (it-t9)% Clo43) +--+ cues)
. = [yt lb +S 425444 164 HUF OF 1) ale
Ce > aE ee Me
= I-46 3
yy : e s-\e
tap = Ns WOT i ae
Fe MMe Foyt $2 ie
Confidence. tunity ave T+ %S. = BE 9.26 = (10.14, 15:26)
4) Aciindom Sample ap loo teachers in a longe metropoltan
anca sertcaled a mean \w2rkly galavy of eS.487 with a 5.0
Aad, Wrth. what clegter of Gontidente, Gn we oere
thak the auercge Weekly salany ofall Bachers in the
wakvopelitan aaa is Hus 472 to $022
SBE Giog t= ts>, ras, n=loo
ee SRT
Th "Wo U8
mp €2u7d then 2 WHIT 3.25 any
Uk
$e K =Gor then 2 =e = 3.5 (xr)
P(-Be0 42) = Ata 4 AC) Bara novtnal ablep
plur22 25m) = ACZIas) +A@.05) ea
= 0,494) + GU) relent
Bech
Confidence = 6,992 Kt % = 19-82 7. Bind 952 Confdonce intorual fry mean of-
Noval distibutin tt Varian 0.25 using a
Mrple af n=100 Valuoy wih mean 212-3.
Salt Geen n= l00
mean op sanple Hi) = 212.3
SD So = (ans 20.5)
(1-0) =95 7 ZB eas
? te
Max Eoaay CE) = 2a,
2
Con¥ idence iwlavua) fay Nas
a Pau, Za = eras ea
en ot cet sa Ge
ala. 3 — 0048 2ML AZ+ 0.098
pee) eee t e) lhe21 Oh
aA aandom Sample aes onl en ercienc|
dowiation of S. Ghat (Me Gan you say about ha
eons oth IS% Confidence.
Sample site (n= 100
$25
Calais < hy, = 196
Marueneery
gd’ “Giuon
pers e ey, % Ke
"
MAX. €9n09,
N
Cree) eee
) Fe 20,98
automobile frsenance @ sandom
y costs hada mean Of
dowiatinn of R38. 62-35.
to tho oud querdze
ana. (se Can awent
Ps. to 2
1) Ina Stady oF an
sample ae $0 body mopai
Rs.472-36 anda slaudard
Te w -& used as Point estima.
tohak ) Type een: (accept Hy iohon it 8 false )
a the Null Cypotats 15 Hy is False but it i
accepted by fast procedera, thon the e707 J Colled . vil
Type 2 Ay (oO) P-€9H0n. Tt 1 also Called -——
as Consurorns Pak.
cviti@l_ Reston : ;
A vegon Convuyponding to Statiatic ‘tin the Sample sprce
1g’ which leads 42 the mcjection oP Hp 13 Gllled a3
caitica) Region 1) Rejection Region.
A cuxicr cakich Loads to the accoptance Of to 4%
called ACtopheince zag.
]
+5 oh
topo
eS fied
Regi
/
“ui | Vir
zy, Oo 2,
Teno -tailed Test One-tailed Test <
2F tho alternative hypothoris op the form
Hy. 2 LLU, (i Zp OF) Me ste) in @ test ap
hegpottosis then dhe tat proces (4 Colled
2 of
The tailed last. a cs
~ a Qo ao
2) kg — Tailed Test :
TE the higjpblker’ Alternctie. Mypotess OF the. ei
form Hy 2 Lally tna fst of hypothesis PA
, hen the bt 4s Callao Piglet Tailed Test. Oo & 9) Let Tailed Test:
De tho Alternatixe hypothasis oP uf |
tho dwm Hy) t Ate Ly in A aoe
task of Aypottasis then at is Called Loft eae catia
Pecans of Testy oF Hypothesis.
Stop At Nall Hypothesis :- Hp + tt = Ho
A Hypothesis wrth no digfetence 1s Collad Null. Hypothesis
ctap 2: Alternative Hyputhasis sy $e7
S=— st is dofnad ax one eta Follooing
D Hy eto (urtte cdo)
2D Ws: ws Lo
yd Wi p< tr
step :- horse OF Sighifiance
memes tales ees i/o OF s7. ov lov
Table value ey, OY Th
Sop y+ Te Tat statistic
‘ee eG)
3. ect)
(A
Sheps Conclusion «
ae
Tp lel < 2a, then ty is trua (or) Hoe accopled
Te Il 72, ‘ton to + folee Gy) Hy we sejected
ae
So, We tho fy 8 true 7
Cota valoty 2
——— Taye of significance
- if ot low
Tiwo tailed Reg |= 258 1.96 cea
ae eit Sq 22.3 ious (1-28
Lott tiled — %e2-2.33 GUS LB
Tesh Lene> Slaps ficanee 6p —SArmple ade
Tet of Significance for Lye Samples
dp the Bangle Rize 730, then we Consider such Samples
as Songe Samples .
Fav Jorge Samples the Sampling dsttibubinn oF OW
sjatstic 13 Appronimaldly a Norma) dlixbri bubinn..
Suppose wie Wish do textthe hypothoals that the
prebabiltty Of success in Buch trial 7s PB Assuming. w® it
be true, Howan 1 athe §.D o the Sampling
datiibution of: 0.0f Suceayen are Mp 2 Py
goxpeckioly |
Tp A be the absevsed Moog Succedses in the
gampe 2 Za the Standard Normal Variate then
es
=
Thus We hate the allowing fest of Signifraance :
i) Dp 12) 21.96, tho difference between the obgevued
and expected ho.of Succnes 13 hat signifraant. Je lel 146, the difterence 8 Significant at
Boe LOS
3) Tp lel 725%, Apo dlifdexence 8 significant at
17. LOS
Assumplbns Poy Lange Somples :
The following Oe dhe axumplions
) The aandom SAbapling. distibuko
properties OF the normal Cove, . This Via
hold good in care OF Smal) eupled -
2) Values Cie, statatie) given by the gaynples HVE
sudgicentty Chee tothe population dJobaoy Che,
Porcrwakey 2) and Can he ed tn ts place dor
Cdadating the dardand een (S-E-) F tha
under tohich
ry op Static has the
nay nok Problems :
$8
) Mean af population =0.100, mean of the Sample <0.742,
Standard dariation of tho Sample = 0.040, Sample size =(0:
Test the to null Fupothasis fer populstion wean = 0-700.
4
sa.
Given te = mean af popN = 6.7
K = Mean OF Sample = 0.142
= Standard dawiation op the Sample = 0.040
N= Sample Spe =I0
The tost statistic 2 X-se
Thy
eet)
-~ooy Te = OOM - oem
OO! = oe
4/6 OOo bo 010126 ai
soa [2] 71.96 ss l= 6.05 97.—-LOS
Ral <1-96
pea Sarmple is Not from the ta > [ef
Pop lhore mean 3, OWT.
2) A die ix txgeo} 956 tings ard 4f terns up with an
ewen digit iso tinos. Te the dio biarod q
sal’ Giuen N=256
P =The probability of getting an euen digit @ ov 4 6x6)
a
Aes iap’ ele
|
ae Sh 22
LZ
w= np =Qs6)t = 18
T= (npr = 88x Lyd = fey -%
= NO.OF succones -150 [reat enumler of |
er
oD 1) Wall Hu podhaxis u,: To die is Unpiased]
2) Altevnative Hypothoss H) 2 The die is biayeel
8) Level ap significance (Lod) ; a = 0.05
W) The test siatatic 2 2 Ao
ae
1S0-128
Peal 4st.
g = as {led 1% |
+ 21 rh96, the hall hypottotis 4p tas te be sgeuecl
ak 5% 108 w wwe Crclude tat tho die is baxed.
J) The Gin (oas es¢ad Yoo fimer Gnd Tekemed hands 916
Hires, Test the hypottasis that tha Gin 13 Ue Unbiaxd «
Use q 0.05 Lowel af Sipiffcan@.-
gals Biter) n= b00
"Pe piotability of getting toad =z
Y= |-P = ee eh
T= (nPF = fwoxdxl < fics -10
go = NP = YO xt -200
H =ho.oP StecCowes = 916
1) well Hypothesis Ho: The Coin 18 unbiated
's) eftemative Hypettesis H,. The Goin ix biaed
3) lowel OP significance (LOS) , «=0.05 (el 146]
4) Tho fet statistic 2 - a4
ae MG -200
lo
a
=o Ce
= cea
lal < 1.96, the hull hypothesis Ho has te be actepled 2
tte. Goncleade that ha. Gin ig Unbiared. an
o~ iid Ay
Under lange Sample Tesz, Loe call gee foun important
OIF to dest the significance -
) Testing af Significance far Single proportion
2» Testing of Signsficanca Ay difference af proportions
3) Testing af Significan® oy Single. moan
4) Testing of BigniFi ane Foy diffesenca af wmeans
Task fr of Significance op @ Ginjle_Mean) Sample mean
A Random vanicble op size ni has the Sample
mean i, Whidh & taken From the ptpulctio”
worth mean LL 2 S.D and Ha Prpuletion
mean yr has Rpecified Value So
) Nal Hypothesis :- Hp: Mee (H= 4p)
2) Altevnabic Prypothsic - yr
4 3 ftu (HH )
Hy > Wet (7H)
Hi mem (wed)
3) lowth oP Signéfrcance -
Table Value Say
Y) Text static
Cees
UW
he 5) Conclusion
Te J ct fe Tree
UE Teal ee Pay Ho fabee. . fs) Si
Problems :
DA ample oP Yoo tems i$ Aiken oma population sae
Stardnd dauintin 1 10. ‘The ean of he Sample 1S YO.
Test chethes tho Sample fas Come fom a population @idh
vouan 38. Alse Glastate, 45% Gngrdence interval for the
population,
a Gwen W= 400, % =o, fee Ey ae 2100
1) Rall typothois Ho: 23g
> etternative Hypothesis Hy . 90439
i SY
Be OCme anno (led = 1-4 | (aera
ot
“) the tat statotic 2 _ Hu
2. 40-38 A "a
an 5 ee
°/ a6 You
2 as 21.46
Sey 7 ag OF ees Fay steed in
“We gojeck the Null Hypothosis Ho. Met Aaa, ite
fees
fe. The Samp 1s not Rr dhe pep)? chose tnem 13 38, |
IS% (ongidenca interval 4, G Bhan a pie ete
Yn NA
ip
NTO 16h ee ss ie so Thel
Wo tb Be a)
= (Uo - Ib x 38, Yor 19ey se)
= (Wo-0.98, go+o98)
= (34.02, 4098) & Jn Cy Yardemby selected hears of production, the
Wan and Hp “Slarderd deviation af dha number OF
acleptance pieces poduced by an automatic Stamping:
madinn axe y= 1-039 ahd o -046.
At the 0.65 Level of Significance dovs hig
Chable us ta owject tho hull hypathosis 1t = 1,000 @gainst
dra allevnptive hypothesis Js p07 1,000 u
gal): lob Given Z = mean of dhe Grople = 1.038
: J =tmean of the pop” = 1.000
T =5.D. af tho pop" =a." 6
N= Sample size =6Y
) bot dhe lukt Hypothesis Hy: ge = [-000
2) Allternakive Hypottosis Het us Looe
ae LOS: & 0.05 (He =1.6Us)
| 4) The fat Setatic 2 Hse
aK
z_ 1.03% - 1.000 a
Tolyey Ot O38 0,038
ay OWS al g25°
= 2,082
% =Q9.082 > ous
2. le vejert tho Null Hypothesis Ho at S7 Los &
Gndydo that themean OF the pop %127 1000, -
3) A Sample of Feo members has q mean of ap 3.4 Cms er
S.D 26) Crt, Ts this Sample has been faken frm & large
population ae mean 8.25 cm cand s.D. 2.6) cms. UF the
. popudlation 18 rpema) ard ity mean is Unknown ind the. 95% Feducial fiwits oP tree means. 3
als & a Sr ae =a
ie EN 7-900 Ab ee tan resp een
L =3N Cy a-=9.6) wy 5=2.6)
) Null typotfesis Hn 2 pgsune that he S4mple has been dracon
from the popn with mean se= 3.25)
2) Alternative fypolosis H) 2 Le 3.25 on = Yming
Pop mean (ir) = 1D Minds D Nu Hypothesis Hy: s2 = 0 ming
2) Aliavnalue Hrypotipsis Hy 3 16 10 mend Gare
3) Lofel of Significance ea
(ea ee oe
fable Value te =4-64S”
u) Test etatist(>c
ae (On| Smee
Se Gs Ube “Te eye | = oe
Ss) Grtluion —
a fe oe
Izj=alS yp (2B) =b64S
Jz) < [2x]
2 Hp ob fae s, $0 = lO bing |
BD A Simple ap cy staderds hove a mean coviglt OF
70 kg. Con Ha le gegarded a a Semple. Fok &
Population Wit mea (eight 56 ER. and atandard
dowsiakinn 2S E98.
, *
gal: sample size 6) ar
pep Serle SD od = 26
poph mean Grd-= 56 ERK
Sample méan &) = 70 kes
D Noll Hypottesis Hp ; jross
>) Alternative Hypollasis ys fs +56
Thon teed Text D Lower of Sigs Ficance
eles 7
Table Valo Fup =(46
Q) Tat Stetiatic
aoa OK
Ue 2G
Ss) Conclusion :
zU.Us 5 Seu =bAG
l2) 7 leap, |
Pe il et
35]g
= 2 LY-UR
Ts
=p i& fae | Hw true
Sit Sb EB
Tho Nall Hypottasis Has Sejecteded . ig
Test of aguality of tivo means cor)
Test of Sighifian@ differen Lotuseen favo means :
D Neale Hetpatlarys 2
(ok X anh GF be Sauple weahy op favo independosd
(ange Ramiples Sizes h, @ N) draum Prem two
Populations harwing moang 6% 2 My With the staudard
Samm Ben ra ize ti eae rie Conor (nos
420 population means axe equal .
1) Nel Hyprthasis :
ie 3 Skt Sa ea) an es
1) Altarnctive typothesis :
Q) Hs $I (Hy >4, Gy) HM aH, ) [Tivo tailed}
Hsu, 7s (Pégld— tailed )
C) Hz Mes Cleft toiled )
tii) Lowel af Significance (ot)
Table value = 2xy (or) By
ww) “Test Statistic
Sa ten E, We of = %p =o Gey)
ieee
—_—+t+—=
oo at
ea
o fet. v) Condasion —
Ge lel hy,
Te (2) 7 tu,
fy
2S. ths tee
alse
ery a tue.
co! Fo,
Beblany =
Y Th means af fo Lange Samples Of 81203 [060 anal g000
membors ane 67.5 inches and 63,0 inches “pospectivcly. Gan
athe Aamplos be gusarcoel @ daaon fom ho dame Population
of 5,0 Qs inchs.
dal), ok u,v th We the moans oF the tase populations .
a gawyle Airey,
Gwen “V,=1000 , N= 2000
Sanple Wiaty 5, = 62.5 inches, = 68 ‘inchos
fo pulation S.D, = 2.5 inches :
) Nall Hypotesis Ho: To Samples have been doaon fom
the Sawa popelation af S.D a5 inches,
(Eee ety he Om HLS) inchos
a) Nizvnatite Hypottosis tH, . ty ty
B) loual of Sigwificance 2 4 25/. (24, 196)
pte tt Aabstic 3, 2 = Hi
oo oe
crea
oe ee a
Le i
Qa, AD @sy (t+ La | ;
ooo ft ee Oe!
oa a 9005
eee oss
ae 0.0015,
Cass n ONSHED @sy=6as
0.069375
age OS
ee TT a a ace
(qost 39s 0.0968
7S 5.16
ve a aaa ——_—a a L le) 7 By, 425%. (1,96)
ee 1 het to i false , VD tue yt by
te, The catcolatsd value aP z > Khe table Ualuc op 2
Te Nu Hypottesis Hy fs Sajecked aes Loses
= Gondtide tat tho gamples ape nok deen ay
the Samo population ap sp, as inches.
clohen the poph proposéions FYB axe™Enown
has Sample Ropoaliong lp Q b, aye Known