Hypothesis Testing Using z- and t-tests In hypothesis testing, one ...

B. Weaver (24-Aug-2010) z- and t-tests ... 28Appendix: The Unequal Variances t-test in SPSSDenominator for the independent groups t-test, equal variances assumed2 2( n1− 1) s1 + ( n2 −1) s ⎛21 1 ⎞SE = × ⎜ + ⎟n1+ n2 −2⎝n1 n2⎠SS ⎛ 1 1 ⎞⎜ ⎟n1+ n2 −2⎝n1 n2⎠within _ groups= × +2⎛ 1 1 ⎞= spooled× ⎜ + ⎟⎝n1 n2⎠(A-1)⎛ss= +⎜⎝ n n2 2pooled pooled1 2⎞⎟⎠Denominator for independent t-test, unequal variances assumedSE⎛ss= ⎜ +⎝ n n2 21 21 2⎞⎟⎠(A-2)• Suppose the two sample sizes are quite different• If you use the equal variance version of the test, the larger sample contributes more to thepooled variance estimate (see first line of equation A-1)• But if you use the unequal variance version, both samples contribute equally to thepooled variance estimateSampling distribution of t• For the equal variance version of the test, the sampling distribution of the t-value youcalculate (under a true null hypothesis) is the t-distribution with df = n 1 + n 2 – 2• For the unequal variance version, there is some dispute among statisticians about what isthe appropriate sampling distribution of t• There is some general agreement that it is distributed as t with df < n 1 + n 2 – 2