Construction and Application of a Statistical Test for Coefficient of Variation on Normal Distributions

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1 Amerc Jourl of Appled cece Orgl Reerch Pper Coruco d Applco of cl Te for Coeffce of ro o Norml Drbuo olly Mho eelee d Gez Rchm Mymbu Deprme of c d Opero Reerch, efko Mkgho Helh cece Uvery, ouh Afrc Arcle hory Receved: Reved: Acceped: Correpodg Auhor: olly Mho eelee, Deprme of c d Opero Reerch, efko Mkgho Helh cece Uvery, ouh Afrc Eml: olly.eelee@mu.c.z Abrc: I h udy ovel cl e derved for he Coeffce of ro (C) uder orml drbuo. Th ewly derved e wh vlue o egeerg cece pec of produco of ccure em. The C c meure he preco of meurg rume, mog oher pplco. I order o deerme rume relbly, r by geerg meure ug he rume. The C he clculed o deerme f he meure geered by he rume re cocered roud cerl po. I ue of orml drbuo preumpo, or ppromo, pplcble propere of he orml drbuo led o volveme of he ch-qure d -drbuo. A C e he coruced, d wo llurve emple coclude he dcuo. Keyword: Cerl Lm Theorem, Coeffce of ro, Lw of Lrge Number, Norml Drbuo Iroduco Bckgroud d Defo of he Problem Le X be couou rdom vrble wh me d vrce. A uully dffcul o kow hee prmeer, mple vlue re ofe ued o eme hem (Hkelm d Kemphore 8; Nchol 6). Coder rdom mple of obervo from X deoed by X 1, X,.., X. Le X d be he mple me d mple vrce. Thee mple eme re repecve emor of d. The qure roo of he populo drd devo. mlrly, he qure roo of he mple drd devo. Revew of Eg Lerure everl uhor (Bee d Brgg 5; Gelm 5; 8; Kleje ; Reed e l. ) defe he populo C : (1) Accordg o everl uhor (Armge 5; Kleje d rge ; Reed e l. ), he C vlue provde he preco of y meurg rume or mplg procedure ued. Kleje () remrk furher h here o kow ec cl e for eg he coeffce of vro. Th remrk muled ere developg uch e. Purpoe of he Pree udy Th pper m o corbue eg hypohee h volve. Coder he gmm fuco whch h he form 1 ( ) k y k y Γ e dy, k >. Before he dcuo ke off, wo umpo re mde. Fr, umed h he mple ze ued h udy re lrge eough o offe he pfll of lck of repreevee h could reul from ome mll mple. The ecod umpo h mple re drw from populo h re ormlly drbued. Formulo of he Hypohe uppoe h eperece how h X h pecfed me ( ) d vrce ( > ). The empg o beleve h he geerl vro repeed meureme ke from X gve by he C: v () To e he umpo h he C v, le rdom mple of ze from X be gve. Alo, le X d be he repecve mple eme of d. The m o e he ull hypohe H gve by: 17 olly Mho eelee d Gez Rchm Mymbu. Th ope cce rcle drbued uder Creve Commo Arbuo (CC-BY) 3. lcee.

2 olly Mho eelee d Gez Rchm Mymbu / Amerc Jourl of Appled cece 17, 14 (11): DOI: /jp H : v v (3) Deoe by H, y lerve of H. The e c eme of v, where: ( ) X X 1 /( 1) ~ -1 (8) (4) Coruco of Hypohe Te for A ch-qure drbuo derve from he quoe of drd orml d ch-qure drbuo. Coder wo depede rdom vrble, Z whch drd orml rdom drbuo d X k, whch h chqure drbued rdom vrble wh k degree of freedom. Pll (16) d Wefll (13) howed h he rdom vrble defed Equo (5) below, h -drbuo wh k degree of freedom, where: Z χk (5) k Th eco ue mplg drbuo of kow c h re oced wh o derve cl e gve by equo (4). Co (6) demore h for y mple ze from orml drbuo, he quy: ( 1) h χ 1 (6) drbuo X d re depede rdom vrble. By mplco, X d re depede rdom vrble. I he forhcomg dcuo, he ymbol ~ hll deoe he phre drbued. Recll he rdom mple X 1, X,.., X from X wh me. Defe: X X for 1,,..., Therefore, he me of X d vrce for ll 1,,...,. Tbchck d Fdell (7) cer h f he mple w drw from orml drbuo, he for y, (7) h pprome drd orml drbuo. The ug equo (5) d (6), he c: where, 1 he -drbuo wh -1 degree of freedom. From epo of (8): X X The pplyg he bove equo o (8) led o: (9) X ~ -1 Recllg Equo (4): X The, equo (9) we wll be ble o coclude d ue he equo: X Add-o opero from Equo (8) d (9) re: ~ -1 ~ ~ re relohp of vere d rformo of rdom vrble e wh he bc mhemcl opero (Forbe e l. ). Aumg h hee mhemcl propere, coupled wh he fored mpulo, re permble wh ~, he from equo (9): ~ ( ) + 1 (1) The clculed vlue of deoed by clc, whle m,p hll deoe he ble vlue from -drbuo wh m degree of freedom d he probbly p o he upper rego of he ble of -drbuo. The e dcuo ue cpl leer for deog he emor d mll leer for deog eme. Emor re c d herefore rdom vrble 15

3 olly Mho eelee d Gez Rchm Mymbu / Amerc Jourl of Appled cece 17, 14 (11): DOI: /jp wh probbly drbuo. mll leer re ued for clculg mple eme from cul d oberved. Theorecl Coruco of Crcl Rego Th eco eded o coruc he crcl rego for he e ed Equo (3). Coder rdom mple for he orml drbuo wh me. The e formo ue equo (5), whch bref, e he fc h f Z, whch drbued drd orml (deoed by Z ~ N (, 1)) d Y X re depede rdom vrble (Broverm, 1), he: Two-ded Alerve Lly, le: H v (16) : clc The, rejec H f: < (17) Or: ( + T Z Y ~ 1 clc > (18) ( + Reerch c (Ble d Khur 1993; Wckerly e l. 7) how h he -e follow. From Equo (1), he e c: T ~ 1 The e procedure follow. The ull hypohe : H : v (11) Upper Alerve : uppoe h he lerve hypohe : H > v (1) The, by vrue of beg drecly proporol wh, we rejec H f: clc < (13) ( + where, he eme of he populo drd devo. Lower Alerve : Le: H < v (14) clc The, rejec H f: > (15) ( + Mehodology Th eco bclly epl he e procedure. The e pproch for he C guded by he followg preme: H : v H : No H I h ce, where v pecfed, bu eher or re ed, he for he e c, hll be emed by ug: (19) v Alo Equo (1), for uffcely lrge mple ze, z p wll be employed he plce of -1,p due o ue of he pprome orml drbuo from he Lw of Lrge Number (LLN) d he Cerl Lm Theorem (CLT) (Bereo e l. 1; Fcher 11; Freedm 5; Kleje ; Reed e l. ). I fc, from LLN d CLT, for lrge (or degree of freedom), he drbuo of he -c pproche drd orml drbuo. Prgmc Reoluo o Crcl Rego Defcecy of he -Te Oe probble problem he -e h he vegor c deerme he gfcce level. Thu, he uer c re or rejec he ull hypohe bed upo her deco (Mkewcz ). Eve hough o proper, ly c poelly fluece he cofdece ervl order o cheve he preferred reul. Aoher cocer wh he -e h he reul re precely ruhful oly wh orml populo (Rju 5). O he oher hd, Rce (6) couel h rel 16

4 olly Mho eelee d Gez Rchm Mymbu / Amerc Jourl of Appled cece 17, 14 (11): DOI: /jp populo re ever ecly orml. The echque for -e re foruely reobly robu g oormly he populo ecep he preece of ouler or gfc kewe. Lrger mple grely mprove he ccurcy of he crcl vlue - drbuo whe he populo o orml. Dodge (8) epl h due o he CLT, he mplg drbuo of he mple me from uffcely lrge mple ppromely ormlly drbued. Alo, he mple ze cree, he mple drd devo coverge o he populo drd devo. wlowky (5) wr ly o be cuou whe pplyg he -e becue of hdde fluece from fluel d. Fy d Proch (1) lo ugge lerve e whe codo for ue of -e re queoble, or he e lkely gog o be ubjeced o mledg effec. I he ce of he e h pper, qured h emerged d here re herefore doub he drec ue of he -e. Thu, he -e propoed for pplco oly o fr rele o oher drbuo(). Approme Coruco Le Z 1, Z,...,Z k be orml, declly d depedely drbued (d) wh me d vrce 1, deoed by Z ~ d N(,1) for 1,,,k. The, Bgdovcu e l. (11) d Lom (7) how h: X Z + () 1 + Z +... Z k ~ χk Th equo fudmel relohp of he orml -, X d F-drbuo (Corder d Forem, 14; Jye, 3). Relohp bewee X d F-Drbuo For depede obervo from N (, ), he um of he qured drd core h ch-qure drbuo wh degree of freedom (Bgdovcu d Nkul 11). Ch-qured drbuo oly deped o degree of freedom, whch ur deped o mple ze. The drd core re compued ug populo d. However, he cul vlue of d re uully o kow. Whe d re emed from he mpled d, he degree of freedom re le h. The F drbuo ued e o compre f wo vrce re equl (DeGroo d chervh 11). The e r wh wo depede populo, Y 1 d Y, ech beg ormlly drbued d hvg equl vrce. The, le Y 1 ~d N ( 1, ) d Y ~d N (, ), d drw wo depede rdom mple fromech populo, wh mple ze 1 from populo 1 d from populo. Accordg o Lre d Mr (1), corucg he F drbuo ug d from ech of he wo mple, eme ug he pooled vrce from mple vrce 1 d. Boh 1 d re rdom vrble. Furhermore, her ro rdom vrble: F eme of (1) eme of 1 1 Th c furher be preeed mhemclly d hu: χ F χ ( ) ( ) / 1 / 1 () Therefore, he rdom vrble F defed from wo depede ch-qure vrble h F-drbuo. Relohp bewee - d F-Drbuo The procedure decrbed re mhemclly vld, hey fy mhemcl prcple. However, rely, ccordg o Abrmowz d egu (1), h F. Th me h he qured -drbuo wh k k 1, k degree of freedom he me he F-drbuo wh 1 d k degree of freedom. F-Te A F-e cl e wh he e c h h F-drbuo whe he ull hypohe rue (Mddl d Lhr 9). I ued o compre cl model h hve bee fed o de o defy he model wh be f o he udy populo. A how, he F-drbuo ro of wo depede ch-qure rdom vrble h re dvded by he repecve degree of freedom. The F-e developed o e equly of wo populo vrce (Bulmer 1). I doe h by comprg he ro of wo vrce. Thu, f he vrce re equl, he ro of he vrce equl 1. If he ull hypohe rue, he he F-e c c be mplfed. The ro of mple vrce he e c ued. If he ull hypohe fle, he he deco o rejec he ull hypohe h he ro equl o 1 d hu he umpo of equly. The F-ble oly gve level of gfcce for rgh led e. ce he F-drbuo o ymmerc, d here re o egve vlue, we do o mply ke he oppoe of he rgh crcl vlue o ob he lef crcl vlue (Crlberg 14). The wy o fd lef crcl vlue o revere he degree of freedom, fd he rgh crcl vlue, d he ke he recprocl of h vlue. Formo of F-Te The F formed by ch-qure (wlowky ) d herefore my of he ch-qure propere crry over o he F-drbuo. I coducg he F-e, he 17

5 olly Mho eelee d Gez Rchm Mymbu / Amerc Jourl of Appled cece 17, 14 (11): DOI: /jp he followg codo re vld (Rybko e l. 4): The F-vlue re o-egve. The F-drbuo o-ymmerc. I me bou 1. There re wo depede degree of freedom, where oe for he umeror d he oher for he deomor. Alo, here re my dffere F-drbuo, oe for ech pr of degree of freedom. The pproch ug he F-drbuo o vod lef crcl vlue (Bgdovcu d Nkul ). Geerlly, he lef crcl vlue re dffcul o clcule. A reul hey re ofe voded. A regy o fluece he F-e owrd rgh led e by gg he mple wh he lrge vrce he umeror d he mller vrce he deomor. Eve hough doe o mer whch mple h he lrger mple ze, mer h he mple hvg he lrger vrce plced he umeror. I developg he F-e, he followg umpo re ecery he ue of he F-drbuo (Ccoullo 1965; Fdem 1). The lrger vrce hould lwy be plced he umeror. The e c : F / where > 1 1 I he ce where wo-ded lerve hypohe would hve bee ppropre, he pproch hould be o dvde he gfcce level (or lph) by d he ob he rgh crcl vlue (Gleck d Burzykowk 13). Whe he degree of freedom re o gve he ble, he he vlue wh he lrger crcl vlue hould be choe. Th he mller degree of freedom d reduce he lkelhood of ype I error. The udy populo provdg he mple hve o be orml (or ppromely orml le) d he mplg mu be coduced ug rdom mplg mehod. Thee rdom mple mu be depede. Numercl Illuro Cr Crh Prce D The followg core Tble 1 re he prce ZAR1 of ccde dmged cr of he me model Dever Compy, Joheburg (The r Clfed 1998:4). Aume h before he ccde, ll hee cr were of equl vlue. uppoe furher h f prce of hee cr re decl, he C epeced o be mo 1%. The queo o be ddreed wheher he dmge cued by he ccde o hee cr re gfcly dmlr. Tble 1: Prce of dmged cr Dever Co I ddreg h queo, he formo gve could be ued o deerme f C of mo.1 ccepble umpo. The wer deduced he followg e of opero: (1) H :.1 v H : >.1 () Level of gfcce, α (3) clc (4) From equo (15), rejec H f: clc < cr ( + 1 Now, 1, α 3, , 1 d 33. The he e c : cr ( ) + 1, α ( ).36 ce clc >, he H hould o be rejeced he 1% gfcce level. I c be decded h he vrce of he prce of hee cr o gfcly lrge compred o he me. Therefore he dmge o he cr were o eprevely dffere. Emo D Howell () provde pobly o e uder he followg codo. Coder echg profeor he Deprme of Pychology kow Uvery who eche group of ude wh lerg dble. he fd rewrdg f fer ech e h bee wre he c deerme f he e w ey or dffcul. Accordg o her, h h pove mplco for her echg pproch of he ubjec well for eme purpoe. Oe of her dcovere h f he ude mrk re he ro of drd devo o me of 3:1 or more, mpoble o deerme he level of emo dffculy. Recely, he emed her cl of 7 ude. The mrk ( percege) produced verge of 53 wh drd devo of The ly hould eblh f he profeor echg c beef from he le e. I repoe o he bove queo, kow h he profeor c beef upo deermg f he e w ey or dffcul. The hypohe h he ro of drd devo o me 3:1 or more mde, whch o ume h he wll o beef from he e. A forml e of hypohe mde. The profeor co y f he e w ey or dffcul f:. 18

6 olly Mho eelee d Gez Rchm Mymbu / Amerc Jourl of Appled cece 17, 14 (11): DOI: /jp H :.3 Th eed g he lerve hypohe h: H : <.3 The lerve hypohe mple h he profeor c deerme f he e w ey or dffcul. Gve h 7, 53 d , he he e follow: (1) H :.3 v H : <.3 () Chooe (3) clc (4) From (17), rejec H f: clc > cr ( z 1.645, v 3 Now, 1,α.5, he e c : cr ( / ( ( ) + ) ce clc > cr, he H hould be rejeced. I eem h he level of dffculy of he e c be cered. I herefore ccepble o beleve h he 5% gfcce level, he profeor c beef from h e for her echg, or from he formo vlble. Cocluo The ero by Kleje d oher reercher h here re o kow ec cl e for he C muled h pper. Aupcouly, he gp defed h bee ddreed wh he work h pper. The pper corbue o kowledge by creg he ool for ly, whch h udy cl e for C uder orml drbuo. Thee e lo evelop pprome ormly, d eed o he CLT d LLN. Ackowledgeme The uhor re greful o he uppor from he Deprme of c d Opero Reerch of efko Mkgho Helh cece Uvery. Fudg Iformo The Deprme of c d Opero Reerch fuded he mucrp cve.. Auhor Corbuo Gez Rchm Mymbu: Mymbu provded he mhemcl mehod for he e c d wroe ome eco of he pper. olly Mho eelee: eelee eleced he ce ude ued he umercl lluro d performed he clculo. Ehc The udy beefed from ecodry d d he ecery ckowledgeme hve bee provded by me of referece. Referece Abrmowz, M. d I.A. egu, 1. Hdbook of Mhemcl Fuco. New York: Courer Corporo, IBN-1: , pp: 146. Armge, C.J., 5. C he heory of pled behvor predc he mece of phycl cvy? Helh Pychol., 4: DOI: 1.137%F Bgdovcu,. d M.. Nkul,. Accelered LIFE Model: Modelg d cl Aly. Tylor d Frc, IBN-1: , pp: 334. Bgdovcu,. d M.. Nkul, 11. Ch-qured goode-of-f e for rgh ceored d. I. J. Appled Mh. c, 4: 3-5. Bgdovcu,., J. Kruop d M. Nkul, 11. No-prmerc e for ceored d. Bordeu, Frce: Ie. Bee, J. d W. Brgg, 5. Ug d Uderdg Mhemc: A Quve Reog Approch. 3rd Ed., Pero Addo Weley, IBN-1: , pp: 74. Bereo, M.L., D.M. Leve d K.A. zb, 1. Bc Bue c: Cocep d Applco, 13h Ed., Pero Hgher Educo AU, IBN-1: , pp: 676. Ble, C. d R. Khur, Fudmel of ocl c: A Afrc Perpecve. Ju d Co, IBN-1: 7194, pp: 367. Broverm,.A., 1. Ace udy mul, Coure 1, Emo of he ocey of Acure, Em 1 of he Culy Acurl ocey. Wed, CT: Ace Publco, IBN-1: Bulmer, M.G., 1. Prcple of c. Edburgh: Olver d Boyd. Courer Corporo, IBN-1: , pp: 56. Ccoullo, T., A relo bewee d F- drbuo. J. Am. cl Aoc., 6: DOI: 1.37/8687 Crlberg, C., 14. cl Aly: Mcroof Ecell 13. Que Publhg, IBN-1: , pp:

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