# Parametric One-sided Tolerance Bounds for Normal Distributions

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.

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## MathML version

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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 $\mathrm{NCT}\left(1.645\sqrt{n},n-1\right)$ 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 $k$
(Here $\mbox{NCT}(1.645\sqrt{n},n-1)$ denotes a noncentral T
distribution with noncentrality parameter $1.645\sqrt{n}$ and $n-1$
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/ .