\01

Chopie' \l IV\iervotl f 5 .., ,·WI l::f i j 0 V"

\Je_ h~"e lect,\")ed ~1 CJ""v' io estiV11qie__ pqr-qvY/efers af a. d;sfr,'h~.~f;oV\ trovv-1 a S'ct Y'Vl/}/~, Thesf:

J '' I f ll b ~si,·vnC1Td\S were. po,·..,-t es;/mCl-fes ec:c::tus~ 3'veV~ +l-le ou-lroVVI€. o{ -1 tH? scr"""f/e; The es-I/1-'Yra.fe-i.s c:t s/l"lyle poi 1V?i- ct nu~ ber,

No~1 rdi~e(

we wa~t

t hq h lo

If l t · /..:7\\ SO< J mJ ~ e 5 J I,( e s 5 1 s , ;

s C'JJ // r fJ,,'I"/ k i-is befc-ve@VJ ~~ ctVIcl {'(,\\

-A ra,·"'i es-1,-"""'C'#io, ,·s ~ rc,Vlclovvt Vc:trlotble) e,J, A'. Th e o 4 f co lA'1 e ( e s j /.,.~~ cr i e ) i 'S Cf n~Ynb~r; fC'~J, It,

AV\ in -lervctl es-"1/rn.c,1o .. ' I .S Cl ranc:lovn a.. I'\ I V1 7 -e r v ct I 1--v' h o s e ! 11 d pa /~s Cl r e vcn-:c.JJ les ~ e. J· (f=t X) :~ X).

;Y-J-/erva/J

rctndoVV1

pre hQb,'l,'ly l"'a111d0VVl JV1if'(IV

_L-f we re peqi t ~e ex per ln-1 ~ v1 t ma v, j f ,·me 5 5o i~qi -~e have Vlita~~'~) 6u·lro~;qr-:5 fc.x -~L-,e

fO;-H::\Ovv/1 !v·1if'\vdl) theVI o{Jffv:X!.VliCJitly rare ~{ ihes-e> ov-lcoVVJe l~l'lec,,als- w;ll c_e;fi~J~,·v1 + h E' r~ct f Cl ~- VI f=' ·t f' \ ,

ExaM p ( e 1/, I . I Let X\ vll N ~~d EX P(K) I ~··;~l,o

the ll~eii~Vle C·IA vv1ovrfhs) r-eprese~-1 t ct L/ 0

elt>ciric~ I pctris.

-Tt c_q"' bt" Sh

Us;~3 so-f-l"'~.--a\(0 o~ +qlofes we c ctV\ f,,tfl d thl5' pe \C f' V1 1,'/e_s 0 f' [}.__ 'JL17fo) \.v ...

55 = p (J:-:~io) t.. '6~ X< ;;t-~,n(

Def: ..L V\ 5 e V\ e r-q (, ;f

IlL

p[.t_,~ ft,eV'I +he C...L- C-ou {d be 1-v r ; +-1 € .., 0.~ (o; uf!)J.

l)ef: A 11 OV\e - s ,·Jed I o V\... e r / 0 () '('Yo CC)Vt f,'deYI c e /t'141,·t" 1 s de ti'Yl ed q 1/1 q / q J o u s I y.

Excp'VI pie II, "Z. I f-ro "'1'1 + ke P'e vi0u s -ex o wtp 1~ .~ .. f,·"' d.. f~e ovte -s;ded 100 d'/o Co""t;clf'v1cf'> /;WI,'-/5 qs w-e f/ Q.S C!V) 11e}vC!II-+e~,Jed'1 +""~~ s:cled C.I.

0:: p ( 'iiOX < ~~('i 0~

- P( 130 X x,;(

I'? fJ/.'J:;t-1"""'1;;;; V./:) fd/f/0:3 '>! ~:'i.L

...so· ~«=> flAb .5h .. =9

~$00'1-; PI h OJ ~/'/\ \..Ad~+ (Olo. ::::1( s (

Sl r? rJ/, J< oo /

J2 = ro-9 ~u a '1 f"VV ( ( og,) :-;(_ /

ry

>(Of,) ~1 d c= 9 xo~ • 5 d ? ,I 0 ~ ::> r:l/1 lC Vv4 ~J'Y-)~ ~/VI (I::> fdf,'.5'-0""4+ 10 J O_:J

.J d ddr, (o$).R-~~

.1/ fdft5-dV\0 a'1f-Sl )( 02> os-

2: \ \

ll!.f

A~ + iVVle tkeno is e1 choice>/ OV\e VVtJht ctsk ,'.f -l-i-terd e~ fA q/--/c,)ed cr .fcl\ I'

/)5

E;.rc, vY1 p /e; I-f (;) IS k vl 0\..."' v> ,..,

Ik-t ~

De-\':

A 11 lt>cqi;o'r'1 1.-vhf'\f' +o n. o-{ de pev1d

1 '' J· r. po< o~·\?)r ·1er 5ct11f}lr-s f(x; e);: foCX-~)

a... G. clev,5,'/'1

/

f1.1 ., c flo v. 1) ~ -1-hcr-f

A Scq/e pqraYYte-ler so/;5{; P r f(~j 6) ~ ~ +l~ J · Tl-tc~-1 ,·s1 ,·-r X 'V f(x_; 6) fhet-, ZX,., -f(x:;ze).

})~

do~s

Th eo 'e WI I [, 3 . I Let X:. ""~~cl Note 1 "'cd e rf e is o,

-f(x;e). 5L(pp()s~ e i'l ct.., foiL£ .f:~ ~.

ExG?II"lp le:

Exa vnf le:

is +~~ CJV1i,1 f::":'lftcJvvd"-lpr tt~('" +'-12 cl.is/nb ... ii•~. ./ A

/oct:t'-lfoVl pc:t..~ctVVIE'-Ier, -f~en 6)-e ··s F''vdal.

T f e ,·s e1 5c- q le pa rq vvre-/er,; f·hf>,.. A -e ' e:- I$' p; VO ~ct ' 1

Le t x~ ~ ~ ~ d IV (A-( I I) . T 1-,e VI X IJ the /11[..£ t"r M a""d A is ~ /ccq/,o~ potrcrvneler-;

- (

Theo\f'vVI 11.~,2_ It

t~E'V\

a..nd

a0d

where

x~ "-~~d ~(X; f), 1 et) el \.sa.. I(Jcq'lio"' Gz. I S" C cl 1 F' IS" C!

pq rct me if r pctr q~ei e r

-U . G ) -I r ~- e,) '

E.xcrWJp/e: 5\..(ppose :;o'-1 hctve HoiA.. CctVJ !jdt.A. Jet or h!hqf Q ho~-1-L e -A?

q rox ro /o C..L

cr tor-r ~ 7 "to\ e_ '

-r-hi-J +r,'c k v-...()' ks L-vheYI e v'f. r t/-Je t 1.-t ., c -{ ioVJ rn o 111 o -1 o Vlic.

A,

. IS'

Exerrt?ple //,3.5 Le -L X: /\,- i ~d B£i

(1! 0tJsnJ·I-K) d 7 5hJ'/ +j > Yr' / --

( JA., 2 _11 7 ~o·l) d ::: 0 b . /

%-w -

(t'O)N ~ jJ (110)N ~ 1J rr-Y

fLAb W-X p p

j_;:/ c)'1+ hiA,IS Yl I n "' ~ ~ ""'~ 1-"' H-" ~ "'N

btl

Pivo-Jq f f

Le1s use X/~VI io C:\eq-}f' C( ci, 1 c '

• cr o = e ( "' (G ) " t,!.. L brb 1 ; vv hE' I(' q(G) ::: 5~ perc e v71/ If" .£

lo tJ3) : ?2) rJ, r~ f(' !"' n -/,'/e 0 -r ,J, ~ Vl I

F,->'1 d C( r e j [) -Xu.,

( 1~)]

7 .~s-= ( ;~JG~)), _ -n(_ c(~-6)

- e_ 6'

) (oct ( '1) J -= - t-~( ~e)-G J _) e"'

-) 8 +- ez lo;(. C()) = q(Q) -h

'5/rn ~ /C( ( /)) t(0) ~ 8-- G -z. f oj(· o:_) h

I::; d 1/-c.-:) d 5 S~"~ 5

s/f"?'JI

0 ( 9 _)OJ \ ~9

~1 ,? 1 h I a s ;;) l11-"" o tt FLAb (~)0 bv,'ff

Jz_s

De.f: If

. VIOj~b~!XoJJdv : slllv

i__ "'~'1"'1 : J f Sf f fw.., (/ _l a f lA 11 }-t-J J ~j-yp o '1 ~ ~M S"oJ/j!MoX.;> D-"'"'{ d>l\o&JJ:7 dL-f{ op rto.( //hOJ /V\oH

he. I

(~--z_o~~w)N

( 1 '-o / 'h')(\J

(~ z_p /zw)~ f~., --v A ( -e ~ l,w) (\J r: .1 /V :X

z.ry_ 'rr ..loj .T~ •o pv,~

\ 2)

Ch«p te' \ L - t-\- ~ ptq{ 1'5 (co"~;devtC e iVl+cnl"' Is). Now, we vVCf"'i +o qVlS"'-'"r -these sor·h

.. ~ ~_.es-flo"'s:

:r s if f I a .., s : b I e t h ~ i 0 ::: 5 7. Ts ,1 p/"'"s;h/e fl-,q{ G ~ 0 1. l~e f'oce~5 o~ q-

Def 12.1. (

Novv1 e-ve r'Jeed "- WC Z..

Det

~ x / x > zJ is fhe reiec 'lio11 "~Jio t1.

It is rass;_h)e -fo cc..-v~e -fc OV\ i Vl co'~' r~c +

121

Def 12.1. 3 For fo be ft,e Hfj.

ex:= j 1 -l.t = Y'V\ :x J:i( -z. ~A:..() :::: £ (- 2 ) = ~ (-u\. A -

ExqWI p/e:

Qu.es4 i o 111:

N(~o,.k·) N(,t{,, -k)

P..o c ,c.{ I

r ~ Ht..

I Z ~

ExQwtp/e 12.2. I X I,,. 'I X 2.~ ,_, ~ c:.d N ( ,t.( I I 6 )

QuestioVJ;

We 'vvq 11t lc +es-t Ho: 1- 1o.

If live IA!o~~tf o +~s-t of' S 1 i!:.e c;:.y' :-, 0 s- vv-e c q P1 )

{,-nd fhe qfprofr,·~-1~ r-f'jectiov~ re-y/on.

~os-= fY\qx A(X > c (A) == p(x-~ 1o > c -I o A= 10) '< ~ 16 j_ 't;>

,......

:; 1-I((_-/o) '1;4

~:; ~ :::: 'C. Cfb' '-f/5"""

=:.) c = {{) +- j_. :c.,~ 5

So wE' will rf'_Jec-l u. It X I 10+~·-c.qs-. The IEs'-lft 's x :- /1, Lf 0 avtd llt.R re ~ ~c + Ho. 'IN h) YJ o-f use a re j l!'c -lloV1 re J /o¥\ of fh~ [K(>

JI"Yle as observe J. H.j

If the pro b b; /,!J d ""f., 11cl s of 1 he s pe(" ,'{',c: d,·5ir,1bl-ff,·olrl iV~ 1-/o J ihe" f~e p-vq/CAe ,.s fhe t'VlQ Y ,·~ u YV?.

I"' 1 ~ r'.s ex-q wr p /e )

p-· vq IIJ( e : rvta x: .. p/x- ~ IJ. '16 1' ""' sl

tsl

1 n +' . '-\'~A es 'OV\ • Does it """aHer 1·5 the bau111d~ o"' o cri~'""l '~J•'OV\ J q l f€ n-• a -I• ve,

Repea.f (:;1' J 4-t.vo r -\--q; \ ed

b J-f- Vvi 1 h ~ 2 tA ~ J< tAOWVI a\ {t2 r V\ a+ ,· ve ,

c:t ill J 0

X~ h ~~d N(~t-t,cr~) l'Sl

ExaWiple; T e5t H o •• o-z. ~ 1 vs. (-tA: cr?.. / I

'Exqlfvtp le. : Repeat +t)' l~ A : a- 2. < \ ExCiVV!fle: R~ rectf f'O, q +IA./o- .fc,'/ecl ct!ieriAafl·ve..

awd ~ SIWtple n....,(f h;pot/.tes•'S

troW1 X.:.. "- C.:~ J 1\) (~., tTi 2) y ~ "'- i .:d J.J ( 1-1 l. ' oz.~)

f--x a ""'f/e : Te.--t H.: o; ~ = o-~ L

rv1'i ~ a_ t~.vo - s; ded o / ter ..,a i 1've,

Exq l-'l1 pIe ~ PQ; red +-test tor 1 n P

l33

u .. 1rl rtol.v ~~ve ~crve Sitmf/J deviv.ed He t.VV1 at +he re j-ec'fioVl reJ,·OVt at..._i o~ -fl11VI o,·r.

ko'-' do we kl'\oV +~e best ch.br·c~ 1,

We w ovd t k e. r-tos t po e-ve> t'u ( b-t possib /e .

For now .I a !:Su me t.v-e

~jf(){~~sis l~o: 8 =G'c.

v._qnf -fo fest a. s,·Y¥1f/e v~. H,A: G =

\31-{

proot: Let C1C~ be_ '' 11 'ca ( r e; i oVls o t s- ; -t:e o( ~.-v~· t.h (.. * of

t ~~ -fun-'1 c~ £ ~ \ ~e0._ ~ k(,) ~ f:J{Jfc ne l G)+ P (l (, c !)c"c / e,) OV1d

-

rrc*'(e.)-nc~')= P( (_~nee I e\) - P( ( (\ c ~· c ( 61)

·~ ~ p(c•nc/e,) -~1 P(cnec{Qo)

"' L ( rrc,(e. l - lTJe.)) ~ ~o

Exo~ylt (2,6./ Den've The WlosT powe(\v \ +~si of Si-l~ o< of l~o:e.::eo vs. HA: $.:-(8, ~\OW\ ~ ral'ldoW\ 5q Mf Je ~+ EX P(~ ).

A S'v.VVIe e, ) eo . -x·

r\ (~)eo/ e,) =- f(~ ~ 6cJ _ f~; e,}

Tf J.. e ~0 t eo n- l e-x~/e.

J_ e~'~

0

-~ ') )(' eoC- ..

e_

l :.k~ Qf\ e $ 1 LX~e I

\Je h,; I ( reject- ~o

) f- LX: is )a 'j e. •

l_X~ ""' GA"'1 (e J n l

2. 'L x~ ~ ~2.(ZV1 J 8

So) \...-.... e \~_)ect Ho

1s "vDu /d do"Je if $1

l3t

Oef lZ. 1. I A test w1 t~ Co.-! posif" 11 ~'J P.o the ~n . ( 5 ca /1 f> d

11 lA VI j +or>"' J'J WI 0 5 f po vvectl.( I {o,

rc)l,ve r t 4 I I f I + I .s 1'7-1 OS 't ct ll e €: ..52., ( Q I\ ross,'b/e o. HernQ-h've.s)

Ex q ~ p I e ( { 2 . 7. I ) F,·VId C( unito\VVl!; W/ost pot-ver.f.., I f.es"t:· iv'l t-he prev,·o...,_r exaY"''{Jie {or H 0 : e = 3 "5, Hit : e / J ;

Ex CtVV\ pie: Repe-crf- .-J; ih Wo: e f:: 3 VS', HA: e / 3.

As "'-"' k.,.v,.. 51"ev> iv> tt.e exa,..,f/er crhave) q_ UMP fest coY\ Some-l-iMes- be C>hjqinF"cf b~ us;'~J +tte N e J""" " - p.,"', ,- oYI Le .t~(Yl q -l-o .f1., d i he /1A cJ s i f • "'-""r .f:, f te~t fo, s;W!f)e hyp~-fl,ese1'

L3t

Theorem (\1.1.1) I\ X,.._; ~)e) hcts ct /Yloa.~ofo"'e 1;/ce/,"~o/Jc:l \q1;,. iVl t(~) J the~ q UMP +es-t of r;~e tX o-f /-1,: ~~eo V5, H~r: (f) ::>eo ; s fn r-e j eci Ito i t t(~) ?. k(O .?£~) e

ar'ld we ~~v\1\ -reje-c.1

ExqiM p IF

Chapfe, IS GoodVIess of F/ t

Uf Lp'd;/ Ylo~..J~ we q t;,s(..(vned f~q-f we

hqve q/Wlos-t qf,_ctJs

d ,· s -f r ; b t:,_~

J r·('~P·) z.

LvE'

15"

(~~-~;--~~~-o((\) <

- (nx -npe)z

r't po

- (hi-n~) 2

r1 pG

? - .... x - e(.) ( - )"t ;; - )z. X - e" ' = \ n X - V1 Pc

+

f

lJ'~P·)) n p.(l-po)

( n y - V1 e~ )M,~ -n·~·( 1- {'cJ)

el1 -V1 X ) - (n - VI p < il2. n ()- po)

- (c.bser-Ve d5 - C' xpec-1 t>d,}2-e.x p'Pc .. , e ds

+ (obsecvPdi= ~ e"'f~"< -1~"' eyp'!?c .f (',.f~

\ - I (oz_- ez )

Exqrv?p/e l3,'1,1 4 6-s;JPd di(O' ,s roll~d 60 II'Yr?t!',S. We V\_. Q VI -1 to -1 e ") t l -r I _) } ') t q I r,

I H 0 ~ p t = I? 2 ~ p J -~ p If ::- p s ::: p 6 -- 6 H A : a + /e c, rt -/tv c ps qre c:J;.f'ferent.

U" d", H o : i (0, -E, J< ~ ~ "- ( s ) L-"1 Et

No-ff? i hcrT tl1e fe Kf (;(Se) +-h C' Sc:;>vY'l F VIO ·fCI'ftth /

Ot... j e i fc.."" bo-1 h r qt·?clovv1 ov-/c omP S • f h P cJ ( t.fe r 'f> s-u· P

l.Jqr;ctj;)f's OVtd Jhf,'r .,f,cv !c/ b~ c lf'Cl r {' rc vn

c o V1 +ex t , A , q /1:: e r VIC! -1 / v e l,.. ~c L(

Here$ ~ +qb t~ of observed Qytd eKfr? ( --I!:> cl

2 3' ,~ Observed 8' I\ 5 12 IS' 1

Exp~c -lee:( I{) ID 10 16 16 16 {t.~~

He ,e IS 0 -Jq b/e o-f a bse rvecl ( E>x pf>c -ied J cD\...1 Vl -f $,

Prese111i A b se Vl t ___ ., RGc e I

I 20 ( \0 ) 30 ( L-1 0 )

l

One

+tJ\ are

Sv w1

odd

IS fu Ylo-f.

rV1 7t,;-s excrl'vtr 1~ p ~ .30. ~€t("' I'S + ~E' -Jq}Jft:= o-f observed avtd (eKft'c -led) Cou n+s :

Present Ahsev? t

RQc ~ I I zo ( \ S" ) 3o (ss- )

Roce Z I 2 5'(5o) t5 ( 1-(J )

RQcf' 31 Is- ( Is- ) ~s- ( 3 s- )

T~e Ot-{"cC)mP o"'f -fhf" -iesi. fYetit"

Exco-vr p /e 13. t. I

l

Lei XL he ft,e oir·plqnP perrt.

~ t'Y¥?P5' for qr, Eyq~pl~ 1~.7.,1 rr pct ,.,

~~= l ,_'~d A. Poi(3)

The repf1lir Yt'YnPS o-f 'lo CCJj-?-t(;Wf';-tfs atr:> i~C,;;r-c/ed,

QU f' S i I 0 VJ : lio~ Cq n l--v

txc. ~pie r~~ 7., CC)Vl'-( 1'vlC-( ed : C)

\

I 3

fc{ ed 2 5,C({,

The part o-f i he ~b/e ihof t;.t:Jt..fVIIS of s· 0 I rY1 ofe

IV1 or de r +o J

ExC(~M pIe /),=f.Z-! F"r-GlW1 f.lo So

il-1€' pre violAS I

th~f .,u- '"1 e Yc:r vvt f If' 1 1'11o d,fy

no-1 q s.51..f 1-rlFd /e,.;

VP knowVJ,

}-tc: Y~ ,J,VA Poi ( P. )

Q lA. e 5 i, 'u VI ; I.J /1 h o vt -1 k n c; ~.v 1' / ;-t-J h ot.v a.-re blvzs 7

c: ho:s e P1 ,

AV\5'-'-e(': Opii

C ~~ p +er ILl Y1ov1 pq ro, me ir,·c. VVle-ft-?ods

UVI-1,'/ -lhf' w~

Ylo~.-v_, ~e ~~ctve yenerct/j crs.rwi/Y;t"d ihQI-f ~1)fP o-·r d/s- ir~bv~on t--vq._r Kridt-...,J-7 qy,d -lt,P"' m_qde- ,·,..,fer f'n

F >da Some cfa,.W7 fhq-1- -k!ler people eC!rn VVIclf' -1~1CtY\ Shorter }?eopl€. A Sctrnplf? o-r n-- 2()

/1 tcr/ler'\ l')eoflf> (b +i +) WG1s oblo/VJed qvtd "ih~,r anVJuc:~f t;cp/qr/es recotded(iYl f~ou-sa11cls)

12 I 1 Is. 9 l

Fx9~p/e 1'-/,Z,/ /V1edict~ ,·V!come £, Y""ll''io'f 5 1'5 ..$2~)(}() Some cla/W7 fhq-1- -k!ler people eC!rn VVIcif> i t1er Yl s he; f -1 e' I? f" 0 p IF, A s C( h-1 p I~ 0 -r i1 -- 2 () /1 tcr/ler'\ l')eoflf> (C +i +) WG1s oblo/VJed qvtd '1h~,r

anVJuo f t;cp/qr/es recotded(IYJ f/tiou-sa~'~cls)

I() I

$JJ~. = I -/Ji>b' 1-tSib• -

[IJSb' -/]-u~th' -1] -I -

[C"'":f; >z•K jJ -[ ~'~/Z:J!]cJ _ / < [J

01"t?'''X) n ~e1'7 oz':Y)]d -( -

~ oz:o~ 7 l>'X) lJ(~'X 7 o"'X jc1 :: ( o.:o')' 7 5,.1 7oz:'X ) d coS

( ; ) (sZ'),(st_~ 77

; ( sz. ~ oz;·,n)d = o-z ? ·o-z · o"Z ( \ sz•x) j ; v·::,r)J )d = \

Excu-.,ple 1'-L"i.l ( Pa,'red s.,VI Tes-t) ~~ (?f'o p /e

fo dC) C?. Q C" f-' r ·Ia /.,..,

we'e SF)rcft:d crf (1.')' '1 do J;V7

s-urvey.

e5/,'J;c /C(Y? [ere t, per:ScY1 }o.ve

q sccH·e before a.. 5pPec ~ Of v'l d ~ ,J ct /.r'l 0 r;~ r ·/ t1 F 5"/'f'Pc h ...

II 5co~ed_

Ll Sc_o r-ed

3 :Sco,ed

f~E' p(Jf,·J,·c,·qy, h,yhf'r ctftf>l~ f t, P Sf/~C t, him /Aer lo~ ~ r.

h _,·VV1 /her ·I /,f' s-~ rv1e,

F, · ~ s i I e -t. Dt = X{z- JL, (s

N de : _:[ f -1 i e s q r e p o H; b le ( e

+ hfVI P(l); ) 0 jl~,) Y>ta) J, e ~ f) ,·s no·f Covt -1 lnttat.~ s )

0 r A a 5 ro 4 VI d;J e f'(o ('

~,· s j '!:t (i\ i~' n_p >o )- P(O~o). Ho~ever 1-t -then~ ct\~ ~E'W iies, ;i 'IYIC(kf:>s /l't+ie ro

z (

z. bu1 Vl-te d;Q..., of d;fter(>ntto_s ~ 3 I~ no+ ~ero

Exa...,pl~ r ~ f'~ t 5E"COiff d .......

B() t" ~tov (.)s. h.ct ... ~ diff LJ 2_ Y'vifd,crV1 hv.1 med;ql'\

~ 3 of J;.ff 15 ..:e.e re>.

5, 6

rreederPencil

rreederTypewritten Texttwo

Paired- Se~VI/1 pie S 1'5 t1 f d - r q vtlo

p-vq{l.{e - r( T ?. I l iiD>:- )oJ j_;d 8~1

I~ Y\ \s

111 i-f' vzs, ve . lq '1 e_ ,

How-ever,

dOVlf"~

the /?- vc,/.., e wi // be C o.-npv-/q-/,g,.,,J /a ry f' - s ~~~··V? f / e Cfff roy, ~q /,o"" 5

'I, i h,·5 • +e si /s less 5c:>ns/ft've fo o{,fflre-rs.

~"'-es-/iovt: W~af a.~ f~E- pros Ji:-. coV\S of i~f' r~ ai fPd Stjn +est vs-, pcti red sijnecl rem k +e 5t ~

AVI5~A..-e(: _j_t 0 is SjVVI~Mei{!c. Qboui i-l-_s VVIedra"'~1 5i

5(\ £t j

dtte +o \q th-e\

SWZt!l/( r- vq/14e Co vt!c/ s~'YI"'p/'j avr ctsymetr/c d,·sfr,'b!.(l,-~111 a-P t h 'f v1 ~ / c c q 1 /o V? S h; { i ,

be D

If U.Si\1\j \'C(VIk..s Cqv1 ,·V)1f(ov~ IJ()I/vfJI' 01;1P shoi..(IJ expec.f +1-tctf LAsinJ +ttf' c::rcl~ctl cl/-fl'~rence..r rY\\'J~t 3''ve ctn 'eveY? VVZcre pv~;vPr·{v~( test.

T: Z iid.~)oJdt

IV\de-ed t~;5 does -9''ve ct~n Q~esoli'VIe +e.si o{

11: r'Y\E'dr.,.., dif{eie~ce = (; (o.d s~ ""'"""~r1) ~A: med;ct"? d,·{-ferevzce < ()

It ,·~ qfso a/ f.er VIQ~,ye

ExctW~ple~ If

~xce!IP"?i -for fhe oih-er OV?e--s,Jf:'J o, ct 1 V1.,..- o s,'Jed q I ferna1tve.

0 bserved cL"ifer e-1c er are --2-o 1 -to./ - ~ - 1-I I 5 I -lj I 21

The VI There q,e z

P"vaiiAe::: pn~po\~J'o"' \ is a.7 /eqst

0 ~ 5 IJ VI a_ ~ )}J 'r'1 WI e 11 Y J -t C \ L

Poe..! Q II 6 bse cv 0i,oV? .~ t .. , X a ~J r~ let \Jx = SL1 W1 of fl-w lo VI ks of the 0 hse r v q-f,·o VI s-

f,om X.

I-\' ~A: mx < 111y 1 the 'vi.,....- f' f't'j-ec-f ll ,·-r

wx ,· .s 5wtqf/.

I~" 1 his example wx;: 12/

Uvtde\ l--lo '"") S~Abse-f o{ Si~e t'\ (V\uVI1ber o{ obsNved {lo"" X) o{ + ~,e \C?11 k..s 1_1,2, ~~ .•• , n5 L¥1;: nx~~J IS e~ v~ll'j /,·ke0 +a he -\roO\ X. ThE' p- va f ~ e 1 I 5 +~f\E"+o((" e ~ v(O/ I +6 t ~E' pro po ,-} i ov1 of 0 l \ f)oss:b!e Wx

i hcri Cl. \~ SWJ.cr/f~\ th(!j+ t "'p o bse f vfd Vct I ~.-,e of l 2 L-1 ..

There a. "f' +cr ~ t~ s ~CJ' c,: ~i(" ct I vo!ttf>S b~-1 ·~ ·'

I ' f' be -1-!-e r /

io \..,{ 5-e" So5t-IVvCI re -/o '} e-1 C(J-7 e'KocT p ~ v~ I ~vt e .

I+ CCIV\ be $ h 0\-V VI -i h 0! T II := \ I - n)

Naif i hqf Ux CoV' be e,{pcessE'd a: s n.x n!f

U, :: I L j_iX; >~j ~;I j~ I

F,o M i h ,· s -far rv1 ; + c q n 6 e S h o ~ n I

1k.,t /)., is qs'jYVIf-/,-1/cq/ly nor""al lAMder 1-l,,

w;i h o. 5'j Wlpfc i,'c VV1Eq V1 n)( Yl_j 2..

Q n d G s j ·h1 p +o -1/ c V Cl f i q rt c e Y\c v'J (VIx + V1J + I) )2.

This 3'ws ern appra>i~q-lf' +e

SLfppo5e "'-'e iv1dfpt'11 deVJ t

WQY?-/ J(., fest ~t: X ovtd Y are V5. 1-+A: X is f05;Jt'veiJ (C\ff'(q1f>c/ Vvi~Vt r.

I-1 S e' e Vl--1 s Co ''e fqJio n

r·easoV?q}/e, -1-u L{se fke Sc;.mpl~ a s o v r if' 5 t •) -Jq -1/s- 1 /c.

/\

r ::::-

We

ctS

\~~ Jvq r(x) Vqr{Y)

(t x, ~t\ - nXj L-1 )

~~xJ ~t~z)-hjz] CQV1 (....( se a rqndd w;/~q l/oV? -fest

fl-le> fe:s-t s -/-c, l/s It' c..

1/Ld 'f h

There o..-e n] foss;ble pcrir>";s ~-(' H., v1 )f'S

""' r

1--v ,' -1 ~ fhe -}hf'' n 'jts, Yov CqV1 think crbo'-1f 1h,·s- Cfs- ;,·n/~rJ

"'- ".s u e ; '"' 0' d p r : X, I X 2 I X t I (" lc • I he V1 I 1- h f \ f' q I e (\ Chokes ~f;)\ t-v~lc ~ j iD eai\ ~;-it-? X,. NC?v!YI) chosf'y; o t'ir''alct' (o, )(, 1 i~frf' are (ri-1) chotces .fo, Q i"'v1o-fc~

-h,, x, , e-1 c.. n; s '" Tiw be,,;, p ri"c ; p lr ,1-\ eoc{"' -/;,J.

The F'o por -lio V1 of -fhese raj\iYLJ S 1A(J+ rest.( It A.

} a 'J E' lo 'Je r fhaV) the ll'\ a.V\ r Q.5 o\ ohs~rve d "" 1he p -vq/IA f. \ IS

lhis I 5 ql/l exac-f -fest-.

Th;s is q -fesi of indeftY~clerlce; bvf ,._, J's r-eet lly ovd j power-f u I 4J q/J--?S t QV] ct/-1-erVtq-l,v-e o-f Ylovt-~ero Co('re/d--f,·oVJ.

Tl11s 1e5"t v.-ol( /cl have -}f' r r; b /~ per-far ynQHc ~ if X ctv~d V were depev?deH r VLt+ wn corr-e/a-fecJ,

If- (~ Y J ,.s Di 'v a. r; q }e Ylo~ Ma I) -fhF Y7

~~

A'-~ A/ t(VI-Z)

lhi3 afsc pro\.ides CtVI

e vev1 wi·f ~ modt: ret-If'

AI'\ "lvl q/o__5dV s h., t ~;; ~·,) )peq rmqn~ f" f'o ~· .-:: o V1- '5 ,

eX c r)!Pt1-t

.sae-~-r/P

co v1 bf c. c f r e let/ o v1

cppro)(/VV~

5pearmqvt5 Corre{q-l;olfl Is compufed b'j r-rrlaci.-.J fhe yy w/ih fhf',·r rqnks ~/, ... , nJ a"~ d b'j dojVl_Y t hf=' ~ QWl e tor -} h P ys. Ti-?e¥~

-fhe c. c VV1 p 1.1 -le

rctv? k.s~

-1he corre lo~ioVI be--lt... ... e t=>,r'

QrAtos-lioVJ ~ Vvh€1'7 ~,v/// t~e 5pea fW?ap1 Ce; rre (of/cv? he .1.. (~, -l).

AVI~~-ve r : \tvhe~-1 ~ is q fHOYlo -lonicq/ly tLA~ttc-f-t'oVJ of X So fhci1 per+'ec f-ly Co rff' /qfed

Inc rt?Q sin/ (c:lt?c re"s'"J) rqn k s ttr e

1\Jde; "'1>'1 C/PfroXJ'.mcde fest caVI h"" donE' as w; t h 'Peq r· so vt -:s co r-re /Qf/a v1,

A shy ~t qJa plrr/lD"' if a set of h as Q + ;e VJ d o c {rom 1~e Some

q /sa dct-Ja

q 1/o ~;v s OV) e to IPs t co /lee -led o v ~ r- 1 /VV?e

,·-r ,·-j I,S 0 r·Qn tloiiV1 5"o M~~~ po pu lalioVl.

I-f X, ,·s co/lee-fed ~C·~st -fhe0 Y~ 1 efc; CoYls-,·Jer -the p 01' f S (x \I \ \ ( x~/ 2) j t I • ~

A'VI.j ~f the q bovQ +esis bQseJ OV1 a Cc'\re(qfi'6V1 C.oet.f,·ciet1t Cetv1 +esi: J./a: ,10 +r-e1-1d, /P x~ ""'~.:,.J

V5, I-lA; i he f"-(? Is 0... +r-end (LA p~.va \ d 0' dC)lA .. d-1 ~a r cl)

Arwi her neo t test vJq/J-Wolto~~vit-e- ruV1

,- S ca./ led f-es L,

Cit

IrnaJ''vte you ctre preseYJied w,·j"" C

C h Q p fe~ \5 - ~e J re s Sian o Ill d L ;.,., e ct r M o cl e f ~

This chapfer- \c:, q6oll1f Mcdef,·vtj the relq-/-/oV15h,p be+~~v~eV\ Vqfiqb/es

Th~ I-f

+ II \\ ' r _/ d e r M \E-5re s sio111 1 s o 1 rf'YJ lAse . C 0\1\1 e S r ro VVl t h E:' l ~ !!J_ c_ f' H f Lf f J

SOV\S

he~~t

) c I e vfh 5 t I F r· Cl V1 c j s G Q I~ 0 V"1 I /rw.l h 6 rl- '4 J; e d he 'J his 0 + +Of 7 he r.s q,., cl . s 0 Vl s,

•

• _3---i

---~~-~----..-~·--·~·

0:.

fctthe("5 ht>,-Jh-t

-t~e

Frq VIc \ s no f, c e d. + ~at +q f { e r tIt C( vr o ""~ 'o. J e +q-t hecs t· f Vld-e::/ fc have +q 1/er +"' ctVI ave rqy ~ 5' c VIs b \.A f -t-~ha -f· the i' s oYl5 -1 e"" de d + o be

c._ \o Sf(" +c q ve \ C

A SiWip/e /iv1eC{\ mode ( IS cf -+ ~P fur Vl-1

\L::: (30 t- 0• x.~ t- £.r

Xt Lo\.1/J 1epresev1t +he f.c,f~er~ he,yht CIV?d Y~ V7{s tJ.te d,'-f-f'~re.-tce be.\;'-'~Vl cb..serv-ed po,·v? f.s OVl d -l??e fr"" e r-eJ"@SSJ'ovt li...,e,

\J e Co ~ l d a_ { s c c o VI s , .d P (" a. VV1 c de ( /.,vl'i h VV'lo re pr-e dtc+c,5 (,·r? df"pe11dent vctrfo.ble.s)~

Y~ ;::: rJo tf3, Xt 1 f- /Jz.Xi2 -t- (->3 Xi3 + £t

I V1 -t t, ,·s rvtoi~er5

VVICld€ I/ heiJht

t.Je rn'J 'l--ri have ~1~er5 he,jht1 Ot1d jrav1cl{c(/hfr:S he!Jh-t.

4hoiher- Wlocle/ ~ct1hf\~ h-ejh-L

YVIijhi /""c /4de -the. s'/;ut:f\e c-f +he" Cf s q pre d/~-1or :

Y: ;; (3. t ft. X;. '2. +- £. L

T h ;_s ,, s s +; II 0. )iV]e'Q\ I(V'l~d€ \; +hE> /r't?e~d ~ t'5 ,·.,

the cce-\-\tc leVI ts.

L -e ~ 5 '~ f-u {' v> f(:) s ,· VVl p f e ' I VI e a , r e:; c -f- 1 hQ f t.vo v / d

bf"' VIO V'/- ~-f'f() / bv-f· hove ,·r ~ sh;{+

As5u.., in) + (o.,-j t h~ Vlex-1 s-fep (?;0 Cll1d (j,.

I s

i he io

a b o-ve rvto de I -e -s-f/,.,.~-~ o -If> i he

•· . .s co-r-rec.-1,~ ra r C(l/}1 ~ -fp r 5~

0¥\P We o-t

j

vY>e-/ h c.cl , s -1 1-, (> VV1 e -} h" d o-f Jeq s i Srfi" q r e

'2. 5\.4 --z ~X?

'gJ = _h_!VI -~r: ~X? (?

0 == 'Z :xz 'U-'F)~! v. + 5 X \A-:'h ~X/ (_::;. " -s.. C) -= 'Z ~x 7 'y-( 1' 'y -F) 3u -: h :x? ~

D = 'x [~x·sJ-•g)-7~] z-~·~ ~ 17)_ 8 =-0~ {_~

" () ~ ~ V '

The 5 (A VY) of deVloit>d as

Noie ,.._ 'l... -a- -

lheo"f' VV1 15. 3. l

') E(~,) ::~I

5V'v_ared , f s-; d t-1 o Is 'S oi'-1-eVI sst ~ lfof" s u IN1 t 1/t?t'qc uVI bl.:~5eJ e~J-,>~~{r (BLLI£) ~0 \ c, ~() i c 2. r, /f ,, gcyf._ i~t~ the 5t'~-J.s~ f haf if ha_s -fltlf:'

.5 t'Y\q 11~5 t Vcn 1q f1C €,

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