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I want to test whether the mean in two samples are different. I don't know their variance and estimate them from the sample. When should I use the Welch-Satterthwaite approximation to the degrees of freedom for the test statistic?

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    $\begingroup$ The Welch approach deals with the case where the variances are unknown and expected to be very different. This is the famous Behrens-Fisher problem. $\endgroup$ – Michael R. Chernick Apr 18 '18 at 13:03
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Welch's t-test does not assume equal variances for the two groups.

Some people argue that Welch's t-test should be used by default. For an example, see the article at this link.

Also note that R uses Welch's t-test as the default for the t.test function. I believe SAS output for PROC TTEST returns both the Student t, and Welch's t ("Satterthwaite"). See Wikipedia for other software implementations.

For a more complete discussion, see this thread on Cross Validated

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