A Confidence Interval is a Single Realization of a Random Interval
- From: "Reef Fish" <Large_Nassau_Gr0uper@xxxxxxxxx>
- Date: 30 Aug 2006 17:22:28 -0700
Reef Fish wrote:
Mike wrote:
I have always accepted the statement that the probability of a false
positive is "vanishingly small" when you implement the same
requirements using two separate and independent implementations.
This statement is one of those ill-defined use of terms that has no
operational meaning.
Today, a colleague asked me to prove it.
It is impossible to prove something that is not well-defined.
Can anybody point me to an
analysis of confidence levels in the case of comparing the results of
dual implementations of the same requirements?
NASA must have done this for Apollo and the space shuttle. Ditto for
the nuclear industry.
--Thank you,
--Mike
You must be referring to something that is NOT related to the classical
(Neyman Pearson; as opposed to Baysian or Fisherian) definition of a
confidence interval, which is the sample realization of a RANDOM
INTERVAL whose probability is derived from the known distribution of
a Statistic. As such, a confidence is nothing more than a
realization
(on a single trial) of a random interval.
The above statement may not be familiar to the general readership of
this group, so a slight elaboration seems advisable here.
Take the SIMPLEST case of the confidence interval for the unknown
mean mu of a normal distribution with known variance V. Then the
statistic (Xbar - mu)/(sqrt(v)/sqrt(n) has a standard normal
distribution Z.
We can therefore find percentiles a and b from the distribution of Z,
so
that we can make the following PROBABILITY statement about Z:
Pr( a < Z < b ) = 100(p), for any give p.
Now consider the interval a < (Xbar - mu)/ (s/sqrt(n)) < b, where s
= sqrt(v).
The interval can be RE-WRITTEN with the unknown parameter in the
middle of the inequaility to read:
a*s/sqrt(n) < Xbar - mu < b*s/sqrt(n)
or Xbar - b*s/sqrt(n) < mu < Xbar - a*s/sqrt(n)
which is a RANDOM INTERVAL with probability p of enclosing
the unknown mu.
This is the LAST time a probability statement can be made about
the statistic Xbar (the random variable).
As soon as a SAMPLE xbar is taken, we say
( xbar - b* s/sqrt(n) , xbar - a* s/sqrt(n) )
is A 100p % confidence interval for mu. The interval is not unique.
There are infinitely many different 100p% confidence intervals for
mu based on the SAME sample.
That is HOW a Confidence Interval originate in Classical Statistics.
-- Reef Fish Bob.
There is no such a thing as a false positive or false negative
associated
with the notion of a Confidence Interval, not in statistics anyway.
-- Reef Fish Bob.
.
- References:
- Prev by Date: Luis A. Afonso posted as a SPOKESMAN for all sci.stat.math IGNORAMUSES.
- Next by Date: Re: Luis A. Afonso's Impertinence and Disruptive Behavior should NOT be tolerated
- Previous by thread: Re: Confidence levels in comparing alternate algorithm implementations
- Next by thread: Looking for a distribution
- Index(es):
Relevant Pages
|
