Chaptcr 2: Pirameter l$timation

1 Inferences on the Parameters

OF oltlr Nsr lir{( qtr.$iors M sltr ld ad&cr5 i" uv r'grsion Nulvsis Dn'blcm

N shcttur ! and ,\ ar hrmLlY Nor i{ied Spfl ili( allv n' R hrcaL rtgrBsi{Jn problcm

tlds qu6tn inNlai$ trtorl',{ktrtg Lor or no1l] ' 0

For cxampic. if J = (). N lr.v.

^(^) - &t o'2!': &

' v-N(" 1)

'll,{'. th. lt,bal)iLilr {lL*'il)tti,i'.I )' Ls n |l)rixl'r ol X S! v an'l I htrwr ro

;:;,:;,;.,, ;,"'",""',,,.",. G^. r - ^,

a, f", n )Example r' h rrf r&al x4 aht .xnlnrL i) - | uoulti nryLu that nr'qc nl

rdjtt art ro! t,rnrl! G'sotLal

Tlua dr hvo nNtntr1 lrdls ir! rtrd[i'B irfrrdt's rbotrl- r'rr(r{ypolh.ses l.sls dnri CoDlidcDcc nrtelvals llowNd bci)n 8'nrs trhrld with

r[Ae proc.drre" sonn' *sn lnir'rs trrtlh be talis0cd

1.2 Assumittons

'll'cn,rr l*sndll\ n! Nsutrrdi!'s ih tr{'lru s'ris6(l Th(Mak:

' Th d.nr ntr tr\ lrlttr I Lrr(krr' 'dtrPLc 'r Luu""''i fxp'iirx'i + 9t(S

. Tlrol,i\\nlnrsJcrLli' t lJ'irtori or c{[ ol'lnr F'ruM!'|l" Mshouu ? lidrx'' datz

u{ nn,liirn,.tari\arntrN {l'jrnsor(x v)t?lLi:l rron' flr's$r: sroF

! y {or /,) ()i )) is liltr ar h,f ls.d !} x ovf r ift drsc or x uluur tn tllc snpri, l;d4l/'i(Si l N. dorcl Dld 10 srY rnvtlnrg abo eh{i halD rs oubidc thsi nngc

i4 " " rJ "'hd '+^'^*5

. Thc dislibulion of y slris for

tbn) stnndard dcvntnrnj whi.h

1.2-1 How to.beck lhc dsumptiods ?

.od psticulN x rdr hN ihc s$ (popula-

w) .dl r dns N ih, lbDr8rmitv Nsunrtrion

The 6rsl tro sumptioM (ra on zrtion dd ndcpcnd.urc ol rh ohscx?in,m) cm

bc hdd to diclk aft cr lh data is collcct.\t. so ii s nnpotunn io c .IuUv d6isn md

.ondu.t st ies t{, cnsur i. sdBn.. thal ihcse ssunDin F{e satGlid TlFre.realso {ays oI.irL{kn{ ihc lL{ ihhi rssnnDiil)$ (lin{iari$. rornDlity, dnd .onstdl*aftlard dcvisrion) {s M will $\i ldl{l

. To chc.k ih. lnrltity asunrlL.nj r rlJ trrko$r ilLii t[ tn)i,'ls ,lor'r f,ftrraN sort of (u^r{ shapf. FisuLo I slors ,ul .yutrd.. ol dtrttr for wli.h this is

not th. ce., nLuins ihai ihc lin.rity Nsunrpinr is lnntrnid.

'"f,";w ?."

violdio 0l th. liNr lr rstrtr{nior

tr$unr|niotr is n litrl. nrre,o!'Dli.nlod

To.hochih.oDinntslariuidc(i tu!strh|nn,n.l,ukntiMvfii..lsprc.doI thc ponns (1hd is, ihc spra(l of ih| v vrhxr)

^nd Rr rl it s snnih tor all

X wluc Fisur 2 shows u, ,ix$,pl. of datu lor vhidi ihs a not ihc Gc,m.lnirg ihd th....si!nl stand d d.virtion rsrnDti,D is \nnrnrl i. rh.ris hL{orceoncity n, ih sdrplc.

?

Frcuht 2: Viuhiior ol rho.orBhlri slandrd dNiriion aunDiion

1.3 Hypotheses Tests for ri : Regresion I Test

$( wil pcrtor! i*! iyDcs of l,JDotlnsG icts fd J : REgrcsion l' tel' 5trd

A\OVA ftesl F'a' v"v'll l.'tl'"l{n''

F,tr isling whotl.r v dkl ,t Nc asocidtcYl tr hN' lho following i'o hJpolb6s

NaU nwLth?s( : lJ0 rJ = 0 - Y dnd X hru tro tiner ::$c'8hotr

ALlemnhr.: hlt1bu'rset 1/,,I .J + 0 = Y .tr'l X trre liDdulv nl6tn

The .bovc .li.raliv. hlpolh.Ns is tr lq*idd ouo' D0p'ldins

{.un! {,siir'lni nh"i l'\''ih$'r''>Jr' fcrr'tiw ^-"4'"t -d.,rr<0 (i.o tqt !e dvJciatilr)

Fjxanpr.2 tlt thr 1!.idnritrnk annpLr" rc n'ln Mvt b test ll ' '

) <0 sinct

utghl Bl mllmlc adctnl\t ln1

Thc l6t stdiisin lor n\iins iln tLbo!' hvpolhcs is eivcn br

tJ0/ =- .'/,

_rLdr// (l)

' \ P""'^t''- + t& "L'''^r'd

(9f )

Horc '$ $ands nr stu&rd oror md 1,., st!.ds for a t distribuiion silh r 2

d$ftr-s ottrtrd.,n h .hn bc sh.wD ihai

Ot,,,t =\lsD

t(x, xl{f,,,n nN un \lsE rtr,t vsE. q1l] rGr ih. i,. tt I s).

oJ lhe Y, s. iI a atlo no,inoLht dtli[ ed

Lt i]" N(3.v(1)) trh.1Rattt r. siaa i| a a thruf 6rn6nntian

uth nnn E(3) . iJ dn,t Mnan.x v43)

(2)

LA*..&)mr'! atutMt nart steanb ;J fu,^NoLe 2 : A3 olut!', Ure L ttati.tic tell:

t ^*t v.,l' q A.

Rene,DLer the dehnirnD.f ilp Evrlue

D.tibitioh 1. ?hr I ,al'r n ,,rr pnbah NtU rl g.rhn! n hn shtLstu tahtr at LNt6 eltftnte 6 rhe di abktun, { tk b t^.e

so if ft olbdvc h t6t stariBiic valL'c 0f / = 2.2, rlur tl'c D'uallrr is rlr pn'l,hl,ilit!

d Bctnrs d, valuo $ovf 2.2, Lclo{ 22, or botl, (d.r.D.lins { th. sign or th.hurrativo I'yl)orlLcns irji>,<,.i) ul(trlht(,'l {s if dr rnll hrDoil,Gis *.r hn.. /ltxft ott that iJ tI. r tat, thcn t h6 u t tL&triLnttor @t tl..t,td oJ trdla,t Er'!ta .ttu,, the Ee&nt d?a'u.\ ol tr..nan J'Ln ttu ANavA Lltt)Lt H.r. s tr quick

'oviowol so'n. of th. Dhp.,ti6.f rh. I dnhibntionl

ll is syufthic a cdtcrd t zdo. wlth Lotl' i t)oslrivf rrlr N8hlin.Inil.

lG fxr.t 3hUc is d.n,nnnni bl iis dostu,s ol frcddn.

. A\horgh it loks likia{s,la(lmnnd distribrtion. at (lisLiburiotr I's thickdr

ttls tld tl( nonnal Howv.r, rs ih. d..gr.c\ oI f'md{'n scts lugc,. ihc l s{ts.lcr io a stndrd nofl.d.

So il our oL$v.l val,( of / is 2 2 u(i tlt dllonhtivc I'vPotl'cvs is iwcsid.d. thr

p-raluc is llk rcDrl,ildl rl $ov. 2.2 and bdo{ 2 2 lor t I .lisiihrtnr with ,lR^

d.srs of frdioD'. r,prNonnal s,rphi(dl) u lisuro 3 Sincd tho i dinibutiotr n

svulnehi.. *. irt)nJly ntr,l il,P t nrbility br jLst om of ihN tils. usDallv the

oN on tlt rishr. (lrl'rL,Ld'bh ii togcl th. rotal lor thtlFvthtr (Olcoufto for

i orcsidd n\r wo Notrl(l h nrnrr\kli in onl! h otr,ltihrl pobabilitY. btrt likr *o

sdid. K uually ,lo tv. snl{{l l(sls tr' r r0slsior eni(xr )

4-20Value ol t

Ii(xiRE 3: ftro iail(al ploliabili|\'.l

IIM.'lon( l'rf r...sr,5tniiii(rl$l6v:nf nJ'alculs!'Fvnlncsn'us wcofni

L f b uso i t ial)h lilc llLc !trtr irL rl'. buk!fotrr rdibool ro 1rv ro lrsul oul rl!'

p\ !.. ANpial /rahl(,hkf rhrorf showrn'lisu5l'NIiodifle,ctr(.4 vrl'r$. $nl'nl rn aDl)'ouitb ro\r rlro t!b[ shuws tlnl n\l statsn'

ralu$ rhst corr.\)o !,.crt r figlrt tLilD!)brbi ts $n ''n N ihis i'lonftirn

r!hsutdur rr al)r)oxnDdorisl( riil l)r)ltl)ilitvlilr ! rdna

dnd!.rhcn d.rhl( t[Nrisl'r tr'lrJx)l]abilitl iosd rlLo p r r (for trLwo'$dd tdil

l\icd s.. ilLir lli. D \zlLr'])!h

. lf lh is htrc..u, tsr rsristi, ('llrrobabl! l)'nnncwhat clon b zoro sdotrr

! valuc wiu pdi rlv lr hrs( *h'ch m*G $$vr' {L'sc PvrrG r:Pr$ni

litilo of no.vidorcc as*n //o )

I IlirsLi,l l/, is rtr. rhd !trr isi shlisin villfr)l)'blv lir fart[cr ilom zcro

ar'l so oif r>$l'r'rill1,'.bil'l( L's all[ vhich ds) r'&r\ v'N (Snrdllcr

p lantrs rt,'is.r1 rrnf friritltr( rgrn'$ //0 'il in suDpo'r of //i )

,UI 0.100 0.050 0.025

3

5

3 073 6.314 31321 63.057

22.:127

10.215

12.?06

2 571

BL.r ni nnd ilDl iI elt*sr savs

lcst is 0 07,1, thd in..Ds it cd.uhtdld.ubld il l,.f{r. n! nrs d'c ars{d

dut il,o ]}vtrluo fo' r hvcsidul n'lrNior I. righntlil n.bahilirv of 0037 r .h.iy

t'iMlly. q. n(!l !) n'ak: r ddsion JDui whcihcr l/o n 'e!n'al'lc.

or Nh.illr rrhlvo.noqh .vn[)n.. ns,nrs /t0 10 h.liov. //" nr$.*t \vo d. il,is l)v .onDi,ftgrh. Fv!h'. t. onr rhcon sisni6(an.r l.vol n t,i1.n lr {rt) nnd D,rkn,s r d..iion thc

. llrhc Fvahr.i3 l(\s rhan or fqMl ro a. v.ni(l //0 atrrl hdi{{ //, instc&t.

. lr ih. pvJ,r is s,on., r[in a. k, fril io ,.jd //o rnd .on.l,d. thd l/0 nrMnablo bacd ou ihc data

t/./In -od\r" i' ' l\t 'l'-/qrl'i v illJind r 'l' tr dr.'b

follo*jrg:

If m d€i /4.. thcn wc rc concludnB dkLt v is lnt h dtpcd.nt on X.

. llwc tail b rojd l10. d'cr lvrrc couclLdn'e thhl it 5 n.un'!1,1,. tl'ht l, da:Dot h*rly,l.p.nl.n t, whid, Knld lraD tl'rt il'c,c Noukl ric m lreni nr

irlin8 t n* ,1 h prcdnl f

Note : IJ E" iJ > 0 utd ue rtrd tL. it MAfNAT ttrtll tttd v "r.1 X mind.?end.nt: (nautu J < 0 L. v ant x hne d nesnttue Nso.t bon) W. .oa anlu

b'rl].d. Ust s1!d he dotr. t?rt s stnns eidenc. thatv otdX daes not hne

1.4 Weight-mileage revisited

Enmple X- Lr:t\ eo lhn,4h ttL u'e ne6 ol a tua'tilled ftgras\or r'tut Ior ba

r Siucr wc rr'<!$nhilg a paiiculsr@ lot, rho rundoh sdDrpling Nunplton

rE] not I'c $rislioJ. How.vor, mgdlivc lirur so.iatior.bciwr *ci8ht-

drilcagc likclt lo hold i)r r rsDdon $l€lion of ds

. wrisl,llil(lsu obsdlrtidB li,r 'hllborl cth r bc {i\u!d lo lt nllLrrlc _

Mldr'|l*drrmqlrr" r'd.vd iur'd 'v'J lrg|'nldlrll r'- IvLttu\t'-J* e"LI v"e'lr.s br dnnr' rlut 'li 'n'lds

!du-'hay d nurnal di"srbu'irto in tr'"

Votical sDr l olihc v $1'(* rlpptlxinuhlv snnihr tor dll X uJt'x (vidc lig

? ir l(,ctm l) '/ i d,onslAnr raridtre asrnprnD seFs r' he srisned

,.( nil.asc d uiglrl hat no lni.ar NooDrror

,,, niihrs, aM uighi aF lincrlv rjscidiri

T6bL I shNt thc $ttwnr'ontUtSo, th. i6t stliisti. suld bc

+,4t

for tlr Rigllttrt cusc d3t5

l - oos:'oTD6

7

Pnlictor E\iinatc Si E,ror Lkl1tr. /'(> r)2.603

0 050 0025 {r.{JnJ 0.005

17 53i 3 30. lt "'-

Th$ is dr au Lololt .r2ta Jsog 32tu, df.r d/ = 23 r. :$ ftnloss :

fr.,nr r,, '-ririt 1 1,, r! !l't r l .i!i Ln .

3 '",,l 'l",r"L|.\'1. truI\li f0or rrdion {4) or ihc rl,ovf stalisrn r ill lr 2t- ,- :23

ir,, \, w rl,2rdl ll,, 'r,tr6

23'lo# r 714 2.000 2.s0{J 2lr0i 3.135

5: ltight tdl Dr! lJil,lid 6r i 2il

1.5 Analysis of Variancc for Rcercssion

l. ordor to pcrto'i, iho L.ebi.. 1. tci df ,, $t nr''l r,) haw

,^nalvsisolV sxc (or NO\A ) tcduntra th.o fxioLln'oi' (s'.$i(tr'ana!

-8.tu

so rhf I|ralu. $!uld hc o

Si'n..nrpnhn.isrhoni0{hu,(.li:ssthd0.05).{rsn.nsl|q.(t so d,oucl c

ditrt tl.n jsrt,lrs.vnldtr..f lntrr as$liaiion Lctrr(r Ncish( in!l nil,.agr

lllrq,vcr. ilM.l, r\hl //,,,iJ <0.11ron. sidcd |}rnlr.Nrld rlso l'{ll!\1t !0(in( thc, $atistic is rcsrtivc), wo still wotrld I'n!.

'!ri,,.r.d /4 rxl ..n.hdc tlut

th.r is stong cvidircc to iJolicvc tld nilcJac and wiisht {c uosdinlr rsociat.d.

Fignrc 6 shoss Nlnih d,n{usiors r ntoDrotaliors go r,,g0du n,r r t"csidcd)

r-t.l SDms ol Squr's

,\ \OVA ftr ,osi lsirn' nN'lv5 111" ' sunLs ol vlud 's J

. Th. total (co.rccred) su'r or squarE or s31t no:L{tr$ iho nhl-

' ','',,.."'. Y\4 kr'l'r '11! I

ss "'

= t{Y -r/f

resression sum or sqn'rcs ur SS I a" DoN$ 's i[c d$ilitv nnc d

Noie I Cnr[lvsr'rkr's St"*' tU"^ l'-' - ' 1""d!n mhavcdonchv

br s,l' r'r \.ni'

I

I

I

ErcuRE lj: Rsnlls d nnorprcl'ations or a (rwcsid'd) rcsris'on I rs!

iNOVA 's

tr strinrn,,l p'di ' of 'lcconr|nBirl8

ih variabrlitv i nttrt r h

,lar.$t nn.ltilliL.rr n r6 I'r r's$snr 'mlvsi\'

vc hw: td '16l

{{[ d'ltcn ft

.n,", r".t,tnr t," hriil)ilirr ol1'rrotr rf vdLid'jiL'tv ol i ab"ul r dc) ri

rn$ oni lltri \. trr {lso.'bri n'i D INOVA iabl| fi)L snnDl' l'lfar rognson io

.nslva ihc vsitLitiiv nr llf r'sponv v i$io (y) vdhB fhc rcarcs\nr ANovA

t.b;. lko -'y dh.L ANOV l'$l$, nrcluds $uns or squ'r6 dcsrccs of tLdl'n'

Tlc reidual sum of squarc, or S,tk, n€sntus lho v{aibiliry of tl| a.ttral

.tt vcd { ulu$ mund thdr pruticki vdtca. Ils lotDula N

"vJrk: zlri - ri)

rl |utuo lhd! JSrd = S.9tus + t9h.

1.5-2 Deg.es of fteedom

Likc m re srcn bi.nr.. ihcr n qufiiiG.tllcd dc$($ olatLd witl crh sun oI squns. Tlrcir fonNla aro

4tu=n-1

Noti& thal d/id - dJhs + 4tu."3.-'r*^**:

Thr han lqftn* d'i j$t llr

wc norer sorry aboui Mird,

n - I b€aw w0 aro coruiing lor tM {hd nnxu.

I baausc thcrc is I cxdmtory wiablc, -Y

n 2 bLrauc !h.tc arc t\h stirn$h l)*dr.t!N (^.nd l)

iuh, f.Ohns divi'l"d Ly ih' r d snd,'ffiucdom:

1,,{^rK {.ll omr n,d it flr Jryrhin€.

Not€: h cdr bo en@n ihsi

. .. a(Vsc).,, . rsb..d., uL.'aa"rnrLr", Jd

. y'ffs;; nak! ! cood dimic oI ', ur populrlioD sidd.rd dovihtion or th.Y rrlus ai cdh X laluc.

. ttMrks'- d-+ D L\^,-^1

$;r'rypic+llt sutrtrn ii l ilris i"ldruiion iho

is ldjd oui ssliwu ni l'ig0e 7

ss

rcgrsiun ANOVA tdblc, uhich

MS

tu,sns\nn, .l q. S,teF itsr@R6id'rl .lrG .tstu, Min..

Ftr;ukE? C 'r rs,sDn A:!OV t4hlt

Th. ANOVA l,Ahlc fu 1lr nril.ag.'Noiglt oxsmPlu is shoun ir Tdblc 3

Df S' n 'fSqtrd,r Mrd' Scu"nl

030.025

T^BLE 2: INOVA tlblo fo! Mi3hrhil06s. d ,

L5,5 R2 thrcLBh ANOV

ln k iur l. wf h f l.utr'{l tbonr rr or thc coedtcient o'deternitation *hich

is a nrarur ol tho pt{li.iiv. rl)ihr "l tho lost vl'8iis rcsrGion lin' wc h6vc

sen lhat ii is tnc squarc oI lhc L$uol 'oneldtiotr

ru'llicie

Fa c dl$ b. d{t{nfird trsi'rs th sun dt vqurs oll}ind ftonr dr ANOVA

rnlic. I', l{t.,r 9!jjs" .9Sh

s tt-nlrho'rhhr ^z ll"i"Rt

rs.! rr,q, ssF. =) x,. -na;-", fi, t. run,r,, "rli 'b w'u rr-pur"utr I Ssr"ur":i I'um

t lfuur rcs!lri,ri r\udinr lrr sood

of SSrd (trNs fion' Sstr,a- $ Rr

pNti.iiv. lbilil, S,9R- is tnall l hcn mo$

l1

ssrvl

. Il our rcgreioh oqualioh h6 poor prc{icliv. .bilily. J,tk is large. Th€n nclor,99tu @Bs 6on "9Sk instcad of SSR.g, s ]]': is .lR to 0.

For iho rcight nilclgc dds $t,

-.7--^f,3o.e10'

rbich s tbcsmc 6 Rhur *c slt n ldtn L

1.6 ReAression F test for P

ln thc plcvioB selior, m IDW sn tL.1

. E(MStuf) = a' + it,)jlx, *),

Il/' = 0, rhc enlpling distributioDs of MSe, and ,tser rculd bc clr sdhoe MStu md M5&, mdd b. clGc kb.

Ird+ o,14,t(x, -t): > o

shificd io lh. lishr ot lhol

rbo somplitrs dishihution of ffS@ mdd bc

MS&.. Hmc nrgr.r muld bc srakT !h&

nD! bo 6 3@d my oti6tilgtl$t trnld bc.n cvidencc of

ttstu, ir wouu nndy P + 0.

TLis sussBts thal A conDri$n .l MStu" nd MStuq

whcthd or ftn , - D. lf MSar ahd ,]I,r'.gtu, drc (lG('

i] = 0. Ilmrcr, if Ms&r tu strbstmiially lalelr thdi

Th. [}Tothl]s Io. tlo rarsion ,. n* m a lolloss:

).2

dd X di(. mt lin.dlv asgi8t0d)

is lhtrdlr:x$.i6n {i!h X)

flr '.slu'u1 | rv .'l'l B' I dr

diforc frorr reb *irhorl sp(itvnrg vl!0tlu ii s grcdct th8

Thcrc's ro $rh tli$ 0s r otr('tid.{l (tsIsion F tlBt

Bdsn on ihc ,nrovc dMnssxtr',

II. is ttue, tho F will Prohblv

Ct",. .1, t (3)

Thu *bo{, t5i r{risti. l.llo*s d F distibtri'on viih 4' = dt&' md '& = dtb i c

(t.n 2) Morl,v$. sirr.. th' '_

slalLslic cbn oDlv bc p6iiivo th' Pvduc will b'i

;strl uiU t" . at "* ltr' ,,txrv{xl l. $txiisiic vdl( und{r d F(l n 2) dnlihuiion

$t do tlis Lv (.ntl,aLns il't I'valtc io

r' wnl pr"bablJ r,, 5mdll, snd if

o$ .\Nr sigrilicaNc lckl a (olic' 0 05)

4-l til l'r"l(i.,i obsF > r" 1.,.-t, {c qcct to^dnd

_

. lfih{, Fldlrt is k5s lhd o' 'qudl

to o

. lr thc Fvdlnc ir srcakr than I (i'c obst < Fq 'r_''

u' rdl bconcl,do tllri 1/0 ir rcNn'rblc l)tr{d on thr dahr

E$nple 4 Milua.lttght n\Lercn

fb rlt N{ cdr c\nnrdc wc

As usu.l K ssu! ro k5i to:lrd tln,l0lloNn€ ANOVA fubkr l

,J , 0 vs H,, : I + 0. Tltu F sldtislic will bc

a,t kg LuI c

1"17=6l tl

l)f S, tr Sqrd,1 \lnr \,rh,,\

'l)\3rr l A:IOVA rdL! tu' qcisht tr l, ,s ,l,t.'

l+{tr tlk F tallc, !r havo a(0.05,1 23) =123. Si,n. ih.lriBlrr rldi 123, F, *rcnslv qf.i ll! rn.l.on.l'rl rl r

lnr{r asdalnD bdwN, *aiAht sn(l Dil,4,)l.istrn 3 shows Nlrid' LonclNioms and nftrpntatid's ao

olN\v{l I st*1isri. is w.v

irnrr i5 r,org(vi,t(1'.d of

n,s.1ltrr frtr iln, '.srNiotr

{r.D.h laLAor dH l)

!E\id.no reanu //o (Lr //^)

Dvid.n& ihnr ; dcDdnls or x

(r dDlr r d r{*s irrr | )

I)!.ridctrtu lganlst /o (for t,)

l'tril b (j..i /Il

N' ""iilN."

rt,.; )' ,1,r, tr,1, d, x

fr,'ir.lr i,l ,'t. ,.,, ',!'!r/','

1.6.? R.lation bctv€n F and t t.stg

I') inrDl(.lirlr n.Bnsi{ri. hr l le{ cxaclly qr,rvrl{tri io n lr(,hnkn / rsl for

rjtri' ntnl r, hl$!s n*ch ih' s.!x .!N l'$nD I', frr. ii ..n h. sho{r ihri t = ll

Example 5. For i[. 'reartniLeqe

e.ur!)le, th? rt7k*tor t sraiinj. urs 3 323

or vtluortg i. n,( hnr( 69 351 u)tn lt * tl t uht : oJ tl) ( l. r at Ln{ nr. hul ahot(

Mortanei. ttE p rt/ats ueft ut sart taa (" 0).

^ a2 ^ln'h,'''h''h,,' i' zt*' -;;" .;2L_. ott 'ia"t,/. _.- ^-r5

E MJ6 MsE

1.7 Confid€nce Inte.€l for B

llrpoth6$ l((s sirnl)l! t(\1s q l!.tlo ir s I o&sn.$1. ll.n /tLe nrersbns ro Jlsurt.our rl'. ni of rll rcNon$lc nn6(on$ructn,s.,.rfi,l.n(. rne,ral.l,l

&

Toa,$tn'.t a,.DIi.l.N. c^?l Ltr /1. *! nrk.d:utly th. smc asrulptions d n,

dnogrlsior I dnl I n.{s Lfl)nl, sc alwlv,liirsirl

lu odor to ron a r00(r ')% (orftle e ir!*val nr ii w{' nak, rlr lnnxtlilv

t | 1,, , .:- L,,, ,lP( i |,p..t (:t)..t < t i;\'i)t,,r,,,)

so. a 100(l o), (or0!h!1 irr.N'l ror rJ s,ll b.

(. ,;, r' .,7 ! n t lrl

sh.r thc staninrl c!o' of I N ih. sdn. $ tlr om NL{ iri r i slsii*ic Th.

t-$n.n ! nnn,ln r lLUr llr. t'l)lc ihat d{cnds or hv) lhir'Bs:

. ft depertls.n Ihf (1,:;d {orli(|dtr. lpv,n Hishor..offido .lcv.ls r.quirc

{idor nnftr?ls. {llnl' n,.n liqor t rors. t5% is tlr n$l conDonly chNn

. Ir de.lrn(l. o! ilF {lesns.f fr.ldo whcn ri.kins (onli.le.n@ inrerwls lor

J qr uv. .nh,, a tl'. qt'qs {)f fr{\ton' i(r t|r rv,o,a Fr r Nivcr (oufi

d.mc ldd ihc r {orf dc.rN\ N il,.llq]r:\.f f'(\lon' 4conftle!(c i"roills.rs hn.Mr 0ru,!c Dd? DRi$) a t|[ siurlc ste t

Sonr. t{bl6. nrludn,s thfotrr tr( will trsi providc. socond ii ololuxr hceliigs

.alldl Conffden.. lny.l. u slD{n in nigue I Thcsc are ,l6ieDcd t. str.:mlinc 1ltploc6 ol liDdnrs ihe aprnlniar| i sor.lor s <orlidcn.e tuerv8l Snnply lilnl Ihe

l5

uv for ihc 4p,or'ian, 4 (l Lh{i (oltrtritr nr rl'c rPl'o|un .r''Jnhft lewl anJ

tll nun'bor i'i th.l)o{lv.fth. tabk]is tIrtiuo tll|Ltslbtrl,l lr s(i n' cotrslrctiIg

05% 93% 00% 1J9.3%

tI! 0 025 00n 0005 0001

r. 3 2132

fi'{ f!.w ,ous ,,1 tr I iaLl( q(lL I'a n'!s l!,r (onli,L{1rr l!\al

Tlrslanda herF.6in,n of s .oniidcftc'i'rle,vd (l.ts n\ 95%) f,r 1r i" ihli K.trf !t% (rnnl,ri tl'al ilx, 1ro vtnr ol 1is r,onydr (lowfr rrDbr) rn,l (hishd

Ntr'l,ct Mdc st)e,,JnrllJ refxarher rl' l)J 'l)5% (or{nk'fr' w. trFan rhd it we

., \. nrl,.r , ,, .. , , .d1, .1, . , r

4,.n., tr.,n ri,.1!,r,n- n'.,..i ".," ""' '1.+.?ia*innlwls would coniani thr itr! vdl'r o1/J.

Vrv.l..splv spp n,s, i' soDx{nr6 hlDs ro i,Fi t|)n'L ofr..ntnlcn,e ntPml for

I 13 iho s.l ot p.ssiblc v,lu.s ot /J t[dt d. r'k{'nhh b,Bd ol th. dala !!o 6)hFrnrc wlai * rrcin hy \."vrrhh r! !li$in,s il,..orlnbDt leld

EanpL 6. Wnaht ,titcnge fthttt.t

anDlo wo.ondtrdod lhl rlnr I is i,ors ovnl{r(c il'd lJ:0{ianre ort wtht $lues.rJ dn,,c,uornblo b! cdlol!1ing r

c /.nln . ,060 Nd r(/r) ll1)0{i. so, a 05, .or6d.n..

- r.061^ - oot r 7'oz/x da)

ltum rhe Ilabk,. w, I'av

/-@L-,

i.rlora! .re.n,Ip.nnd.f th.\dshtoidcsr.ni& .tol nril':\ dll ar 1,.{n , r, /i,, w,ighr. iis rv.itgc n,ihqrr. will d('cu b! ni

its svrf,agN rnilcssc will dccrm by

nril,r if fir s 10S0 po0d nucMn,,* I m,r6 d,r { rd

7Ihm, bolb rh.rwo sided sisriicdre iest aM thecorlide(eir eml srv6 uslh.

sdno corclbior $ort rho sl(4!..

r.8 Inference for o

\!n idr r Do urn i"linftrLidlDro.(ltrrs tor d bNaus. ofic. its inrolprcbtioi is ior!.{listi. Houcvcr. llvrorl,enis {csts d .onndon.c ntcNal pnccdurB for a rcrkqdtly ihc sanc ury tr ihG. for /t.

1.a.1 Tets and Conffdence Inier€ls

Akl%aE to th. .,w. of J. tl(. i($i s istic ft,r n\hng

Ih.,,'\) D3 It": t to

/-.H.r,- (5)selal

d n 100(L -1)% (o i(lt c i',rdvalror a will hc

(6)

r.9 Inferences on r(Y)R.thcr rl$n {skn'B qtr,ftnr$ al)otrr rlLe .unll Rlationshil) l)'r"tirr X md v. wc

n'igbt nNio l*strt!)PnksiXv I Gn!, x) Nd 8k how v l'rih{v6 nn individu3l'

wnh tlEl,{ hhn Tliis is s!uo.!nb$ b ronsiruciils s confddce inte'td lor

(,i. /,r,,. 1a(,t,,t.;(,r,,p' ,)

"h". i(.1) = /t(.t. rr n&rdr, (5) dnd (6) @ bc usd 1o

orliden.2 inlclnl procdns or o .xdily as rc did {o! 1r'

b x = flt-' eLra/-,..L.

t7

) trtca-Jt-I r4'/.6L

v(].l)

t(Y*)

r.o.3 Conffden.a Inl,erql

E q) "r[^.

"[;

I.r/sD |

: 1

t-

v.t--, sL auJ n,1.1, s+.J'.tt 6L' e-ai-

; 4 h^'tuk^- a./ w. r^^./ / |lr ih /"^t"-42'' v,Lt {e"'} x'*

Srru,"{, rutr tu\uin.irr is Cllivxnd Tlril Bl,aL \vhil &iAhi 1612 pands ltnal hc of n,iosr b khv thf rc.w,'abl( valrsolrlr rv.rgf rnil(rg..f.\l,L.Ns\drh ihe sanF veishr Tl,is is svn.nynnnE b .al, nlabrA the o itler.s nteNRl of

rr(r = aol2) Nou we willl.rrr hd! !),lo this

1.9.2 Samplingdistributio.

Sru)oY r$ iix ! !al'r' srt Xr of X dll {dri ro n,i' rhotri rlr..rrsp.ndn,st,or rtion ndi nbl,oH. t(Yr) snd hy

E(Yr) -,,+lrxi

l his .u l{.iono tv pLr to,n l|] n,trnftls on vr Bnfr l"

, t+,rxL

E(i) ' E6)t xrtu pro.Rl vnh rhf (.nhdo,r irnrul |n(l,m Kcorospondnrs $ddrrd orors $i l,rR lhr fdlo{n,g

(,\, jl'

lln 6l',nah1|dion( aI

vr r(ri)rdt-"''

*h0! O(i,

TIb , r00(r ,,)'/",, ffd. ..idehtror 6(yr) wi h,l

r,,/r,, i(v*),y|1 ;(vr)r"/: "-?)

(7)

a 95% (nrfi{renf n,t.rnr ot rhc rnean/averase nil.ase ot a .66 *€ish-

Th. prdickt ( ,N, Drilo4c of ALl, cd s whoi, uiighi ir 4612lbd{ill

Y(4612)

201l'f,3{w.tt)Drem uile€t of dl .nB ucigbi4 obou! 4612

- 1r'6t: 2251,

. MSE = 9 097

. (x( tf ({612 4a37 t-6)r - 3297516

. )_{x, xr r02rir5r'xi.

,., . ., I Dq71l6l ".,-t.

Ale, t EA .. 2 {rljg. S.. ihc rqufirl 057,ao idence nnei*l oI !([) will b{i

t2 t1 !" (ai3/

t t" ,r Ar93 dt'r 23 t'-nn''

1.10 Prediction of a new observation

Oltd wc mey bc i' .rcst d n' Dk{iciins thc y nluo ol d ncw ob$rEiion lor sivcn

qhK or x. Tlr n.w psir oI ohqwtiots, say. (1 ,.,",,Y'-.) is coftid.rod indopcnd'nr

.t thc obsomtioN (or d.lr) on wl'nj' il'c nsr.Rsbi ltlnxiiur is b'6fd $! car do

this by.ondn'd;lg r prodi.tior nt.rrd n,r Y,"".

Lrnliko tlt proviors(Nc (n,rfnlrc oD lrr nnn ort|r )'distribdiu. l'(vr).hrc {o rc predictnig d individudlonr..nni lroD tlr disribtrtn,' of v ll'isj5tl4LNic dillorrco L.LwFtr rlre l{r t)nrldures

suppM your tricnn wan9lo bry { Pqsh. vho$ rrigl,l is .1232 lbs.lic innis rcu n)

rol hnn a rmsc ol lD$iblo vxllcs.I tlc nileA8e ol tlp.rr lrfor I'r Dkkh a 6nal

d(ision. Usins iho p(diciioD irnl oI I,.,,, you villLc dl,l. ro L.llhnn (wiih ihfEquirdl ,n,ounr or.on6dcn..) ihc rcasonlblc rnh.s .r n,ilmse 6r th€ Po$h. (a,d

d$ ror my rundor,lr chNl ca, srxlr _{ds[l is,1232 lbs)/

r,r0,2 Smpling disi.ibufion

Fd siv.n iklnc of X,,.,, tho pnrlnr\l vaLrt of {,, \ill l)r

]i ,", ' .t + ix,,""

ohvidsl! 1." N,n rDbi{qi cstnnatu or r()i"") brnui,

rtC \, Fft\*x E(A\.'l.ka\|!/

' cr \Fxr.." : Elv 1'rlr pr.diclion innrval ol ]i

"- wou tollo{ tou tl[ klrin

'

ii ( ),"," la,)(3)

\ob tlut llc trun'rraL,r ot (8) tupr{ris llo* frr thf rur .lHrntior 1,., aill.lNron.tom its c*n.nt i,., h{v{l on tlt oisn'J (ldn (rl'( oi'str,rl r plns or

oh{lMin'$). lhis niftr.tr.. (1,, J'".,,) is $.lld th p44icln oro, Tho dc'

nonft,ator oI (3) is tn. Lstnrulod stunda'd c'ror or il'e u(ticicrsilJ- .bt!in.d hy uiiliznlg tho jnd.Dcnd.n.! bct{cu ]i,- ao(t rhc oLigilrl r PAirs

r\s aluys, ir ordd 10 tornr itu p(hcibn rntoN"L. *1r &\r ro l(]ls .e(vr.! -)j,, , ). Fd ihut. $c hnvo rlr nnL{n,s rsull

r'()1,.,,, ::rl

hav te 6hn t d an6ne aJOt ftalatrts a'| bv MSE. in

t(la-, - )j.,,)t(x'

1.10.3 Predicrion lnt€.v.1 ol l,,,,

Bsed on tho rbovc r6nlts. a 100(1 a)% prGlicliotr ir'tdkl tu v".- uill hc

(r",, r"r,,,,"?tr,,, i,,,"),i,^. Av""", y, ",)r"t,,,.)

*rM, i(r,,.", r;,,,,) - v(t(L,,, t.,,,))

Lefsfild a!5% p'rtliclior nncnal oftic milo.gc olrPoahe mishins 4232

Thc prcdicicd mikag. ol th. PoB[. vill bo

(e)

Rault 3.

L'l'11

utl,r",),'lr-:+t

,,.,, rf

-\),r*-!*itst)

\ \42J4. lt .i' oos^ 1Lt2-'

i(y,,,, i_"J _ 1,,1LA1$,4rds8 2060. So.1,1,f r.(trir.(I95% pKliciiuD inlorval oI v,-,, will bc

41 fit 206?'

= | It.tJ.3..8t )

So, we arc 95% conlidcDt ib.i thc niileage ol r rAn(loDlv chasen car rei8hinA {232

!,{dsw.u.,lh h'rtr r lt lf r.'iii'rJ l, t} r.i6NOTE : rhs ,'d!1itr mr.'r,al Jo1 t $agLe olxctudlNor v is mvl udd thor Uu

nt$.t n.. int.n'nt oJ tl9 Pnnt^tiar ntur aI Y 7:h6, ue @ estitnou pop'lohDn

,nd6 aort pflaa! ,Jraa inl'i,n

2i