Forest Products Laboratory





Parametric One-sided Tolerance Bounds for Normal Distributions

Here is a form that implements the theory described below.

For a complete discussion of tolerance regions see Irwin Guttman's Statistical Tolerance Regions: Classical and Bayesian, 1970, Hafner Publishing Company, Darien, Connecticut. Here we work through the calculations needed to obtain a one-sided, lower, 95% content, 75% confidence tolerance region for a normal distribution.

PostScript version

Here is a PostScript version of our explanation of what the Web program is doing.

PDF version

Here is a PDF version of our explanation of what the Web program is doing.

MathML version

We first present the mathematics in a MathML form. As of 3/21/2002 most browsers cannot render this markup language into mathematical symbols (and I probably don't have it right anyway). If it is not rendered by your browser then do not attempt to read what does appear. Instead look at the LaTeX section below or obtain the PostScript version above.

For a one-sided, lower, 75% confidence bound on the fifth percentile of a normal distribution (which corresponds to a one-sided, lower, 95% content, 75% confidence tolerance region) we have .75 = Prob ( \bar{x} - k s μ - 1.645 σ ) = Prob ( \bar{x} - μ + 1.645 σ k s ) = Prob ( ( \bar{x} - μ + 1.645 σ ) / s k ) = Prob ( ( \bar{x} - μ + 1.645 σ ) / ( s / n ) k n ) = Prob ( [ ( \bar{x} - μ + 1.645 σ ) / ( σ / n ) ] / ( s / σ ) k n ) = Prob ( \mbox{N} ( 1.645 n , 1 ) / χ_n-1^2 / ( n - 1 ) k n ) = Prob ( NCT ( 1.645 n , n - 1 ) k n ) , or NCT^-1 ( 1.645 n , n - 1 ) ( .75 ) = k n , NCT^-1 ( 1.645 n , n - 1 ) ( .75 ) / n = k . This last equation can be used to calculate the exact value of k provided that one has access to a noncentral T inverse routine. (Here NCT ( 1.645 n , n - 1 ) denotes a noncentral T distribution with noncentrality parameter 1.645 n and n - 1 degrees of freedom.)

FORTRAN and C code to calculate the noncentral T inverse can be found in the DCDFLIB library of probability distribution functions. DCDFLIB is a public domain library of ``routines for cumulative distribution functions, their inverses, and their parameters.'' It was produced by Barry Brown, James Lovato, and Kathy Russell of the Department of Biomathematics, M.D. Anderson Cancer Center, The University of Texas. DCDFLIB can be found at http://odin.mdacc.tmc.edu/anonftp/ .

LaTeX version

For a one-sided, lower, 75\% confidence bound on the fifth percentile of a normal
distribution (which corresponds to a one-sided, lower, 95\% content, 75\% confidence
tolerance region) we have
\[
.75 = \mbox{Prob}(\bar{x} - ks \leq \mu - 1.645 \sigma)
\]
\[
= \mbox{Prob}(\bar{x} - \mu + 1.645 \sigma \leq ks)
\]
\[
= \mbox{Prob}((\bar{x} - \mu + 1.645 \sigma)/s \leq k)
\]
\[
= \mbox{Prob}((\bar{x} - \mu + 1.645 \sigma)/(s/\sqrt{n}) \leq k\sqrt{n})
\]
\[
= \mbox{Prob}([(\bar{x} - \mu + 1.645
\sigma)/(\sigma/\sqrt{n})]/(s/\sigma) 
\leq k\sqrt{n})
\]
\[
= \mbox{Prob}(\mbox{N}(1.645\sqrt{n},1)/\sqrt{\chi_{n-1}^2/(n-1)}
\leq k\sqrt{n})
\]
\[
= \mbox{Prob}(\mbox{NCT}(1.645\sqrt{n},n-1)
\leq k\sqrt{n}) , 
\]
or
\[
\mbox{NCT}^{-1}(1.645 \sqrt{n},n-1)(.75) = k \sqrt{n} ,
\]
\[
\mbox{NCT}^{-1}(1.645 \sqrt{n},n-1)(.75)/\sqrt{n} = k .
\]
This last equation can be used to calculate the exact value of $k$
provided that one has access to a noncentral T inverse routine.
(Here $\mbox{NCT}(1.645\sqrt{n},n-1)$ denotes a noncentral T
distribution with noncentrality parameter $1.645\sqrt{n}$ and $n-1$
degrees of freedom.)
FORTRAN and C code to calculate the noncentral T inverse can be found in the DCDFLIB library of probability distribution functions. DCDFLIB is a public domain library of ``routines for cumulative distribution functions, their inverses, and their parameters.'' It was produced by Barry Brown, James Lovato, and Kathy Russell of the Department of Biomathematics, M.D. Anderson Cancer Center, The University of Texas. DCDFLIB can be found at http://odin.mdacc.tmc.edu/anonftp/ .
For questions or comments about this Web page, please contact Steve Verrill at sverrill@fs.fed.us or 608-231-9375.

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Last modified on 4/1/2003.

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