Martin Weiss wrote:
>
> Please explain!
Take the simple but illustrative case where X & Y are
independent, and have the same standard deviation SDx=SDy.
Since they are independent,
var(X-Y)=var(X)+var(Y)
= 2*SD^2
hence
sd(X-Y)= sqrt(2)*SD = ~ 1.414SD
so the CI for X-Y is going to be 1.414 times
as wide as the CI for X or Y, not twice as wide.
As long as the difference between X and Y is somewhere
between 1.414 and 2, the CIs will overlap but the CI of
the difference will not include zero.
To wit, suppose mean(CI) for X is 0(-1,1) and for
Y is 1.5(0.5,2.5). They overlap, but the mean(CI)
of X-Y is going to be 1.5(1.5-1.41,1.5+1.41),
or 1.5(0.09,2.91). So the difference is significantly
different from zero, even though they CIs overlap.
HTH,
Jeph
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