| tol.lim.fac {spc} | 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 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 p
with confidence of at least 1-a. |
mode |
distinguish between Wald/Wolfowitz' approximation
method ("WW") and the more accurate approach ("exact")
based on Gauss-Legendre quadrature. |
m |
number of abscissas for the quadrature (needed only for
method="exact"), of course, the larger the more accurate. |
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)