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Confidence intervall

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67658] Confidence intervall
  • From: dh <dh at metrohm.ch>
  • Date: Tue, 4 Jul 2006 01:56:23 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,
I have a problem that I thought should be rather standard, but I am 
looking for quit some time for a good answer and I am still at it. As I 
know that this group is read by quite a few knowledgable people I give 
it at try.
Background:
--------------------------------------------------------------
A calibration curve y=f(x) is least square fitted to some calibration 
points ci=(xi,yi) where y is a random variable and x is known
A sample is repeatedly measured: ysi, and xs is determind, so that 
Mean[ysi] = f[xs]
A confidence interval is searched for xs, taking into account that we 
have small sample sizes.

Ansatz:
By error propagation we get that part of the variance of xs that is due 
to the calibration points: varc and that due to the sample: vars.
confidence intervall: student Sqrt[varc+vars]
where student is the student factor to some confidence level with a 
unknwon degree of freedom.
Now the questthe sample size as well as the variance of calibration and 
sample are different.
---------------------------------------------------
I think the above problem is equivalent to:
get a confidence interval for the sum of two (normal)  random variables 
x1,x2 with variances v1,v2 and degree of freedoms:dof1,dof2, where the 
variances and degree of freedoms differ.
In the literature I found the approximation of  Welch-Satterthwaite but 
I found by simulation that this is far from perfect.
Does anybody know if an accurate solution or a better approximation is 
known?
Daniel Huber


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