Efficiency of Modified Lord s Test in Testing Equality of Means: An Empirical Approach through Simulation with Theoretical Proof!

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1 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V54 Effcecy of Modfed Lord Tet Tetg Equalty of Mea: A Emprcal Approach through Smulato wth Theoretcal Proof! Aldw M. Teve, Cetral Phlppe State Uverty, Phlppe. Emal: joowe@yahoo.com Kare Luz Y. Teve, Cetral Phlppe State Uverty, Phlppe. Emal: kartev@yahoo.ca Abtract The tudy ted to vetgate the power of the modfed Lord tet ad the clacal t tet baed o ther ablty to reject fale ull hypothee. The tudy reveal that the ue of mea abolute devato a replacemet the etmato of the pooled etmate of varace of the clacal t-tet comparg two mea work pretty well. I tace where the abolute dfferece betwee mea were le tha or equal to oe tadard devato, the modfed Lord tet rejected 65 out of 40 data et, whle the clacal t-tet rejected 4 out of 40 data et. Th mple that the modfed Lord tet reject fale ull hypothee about 46% whle the clacal t-tet reject fale ull hypothee of 9% at level of gfcace. Moreover, whe the two tet were appled to data et where ther abolute dfferece betwee mea wa greater tha oe tadard devato, the modfed Lord tet rejected 9 out of 70 data et. O the other had, the clacal t-tet rejected 0 out of 70 data et. The codto mple that the modfed Lord tet 76% more lkely to reject the fale ull hypothee. Lkewe, the clacal t-tet ha a 65% more lkely to reject the fale ull hypothee at gfcace level. Coequetly, the mulated reult revealed that for data et whoe mea dfferece wa oe tadard devato or le, the modfed Lord tet wa 7% effcet rejectg fale ull hypothe tha the clacal t-tet. For data et whoe mea dfferece wa wder tha oe tadard devato, the Lord tet wa % effcet tha the clacal t-tet. Th modfed t-tet ha a hgher t-tattc wheever the data do ot cota etreme obervato thereby cled to reject the ull hypothe. It ha a hgher power compared to the uual t-tet for depedet ample of ay other alteratve tet. Thu, clam to be more effcet.

2 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V54. Itroducto Comparatve tude are deged to dcover ad evaluate dfferece betwee effect rather tha the effect themelve. A commo applcato of the ature of the problem comparo betwee two et of data term of ther locato parameter. Theoretcally, the dfferece the populato parameter are teted baed o the avalable formato comg from ther repectve ample. Gve two depedet ample wth mea ad that are etmate of ther repectve populato mea,, are ofte codered approprate tattc. I practce, the tet of gfcace ad cofdece terval for mall ample cocerg the true populato dfferece - are baed o the t dtrbuto, where the clacal t tattc ha the form t c [ / / / / ]. Eq. It aumed that ad are depedet ad ormally dtrbuted. By theory, ther dfferece alo ormally dtrbuted o that the umerator of t ormal wth mea zero. The deomator of t a ample etmate of the tadard error of -. The populato varace defed a the average, over the populato, of the quared devato from the populato mea. The ymbol avg deote the average of. Thu we may wrte [ ] Eq. avg I mathematcal tattc the varace of derved the form E, Where E the epectato. However,. hece, by takg the epected value, we have, E[ ] E[ ]. Eq. 3 By defto of populato varace, avg Ad avg, the thrd term equato 3 vahe. Th paper attempted to eplore the power of the modfed Lord tet a way that the pooled etmate of the tadard devato wa replaced wth the weghted abolute mea of the ample. The orgal Lord tet utlzed the average rage from the two ample the form t c [ / w w / / / ] Eq. 4 Where:

3 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V54 w - The rage the frt ample data et w - The rage the ecod ample data et. Here, the computed t tattc alway flueced by the ature of the data et coderg that the rage ealy affected by etreme obervato. Bede, mathematcal tattc how that the rage doe ot provde a uffcet tattc for t fucto of oly two etreme obervato. O the cotrary, the modfed Lord tet th tudy ue the weghted mea abolute devato a replacemet to the pooled etmate of the ample tadard devato. I the cae of equal ample ze, the average of the two mea abolute devato replace the pooled etmate of the ample tadard devato. The tet tattc of the form t c [ / m m / / / ] Eq. 5 Where: _ / m, the mea abolute devato ample _ / m, the mea abolute devato ample The ovato make ee coderg that the mea abolute devato a uffcet tattc. It a fucto of the etre et of obervato from a ample. Thu, th tudy wa coducted to ae t power to reject the ull hypothe whe t fale.. Objectve of the Study Th tudy coducted wth the followg objectve: To utlze the mea abolute devato a replacemet of the rage the Lord Tet Modfed Lord tet; To determe the power of the modfed Lord tet ad the clacal t tet; 3 To determe the effcecy of the modfed Lord tet over the clacal t Tet. 3. Methodology Th tudy utlzed mulated data. Te ru of mulated data were produced wth the ad of tattcal oftware. I each ru, t cotaed dfferet umber of data et. I each data et, t geerated two ormal dtrbuto wth dfferet mea but had equal varace. The power of the tet wa determe by t ablty to reject the fale ull hypothe at pecfed gfcace level. I th tudy, the ull hypothe aumed that agat t alteratve that. Two dfferet method were ued to tet th coteto. The clacal t tet ad the modfed Lord tet ug equato ad 5, repectvely. Ther 3

4 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V54 repectve power wa determed by the ablty to reject the ull hypothe at level of gfcace. Ther correpodg umber of data et whoe ull hypothe wa rejected wa recorded. Thee were ued a dcato of ther effcecy. The effcecy wa meaured baed o ther ablty to reject fale ull hypothe ad wa compared wth the ue of pared t tet. 3. Tme ad Place of Study Th tudy wa coducted from July, 007 at the Negro State College of Agrculture, Kabakala Cty, Negro Occdetal. Cotue refemet ad theoretcal proof for teratoal publcato ha bee ought. 3. Reearch Deg ad Data Gatherg The truth or falty of ay tattcal hypothe ever kow wth certaty ule a thorough vetgato performed the etre populato or populato. Th proce eem to be mpractcal mot tuato Wapole, 000. Itead, a ample take from the populato of teret ad make ue of the formato cotaed a ba formulatg deco of whether a tated hypothe rejected or retaed. Other method volved mulato techque where the real tuato mmc through the ue of probablty model wth a ad of moder computer program. The reearch deg ued the coduct of th tudy wa mulato techque wth the ad of computer oftware. Comparatve aaly o the uual tet had bee carred out agat a alteratve method propoed th paper. The method volved were the clacal t tet ad the modfed Lord tet comparg equalty of two mea. 3.3 Stattcal Treatmet of Data Data et were geerated ug tattcal oftware comg from kow dtrbuto. Gve two populato wth mea ad wth varace ad, repectvely, tetg the hypothe that agat t approprate tet alteratve wa performed ug the dfferece of ad, the approprate tattc uder the pecfed problem. Here, the problem poed to tet equalty betwee two mea. The tated ull hypothee were formulated wth the hope that they be rejected ug the clacal t tet a well a the propoed modfed Lord tet. The ablty of the tet to reject the fale ull hypothe determed the power of the tet. 4. Reult ad Dcuo Reult from Smulato how table below. I th tudy, the clacal t tattc wa ued to tet dfferece betwee two populato mea. A alteratve Lord tet wa alo ued gve the ame data. The power of ther tet were compared a how below. 4

5 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V54 Table : Number of fale ull hypothe rejected by the clacal t tet ad modfed Lord tet of mulated data et from ormal dtrbuto wth uequal mea but equal varace Smulato Number of Data Set Populato Dtrbuto Number of Null Hypothe Rejected N~,, N~,, Clacal Modfed t-tet Lord Tet 0 6,6, 0 0, 6, , 4, 0 9, 4, , 4, 5 5, 4, , 4, 0 9, 4, , 6, 0 4, 6, , 4, 0 8, 4, , 6, 0 0, 6, , 9, 5 8, 9, , 4, 5 8, 4, , 6, 5 5, 6, I mulato, 0 data et were geerated wth two ample each of uequal mea. The aalye revealed that oly four data et were detected whoe mea uequal were ug the clacal t tet, whle fve of thee were detected wth the ue of the modfed Lord tet. The gfcat level ued the crtero for rejecto of the ull hypothe wa at 0.005%. I mulato, 0 data et were geerated wth two ample each of uequal mea. The aalye revealed that the two tet had the ame umber of data et detected whoe mea were uequal at 0.005% level of gfcace. I mulato 3, 0 data et were geerated wth two ample each of uequal mea. The aalye revealed that there were fve data et detected whoe mea uequal are ug the clacal t tet, whle of thee were detected wth the ue of the modfed Lord tet. The gfcat level ued the crtero for rejecto of the ull hypothe wa at 0.005%. I mulato 4, 30 data et were geerated wth two ample each of uequal mea. The aalye revealed that there were 3 data et detected whoe mea were uequal ug the clacal t tet, whle 4 of thee were detected wth the ue of the modfed Lord tet. The gfcat level ued the crtero for rejecto of the ull hypothe wa at 0.005%. I mulato 5, 30 data et were geerated wth two ample each of uequal mea. The aalye revealed that there were 3 data et detected wth mea whch were uequal ug the clacal t tet, whle 7 of thee were detected wth the ue of the modfed Lord tet. The gfcat level ued the crtero for rejecto of the ull hypothe wa at 0.005%. I mulato 6, 30 data et were geerated wth two ample each of uequal mea. The aalye revealed that there were 7 data et detected whoe mea were uequal ug the clacal t tet, whle 0 of thee were detected wth the ue of the modfed Lord tet. The gfcat level ued the crtero for rejecto of the ull hypothe wa at 0.005%. 5

6 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V54 I mulato 7, 40 data et were geerated wth two ample each of uequal mea. The aalye revealed that there were 6 data et detected havg mea that were uequal ug the clacal t tet, whle 4 of thee were detected wth the ue of the modfed Lord tet. The gfcat level ued the crtero for rejecto of the ull hypothe wa at 0.005%. I mulato 8, 40 data et were geerated wth two ample each of uequal mea. The aalye revealed two data et detected wth ther mea that were uequal ug the clacal t tet, whle of thee were detected wth the ue of the modfed Lord tet. The gfcat level ued the crtero for rejecto of the ull hypothe wa at 0.005%. I mulato 9 ad 0, 40 data et were geerated wth two ample each of uequal mea. The aalye revealed that the clacal t tet ad the modfed Lord tet rejected the ull hypothee that ther mea were equal. The gfcat level ued the crtero for rejecto of the ull hypothe wa at 0.005%. For mulato, 3, 5, 6 ad 8, the abolute dfferece of the mea were le tha or equal to oe tadard devato whch rejected the ull hypothee of 65 out of 40 data et ug the modfed Lord tet. Ug the clacal t tet, t rejected 4 out 40 data et. For mulato, 4, 7, 9 ad 0, the dfferece betwee the mea wa greater tha oe tadard devato. The modfed Lord tet rejected the fale ull hypothee of 9 out of 70 data et, whle the clacal t tet rejected the fale ull hypothee by 0 from 70 data et. A oberved, the ull hypothe rejected by the clacal t tet wa alo rejected by the modfed Lord tet. Further aaly whether the umber of fale ull hypothee were gfcatly dfferet term of the two tet employed, the pared t tet had bee ued. Table how the detaled formato. Table : The power of the clacal t tet ad modfed Lord tet gve the abolute dtace of the mea ad tadard devato Smulato Number of Number of Null Hypothee Data et Rejected Abolute Dfferece of Mea Stadard Devato Modfed Lord tet clacal t- tet t =.37 p-value = 0.04 Mot data et whoe ull hypothe wa rejected by the clacal t tet were alo rejected wth the ue of modfed Lord tet. For data et whoe abolute dfferece betwee mea c 6

7 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V54 wa le tha or equal to oe tadard devato of ther dtrbuto, the modfed Lord tet ably determed the gfcat dfferece betwee mea compared to the clacal t tet. Th wa evdet for mulato, 3, 5, 6, ad 8. Th mea that the modfed Lord tet wa more etve to reject fale ull hypothe compared to the clacal t tet. 5. Cocluo The tudy teded to vetgate the power of the modfed Lord tet ad the clacal t tet baed o ther ablty to reject fale ull hypothee. The tudy revealed from the mulated data et that the modfed Lord tet wa etve to determe equalty of mea compared to the clacal t tet. However, for data et whoe tadard devato wa maller tha the mea dfferece, the clacal t tet ted to approach the effcecy of the modfed Lord tet. Thu, th tudy revealed that the ue of mea abolute devato a replacemet the etmato of the pooled etmate of varace of the clacal t-tet comparg two mea worked pretty well. I tace where the abolute dfferece betwee mea were le tha or equal to oe tadard devato, the modfed Lord tet rejected 65 out of 40 data et, whle the clacal t-tet rejected 4 out of 40 data et. Th mpled that the modfed Lord tet rejected fale ull hypothee of about 46% whle the clacal t-tet rejected fale ull hypothee of 9% at level of gfcace. Moreover, whe the two tet were appled to data et where ther abolute dfferece betwee mea wa greater tha oe tadard devato, the modfed Lord tet rejected 9 out of 70 data et. O the other had, the clacal t-tet rejected 0 out of 70 data et. The codto mpled that the modfed Lord tet wa 7% more lkely to reject the fale ull hypothee. Lkewe, the clacal t-tet had a 6% more lkely to reject the fale ull hypothee at gfcace level. Thee mulated reult revealed that for data et whoe mea dfferece wa oe tadard devato or le, the modfed Lord tet wa 7% effcet rejectg fale ull hypothe tha the clacal t-tet. For data et whoe mea dfferece wder tha oe tadard devato, the Lord tet wa % effcet tha the clacal t-tet. Fally, the modfed Lord tet ehbted hgher power rejectg fale ull hypothe tha the clacal t tet. 6. Recommedato The followg recommedato are hereby propoed for reearche that may be coducted the future: Ue large umber of data et ad alpha level hgher tha 0.005; determe a proporto betwee mea dfferece ad tadard devato; 3 Ue larger meaure of data; 4 Ue other mlar methodology that tet equalty of mea for comparo purpoe; 7

8 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V Ue a dfferece betwee mea that le tha oe tadard devato. Referece Davd Brk ad Yadolah Dodge 993. Alteratve Method of Regreo. A Wley-Iter cece Publcato, Joh Wley & So, Icorporated. Hogg, R.V. ad A.T. Crag 970, 3 rd Ed. Itroducto to Mathematcal Stattc. The Coller-Mcmlla Publher, Lodo. Mum W. Edward. 970 d Ed. Stattcal Reaog Pychology ad Educato. Joh Wley ad So, Icorporated. Padua, R.N Elemet of Reearch ad Stattcal Model. MPSC Publhg Houe, C.M. Aveue, Lapaa, Cagaya de Oro Cty. Roald E. Walpole 00 3 rd Ed.. Itroducto to Stattc. Pearo Educato, Aa Pte Lmted. Sedecor, George.W. ad Wllam G. Cochra th Edto. Stattcal Method 980. The Iowa State Uverty Pre, USA. Staudte, R.G. ad Smo J. Sheather 990. Robut Etmato ad Tetg. A Wley-Iter cece Publcato. Joh Wley & So, Icorporated. Apped Proof of the Modfed Lord Tet. by Cauchy Crtero / /

9 Proceedg of the Secod Aa-Pacfc Coferece o Global Bue, Ecoomc, Face ad Socal Scece AP5Vetam Coferece ISBN: Daag-Vetam, 0- July, 05 Paper ID: V54. Whe too large, the clacal t-value aymptotc to the Modfed Lord tet, where the term approach to a value of. However, for mall ample the term ot eglgble makg the Modfed Lord tet cled to reject the ull hypothe. Th mea that the t-value the computed tattc of the modfed Lord tet greater tha or equal to the clacal t-value. 9