Research Design - - Topic 2 Inferential Statistics: The t-test 2010 R.C. Gardner, Ph.D. Independent t-test
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1 Research Desig - - Topic Ifereial aisics: The -es 00 R.C. Garer, Ph.D. Geeral Raioale Uerlyig he -es (Garer & Tremblay, 007, Ch. ) The Iepee -es The Correlae (paire) -es Effec ize a Power (Kirk, 995, pp 58-6; Cohe, 988, Ch. ) igle ample -es (Gosse, ue,908) Two sample -es (Fisher,95) ( X µ ) ( X µ ) saar error of he ifferece ( X Iepee -es µ ) ( X X X Whe H o : True X X µ ) X X X X X X X µ s / If variaces are heerogeeous X X a egrees of freeom are esimae usig he Welch esimae If variaces are homogeeous, compue a poole esimae ( )² ( )² X X X X p The: X P X P wih egrees of freeom - Tess for Heerogeeiy of Variace Levee s (960) es of Heerogeeiy of Variace ivolves a aalysis of variace of he absolue eviaios of each score from is group mea. If he mea absolue eviaios iffer sigificaly i his wo group case, i suggess ha he variaces iffer sigificaly. Uer his coiio, oe shoul use he -es for iepee variaces; oherwise, he -es wih poole esimaes shoul be use. Degrees of Freeom Welch (98) egrees of freeom for iepee variace esimaes f ( ) ( ) Degrees of freeom for poole variace esimae f
2 Daa for he Iepee -es Daa Eior for he Iepee -es Group 6 Mea aar Deviaio Usig CLOPE o ru P -es Clope Click a hope ha you o wha you wa o o. Eer P, Pu aa i he Daa Eior, Click o: Aalyze Compare Meas Iepee-amples T es. This preses he followig wiow Whe you specify he values ieifyig aa i groups a, he Coiue box will arke, a whe you click i, he program reurs o he previous wiow. Clickig o OK prouces he followig resuls. 7 8
3 P Ru for a Iepee -es GET FILE'F:\PYCH50\aaforiepeees.sav'. DATAET NAME Daae WINDOWFRONT. T-TET GROUP gp( ) /MIING ANALYI /VARIABLE x /CRITERIA CI(.95). x gp Group aisics. Error N Mea. Deviaio Mea Assumpios: Iepee -es Iepee Raom amplig: The samples are iepeely a raomly obaie from he populaios of ieres. Normaliy: The wo populaios are each ormally isribue. Homogeeiy of variace: The variaces are equal i he wo populaios. Iepee amples Tes x Equal variaces assume Equal variaces o assume Levee's Tes for Equaliy of Variaces -es for Equaliy of Meas 95% Cofiece Ierval of he Differece Mea. Error F ig. f ig. (-aile) Differece Differece Lower Upper Null Hypohesis: The populaio meas are ieical. Tha is: H 0 : µ µ 0 X X X X X Paire -es ( X Bu he aa are correlae, hus: Therefore X r r µ ) ( X X X XX X µ ) X Mea Daa for he Paire -es X X where f
4 Daa Eior for he Paire -es Usig CLOPE o ru he Paire -es Eer P, Pu aa i he Daa Eior, Click o: Aalyze Compare Meas Paire-amples T es. This preses he followig wiow P Ru for a Paire -es GET FILE'F:\PYCH50\aaforpairees.sav'. DATAET NAME Daae WINDOWFRONT. T-TET PAIR x WITH x (PAIRED) /CRITERIA CI(.95) /MIING ANALYI. Pair x x Paire amples aisics. Error Mea N. Deviaio Mea Pair x - x Paire amples Tes Paire Differeces 95% Cofiece Ierval of he. Error Differece ig. Mea. Deviaio Mea Lower Upper f (-aile) Paire amples Correlaios Pair x & x N Correlaio ig
5 Assumpios: Paire -es Iepee raom samplig: The pairs of observaios are iepeely a raomly obaie from he populaio of ieres. Normaliy: The iffereces bewee he pairs of observaios are ormally isribue i he populaio of iffereces. Null Hypohesis: The mea ifferece i he populaio is 0. Tha is: H 0 : µ 0 Or is equivale: H 0 : µ -µ 0 7 Effec ize a Power Cohe (988) saes i is coveie o use he erm effec size o mea he egree o which he pheomeo is prese i he populaio or he egree o which he ull hypohesis is false. (p.0-). Wih respec o he -es, he propose: µ µ σ Esimae for he Iepee -es Esimae for he paire -es Where: mall.0 Meium.50 Large.80 X X X X poole 8 Power esimaes ca be obaie usig he Cohe (988) Tex or compue usig he GPower. program which ca be owloae from: hp:// GPower. calculaes power esimaes for mos saisics of ieres o psychologiss. I has wo ypes of applicaio:. Poshoc permis oe o eermie he power associae wih a give sample a effec size.. A priori permis oe o eermie he sample size for a give power a effec size (o available for all proceures). 9 Refereces Cohe, J. (988). aisical Power for he Behavioral cieces ( e.) Hillsale, NJ: Lawrece Erlbaum. Fisher, R.A. (95). Applicaios of ue s isribuio. Mero, 5, Levee, H. (960). Robus ess for equaliy of variaces. I I. Olkis (e.) Coribuios o probabiliy a saisics. afor, CA: afor Uiversiy Press. ue (908) The probable error of a mea. Biomerika, 6, -5. Welch, B.L. (98). The sigificace of he ifferece bewee wo meas whe he populaio variaces are uequal. Biomerika, 9,
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