The assumption of equal variances, which was made in Exercise 8.41, is not always tenable. In such a case, the distribution of the statistic is no longer a t. Indeed, there is doubt as to the wisdom of calculating a pooled variance estimate. (This problem, of making inference on means when variances are unequal, is, in general, quite a difficult one. It is known as the Behrens-Fisher Problem.) A natural test to try is the following modification of the two-sample t test: Test
H0:μx = μY versus H1 : μx ‰  μY,
where we do not assume that σ2x- σ2Y, using the statistic

Where

The exact distribution of Tʹ is not pleasant, but we can approximate the distribution using Satterthwaite's approximation (Example 7.2.3).
(a) Show that

where v can be estimated with

(b) Argue that the distribution of Tʹ can be approximated by a t distribution with u degrees of freedom.
(c) Re-examine the data from Exercise 8.41 using the approximate t test of this exercise; that is, test if the mean age of the core is the same as the mean age of the periphery using the T' statistic.
(d) Is there any statistical evidence that the variance of the data from the core may be different from the variance of the data from the periphery? (Recall Example 5.4.1.)

Transcribed Image Text:

X-Y (화끎 3 TL (approximately), ~ v SY Ti TL n2 (n m(m-1)