Givenx1 andx2 distributions that are normal or approximately normal with unknown σ1 and σ2, the value oft corresponding to x1 − x2 has a distribution that is approximated by a Student'st distribution. We use the convention that the degrees of freedom is approximately the smaller ofn1 − 1 andn2 − 1. However, a more accurate estimate for the appropriate degrees of freedom is given by Satterthwaite's formula:
wheres1,s2,n1, andn2 are the respective sample standard deviations and sample sizes of independent random samples from thex1 andx2 distributions. This is the approximation used by most statistical software. When bothn1 andn2 are 5 or larger, it is quite accurate. The degrees of freedom computed from this formula are either truncated or not rounded.
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