David Heiser wrote in message ...
>
>
>-----Original Message-----
>From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]]On
>Behalf Of Alan Miller
>Sent: Tuesday, April 30, 2002 4:32 PM
>To: [EMAIL PROTECTED]
>Subject: Re: Is Hogg and Craig Wrong?
>
>
>David Heiser wrote in message ...
>>Been doing a little cross checking between sources on the confidence
>>interval about sample means, where the sample (n) comes from a normal
>>population with unknown mean and unknown variance.
>>
>>Several textbooks give the confidence interval about the sample mean as
>>t*s/sqrt(n), where n is the number in the sample, t is the t distribution
>>value at n-1 df, and s is the sample unbiased standard deviation.
>
>Using either n or (n-1) in the denominator gives a BIASED estimate of the
>standard deviation.
>Using sumsq/(n-1) gives an unbiased estimate of the variance.
>Using sqrt( sumsq/(n-1.5) ) gives an estimate of the standard deviation
>which is almost unbiased.
>-----------------------------------------------------------------------
>Interesting idea. Did you mean 0.5 or 1.5? I can see the 0.5, but not the
>1.5. Has this been looked at for small samples via Monte Carlo approaches
to
>determine if it gives a better approximation to the hypothetical population
>normal distribution sigma value?
>
>DAHeiser
>
Take a look at the case in which the true sigma = 1.
Then Z = Sum (x(i) - xbar)^2 has a chi-squared distribution with (n-1)
d.o.f.
It is fairly straightforward to show that:
E(sqrt(Z)) = sqrt(2).Gamma(n/2) / Gamma((n-1)/2)
Thus if we want an unbiased estimate of sigma then we should divide by this
quantity after taking the square root of the sum of squares. Here is a
small table of values to show how good the sqrt(n - 1.5) formula is.
n 3 4 5 8 10
20
E(sqrt(Z)) 1.255 1.595 1.880 2.553 2.9180 4.3019
sqrt(n - 1.5) 1.225 1.581 1.871 2.550 2.9155 4.3012
But then, we have come to live with biased estimates of standard
deviations - except perhaps in the quality control field where the range is
often used to get an unbiased s.
--
Alan Miller (Retired from CSIRO Mathematical
& Information Sciences)
http://www.ozemail.com.au/~milleraj
http://users.bigpond.net.au/amiller/
.
.
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