tol.lim.fac | R Documentation |
For constructing tolerance intervals, which cover a given proportion p of a normal distribution with unknown mean and variance with confidence 1-a, one needs to calculate the so-called tolerance limit factors k. These values are computed for a given sample size n.
tol.lim.fac(n,p,a,mode="WW",m=30)
n |
sample size. |
p |
coverage. |
a |
error probability a, resulting interval covers at least proportion |
mode |
distinguish between Wald/Wolfowitz' approximation method ( |
m |
number of abscissas for the quadrature (needed only for |
tol.lim.fac
determines tolerance limits factors
k
by means of the fast and simple approximation due to
Wald/Wolfowitz (1946) and of Gauss-Legendre quadrature like Odeh/Owen
(1980), respectively, who used in fact the Simpson Rule. Then, by
xbar +- k s
one can build the tolerance intervals
which cover at least proportion p of a normal distribution for
given confidence level of
1-a. xbar and s stand
for the sample mean and the sample standard deviation, respectively.
Returns a single value which resembles the tolerance limit factor.
Sven Knoth
A. Wald, J. Wolfowitz (1946), Tolerance limits for a normal distribution, Annals of Mathematical Statistics 17, 208-215.
R. E. Odeh, D. B. Owen (1980), Tables for Normal Tolerance Limits, Sampling Plans, and Screening, Marcel Dekker, New York.
qnorm
for the ”asymptotic” case – cf. second example.
n <- 2:10 p <- .95 a <- .05 kWW <- sapply(n,p=p,a=a,tol.lim.fac) kEX <- sapply(n,p=p,a=a,mode="exact",tol.lim.fac) print(cbind(n,kWW,kEX),digits=4) ## Odeh/Owen (1980), page 98, in Table 3.4.1 ## n factor k ## 2 36.519 ## 3 9.789 ## 4 6.341 ## 5 5.077 ## 6 4.422 ## 7 4.020 ## 8 3.746 ## 9 3.546 ## 10 3.393 ## n -> infty n <- 10^{1:7} p <- .95 a <- .05 kEX <- round(sapply(n,p=p,a=a,mode="exact",tol.lim.fac),digits=4) kEXinf <- round(qnorm(1-a/2),digits=4) print(rbind(cbind(n,kEX),c("infinity",kEXinf)),quote=FALSE)
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