Essentials

THE TWO-GROUP t TEST 53sample case. Before it became routine to calculate t tests by computer, researchersrelied on a table of critical values for the t distribution to make their decisions. Sucha table is included in Appendix A. Looking at Table A.2, you can see that as df getslarger (because N is getting larger), the critical values get smaller (because the tailsof the distribution are getting thinner) until they are very similar to the critical valuesof the normal distribution. In fact, when df becomes infinitely large (indicatedby the infinity symbol, ), the t distribution becomes identical to the normal distribution;the bottom row of the t table contains the critical values for the normaldistribution. Notice that as alpha gets smaller, the critical values get larger—youhave to go further out on the tail to reduce the area beyond the critical value. Alsonotice that the critical value for a .025 one-tailed test is the same as for a .05 twotailedtest (except that for the two-tailed test you actually have two critical values:the value in the table preceded by either a negative or a positive sign).The Separate-Variances t Test FormulaStrictly speaking, Formula 3.3 follows a normal distribution only for infinitelylarge samples, but it always follows a t distribution. So a more general formulawould change the z to t:X 1 X 2t (3.4)s 2 21 s2 n nThe only question is which t distribution is appropriate for this formula—thatis, what are the df in this case? You might think that for two groups, df n 1 n 2– 2, and sometimes that is true, but that value for df is usually not appropriatewhen using Formula 3.4. This formula produces what is called the separatevariances(s-v) t test, and unfortunately, finding the df for this test can be tricky.Generally, the df must be reduced below n 1 n 2– 2 for the s-v test (the df can getas low as n 1– 1 or n 2– 1, whichever is smaller). The logic of exactly why this correctionis necessary and the details of its calculation are beyond the scope of thisbook, but these areas are covered well in other texts (e.g., Howell, 2002). Althoughcomputers can now easily find an approximate value for df to test the tfrom Formula 3.4, the more traditional solution is to calculate the t test a differentway, so that the df come out to a simple n 1 n 2– 2. This solution requires thatwe assume that the two populations from which we are sampling have the samevariance. This assumption is called homogeneity of variance, and we will have a gooddeal to say about it shortly.12