Welch’s t Test

Mean comparison is the central theme of many classical statistical procedures. The well-known independent-sample t test is often used to test the equality of two means from independent populations with equal variances, whereas Welch’s t test is generally preferred when the variances are not equal.

Let y11,y21,,yn11 and y12,y22,,yn22, be two independent random samples from two populations with means (or expected values) µj=Eyij and variances σj2=Var(yij),j=1,2. The sample counterparts of µj and σj2 are y¯j=1nji=1njyij, and sj2=1nj1i=1nj(yijy¯j)2, respectively. Because y¯j is an unbiased estimator of µj, any difference between µ1 and µ2 should be reflected by y¯1y¯2. Of course, even when the null hypothesis


holds, y¯1y¯2 does not hold in general because of the sampling error, which is characterized by the variance


Notice that, when σ12=σ22=σ2,ν2 will be ...

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