In article <[EMAIL PROTECTED]>,
Michael Granaas <[EMAIL PROTECTED]> wrote:
>On Thu, 13 Apr 2000, Robert Dawson wrote:
>> Michael Granaas wrote:
>> > If n = 10 and I cannot reject a null of 100 I certainly agree that the
>> > corroboration value is low. But, if n = 100 and I can't reject a null of
>> > 100 I am starting to see support for 100 as a correct value. If n = 500
>> > and I cannot reject a null of 100 would you still demand that I had no
>> > evidence supporting the null?
>> Yes, given that mu=100.5 is part of the alternative
>> How about if n = 1000?
>> Yes, given that mu=100.1 is part of the alternative
>> 10,000?
>> Yes, given that mu=100.05 is part of the alternative.
>> And note that repeating any of these will give the same result with high
>> probability: you cannot, then, assume that a long sequence of tests with
>> n=10,000, most failing to reject mu=100, provide any evidence whatsoever
>> that mu is not equal, say, to 100.1 .
>And so we have reached the logical conclusion that we can never know
>anything. Even with data from the full population we have the problem of
>measurement error. Although I suppose for the purposes of discussion we
>could assume that measurement error perfectly cancels out so that we can
>at least know something if we have data from the full population.
>So, having eliminated all the choices outside of some limited range,
>perhaps 99.9 - 100.1, and absent population data, what do we do?
Now suppose we look at the problem carefully. If we assume
that the distribution of a single observation is normal with
mean mu and variance V, then from a sample of size n, the
mean, which is a sufficient statistic, so nothing else has to
be considered, the density function is that of a normal
distribution with mean mu and variance V/n. This means that
the likelihood function will look like a normal distribution
with mean M and variance V/n. This density is positive on
the entire real line, so nothing can be positively excluded.
To proceed further, we need the assumptions from the
investigator. This is unavoidable.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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