On Mon, 27 Mar 2000, M. Wallace wrote:
> I recently received a report from an environmental lab comparing fish
> weights between a "control group" and a "treatment group". The t-test
> indicated there existed a significant difference between the two groups
> (P<0.05)- the Control Group fish were heavier. However, the report
> also included a table showing the upper 95% CI of the Treatment Group
> overlapping the lower 95% CI of the Control group! Can this be
> correct??
Certainly. The hypothesis test asks whether the two
sample means represent values reasonable to observe if the two
population means are identical: it does not ask whether the two separate
CIs overlap. Equivalently, the null hypothesis is that the difference in
the two population means is zero. The corresponding confidence interval
is the CI on (treatment mean minus control mean).
Notice that, in round numbers, either end of a 95% CI on a sample mean
is 2 s.d. from the sample mean; so in order for the two separate 95% CIs
not to overlap, the sample means must differ by 4 s.d. In order for the
sample means to differ significantly at the 5% level, they must differ by
2 standard errors of the difference in means; the std. error = s.d.
times sqrt(2), so the difference required is 2.8 s.d.
(All this supposes that the sample sizes and the s.d.s are the same in
the two groups. If not, the numbers 4 and 2.8 are still approximately
true for the pooled standard deviation if the sample sizes are equal, and
in any case the difference required for the point estimates to be
significantly different is less than the difference required for the
separate CIs not to overlap.)
> If it is correct- What is the purpose of calculating a 95% CI??
In general, to find out what values of an hypothetical population
mean are consistent with the observed value of a sample mean.
> And, what inferences can be made from the 95% CIs???
For any value within a 95% CI, one can reject (at 5%) the null hypothesis
that the population mean has that value.
Notice that this is NOT the question you are implicitly asking
when you compare TWO confidence intervals to see whether they overlap.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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