If I understand correctly, the question asks the required sample size
out of the 50 (or so) objects in the box.
Unless some probability is, at least implicitly, specified, I do not see
that this is a statistical question. To be certain whether the "outside"
object has a composition that matches one of the objects in the box, one
must determine the composition of all the objects in the box, unless
there is more information than has been supplied about the objects and
the number of different compositions represented by them.
If it is possible that one possible composition is represented by
only one object, it is possible that that object is the last one tested.
On the other hand, if it is known that each composition is represented by
at least 5 (say) objects, one need test only (50 - 5 + 1 = 46) objects to
be sure of finding all the compositions present.
If it is known that there are exactly six different possible
compositions, one can test until all six have been observed, which might
require fewer than 50 tests. But if the number of possible compositions
is not known, and the minimum number of replicates of any composition is
not known, I do not see how one can reasonably address the question
without examining all 50 (or whatever) objects.
I have the strong impression that, if this is a real-world
problem, there must be some additional constraints or conditions on the
system that have not so far been specified; whether any of these
constraints or conditions would turn the question from a purely logical
one to a statistical one I cannot guess.
What are the consequences of error in deciding whether the object
could have come from the box? (I do not see how one could possibly tell
whether the object DID come from the box, at least not on the basis only
of chemical composition and the information so far provided.) What are
the costs of error (in either direction), and what is the cost of testing
one additional object from the box? Is N really 50, or was that just a
"for instance"; and is it desired to determine, or estimate, the needed
sample size as a function of N (and other parameters that I have
mentioned above)?
On Tue, 13 Jun 2000, Alan McLean wrote:
> A friend of mine has a problem. The following is my understanding of
> the problem.
>
> She has a box of, say, 50 physically identical (to the eye, anyway)
> objects, but they vary in chemical composition - there may be half a
> dozen or so different compositions in the box. She has another of these
> objects, physically similar to those in the box. She needs to test the
> objects in the box to determine if the single object came from, or could
> have come from, this box. If one of the box objects matches the single
> object in chemical composition (presumably this match is within some
> level of precision) then she will be able to say that the object
> (probably) came from the box (or could have come from the box). If none
> of the box objects matches the single object, then the latter could not
> have come from the box.
>
> She has been asked to give a statistical formula to identify the sample
> size she will need to take to answer the question.
>
> The problem is not very clear - I think the people asking for the
> formula are not statisticians, but managers who think that anything that
> can possibly be quantified should be quantified. But maybe someone cna
> come up with a suggestin I can pass on.
>
> All the best,
> Alan
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
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