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10-16 math - Notation and Equations for Paired and...

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Notation’and’Equations’for’ Paired-’ and’ Independent-Samples’t -test’ Lecture’ 10/16’ Notation:’ n A ,’ M A ,’ µ A ,’ n B ,’ M B ,’ µ B :’Statistics’and’parameters’for’samples’A’and’B’and’populations’A’and’B in’an’independent -samples’t -test’ :’Standard’error’of’the’difference’ between’sample’means;’standard’deviation’of’the’ sampling’distribution’of’ M A ’ –’ M B ,’according’to’the’null’hypothesis MS :’Mean’square.’This’is’a’generalization’of’the’idea’of’sample’variance’th at’ comes’up’in’ more’complicated’statistical’tests,’as’a’way’of’e stimating’the’population’variance,’ σ 2 .’ X A ,’ X B :’The’two’scores,’or’samples,’in’a’paired -samples’t -test’ X diff :’Difference’score’in’a’paired -samples’t -test.’ Formula’for’difference’score’in’paired -samples’design .’When’each’subject’(or’event,’etc.)’ has’two’ scores’on’the’same’variable,’we’can’compute’a’difference’score’by’subtracting’one’ from’the’other. ’ X diff = X A ! X B (1)’ t’statistic’for’paired’samples .’A’paired -samples’t -test’works’just’like’a’single -sample’t -test’ applied’to’the’difference’scores. ’ The’null’hypothesis’states’ μ A ’ =’ μ B ,’which’implies’that’the’ mean’difference’score’equals’zero.’Therefore,’we’use’the’formula’for’a’single -sample’t,’ with’ μ 0 ’ =’0. ’ t = M diff s diff n (2)’ It±s’important’to’keep’in’mind’that’everything’in’this’formula’i s’based’on’the’difference’ scores,’ X diff .’That’is,’ M diff ’ is’the’mean’of’ X diff ,’ s diff ’ is’the’sample’standard’deviation’of’ X diff ,’and’ n ’ is’the’number’of’difference’scores’( not ’ the’total’number’of’observations;’that±s’2 n ).’ Mean’Square ’ for’independent -samples’ t-test.’ Like’a’single -sample’t -test,’the’independent -
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