Description Usage Arguments Details Value Note Author(s) References See Also Examples
Compute the sample size necessary to achieve a specified halfwidth of a tolerance interval for a normal distribution, given the estimated standard deviation, coverage, and confidence level.
1 2 3  tolIntNormN(half.width, sigma.hat = 1, coverage = 0.95, cov.type = "content",
conf.level = 0.95, method = "wald.wolfowitz", round.up = TRUE, n.max = 5000,
tol = 1e07, maxiter = 1000)

half.width 
numeric vector of (positive) halfwidths.
Missing ( 
sigma.hat 
numeric vector specifying the value(s) of the estimated standard deviation(s).
The default value is 
coverage 
numeric vector of values between 0 and 1 indicating the desired coverage of the
tolerance interval. The default value is 
cov.type 
character string specifying the coverage type for the tolerance interval. The
possible values are 
conf.level 
numeric vector of values between 0 and 1 indicating the confidence level of the
prediction interval. The default value is 
method 
character string specifying the method for constructing the tolerance interval.
The possible values are 
round.up 
logical scalar indicating whether to round up the values of the computed sample
size(s) to the next smallest integer. The default value is 
n.max 
positive integer greater than 1 specifying the maximum possible sample size.
The default value is 
tol 
numeric scalar indicating the tolerance to use in the 
maxiter 
positive integer indicating the maximum number of iterations to use in the

If the arguments half.width
, sigma.hat
, coverage
, and
conf.level
are not all the same length, they are replicated to be the same
length as the length of the longest argument.
The help files for tolIntNorm
and tolIntNormK
give formulas for a twosided tolerance interval based on the sample size, the
observed sample mean and sample standard deviation, and specified confidence level
and coverage. Specifically, the twosided tolerance interval is given by:
[\bar{x}  Ks, \bar{x} + Ks] \;\;\;\;\;\; (1)
where \bar{x} denotes the sample mean:
\bar{x} = \frac{1}{n} ∑_{i=1}^n x_i \;\;\;\;\;\; (2)
s denotes the sample standard deviation:
s^2 = \frac{1}{n1} ∑_{i=1}^n (x_i  \bar{x})^2 \;\;\;\;\;\; (3)
and K denotes a constant that depends on the sample size n, the
confidence level, and the coverage (see the help file for
tolIntNormK
). Thus, the halfwidth of the tolerance interval is
given by:
HW = Ks \;\;\;\;\;\; (4)
The function tolIntNormN
uses the uniroot
search algorithm to
determine the sample size for specified values of the halfwidth, sample
standard deviation, coverage, and confidence level. Note that unlike a
confidence interval, the halfwidth of a tolerance interval does not
approach 0 as the sample size increases.
numeric vector of sample sizes.
See the help file for tolIntNorm
.
In the course of designing a sampling program, an environmental scientist may wish
to determine the relationship between sample size, confidence level, and halfwidth
if one of the objectives of the sampling program is to produce tolerance intervals.
The functions tolIntNormHalfWidth
, tolIntNormN
, and
plotTolIntNormDesign
can be used to investigate these
relationships for the case of normallydistributed observations.
Steven P. Millard (EnvStats@ProbStatInfo.com)
See the help file for tolIntNorm
.
tolIntNorm
, tolIntNormK
,
tolIntNormHalfWidth
, plotTolIntNormDesign
,
Normal
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109  # Look at how the required sample size for a tolerance interval increases
# with increasing coverage:
seq(0.5, 0.9, by = 0.1)
#[1] 0.5 0.6 0.7 0.8 0.9
tolIntNormN(half.width = 3, coverage = seq(0.5, 0.9, by = 0.1))
#[1] 4 4 5 6 9
#
# Look at how the required sample size for a tolerance interval decreases
# with increasing halfwidth:
3:6
#[1] 3 4 5 6
tolIntNormN(half.width = 3:6)
#[1] 15 8 6 5
tolIntNormN(3:6, round = FALSE)
#[1] 14.199735 7.022572 5.092374 4.214371
#
# Look at how the required sample size for a tolerance interval increases
# with increasing estimated standard deviation for a fixed halfwidth:
seq(0.5, 2, by = 0.5)
#[1] 0.5 1.0 1.5 2.0
tolIntNormN(half.width = 4, sigma.hat = seq(0.5, 2, by = 0.5))
#[1] 4 8 24 3437
#
# Look at how the required sample size for a tolerance interval increases
# with increasing confidence level for a fixed halfwidth:
seq(0.5, 0.9, by = 0.1)
#[1] 0.5 0.6 0.7 0.8 0.9
tolIntNormN(half.width = 3, conf.level = seq(0.5, 0.9, by = 0.1))
#[1] 3 4 5 7 11
#==========
# Example 173 of USEPA (2009, p. 1717) shows how to construct a
# betacontent upper tolerance limit with 95% coverage and 95%
# confidence using chrysene data and assuming a lognormal distribution.
# The data for this example are stored in EPA.09.Ex.17.3.chrysene.df,
# which contains chrysene concentration data (ppb) found in water
# samples obtained from two background wells (Wells 1 and 2) and
# three compliance wells (Wells 3, 4, and 5). The tolerance limit
# is based on the data from the background wells.
# Here we will first take the log of the data and then estimate the
# standard deviation based on the two background wells. We will use this
# estimate of standard deviation to compute required sample sizes for
# various halfwidths on the logscale.
head(EPA.09.Ex.17.3.chrysene.df)
# Month Well Well.type Chrysene.ppb
#1 1 Well.1 Background 19.7
#2 2 Well.1 Background 39.2
#3 3 Well.1 Background 7.8
#4 4 Well.1 Background 12.8
#5 1 Well.2 Background 10.2
#6 2 Well.2 Background 7.2
longToWide(EPA.09.Ex.17.3.chrysene.df, "Chrysene.ppb", "Month", "Well")
# Well.1 Well.2 Well.3 Well.4 Well.5
#1 19.7 10.2 68.0 26.8 47.0
#2 39.2 7.2 48.9 17.7 30.5
#3 7.8 16.1 30.1 31.9 15.0
#4 12.8 5.7 38.1 22.2 23.4
summary.stats < summaryStats(log(Chrysene.ppb) ~ Well.type,
data = EPA.09.Ex.17.3.chrysene.df)
summary.stats
# N Mean SD Median Min Max
#Background 8 2.5086 0.6279 2.4359 1.7405 3.6687
#Compliance 12 3.4173 0.4361 3.4111 2.7081 4.2195
sigma.hat < summary.stats["Background", "SD"]
sigma.hat
#[1] 0.6279
tolIntNormN(half.width = c(4, 2, 1), sigma.hat = sigma.hat)
#[1] 4 12 NA
#Warning message:
#In tolIntNormN(half.width = c(4, 2, 1), sigma.hat = sigma.hat) :
# Value of 'half.width' is too smallfor element3.
# Try increasing the value of 'n.max'.
# NOTE: We cannot achieve a halfwidth of 1 for the given value of
# sigma.hat for a tolerance interval with 95% coverage and
# 95% confidence. The default value of n.max is 5000, but in fact,
# even with a million observations the half width is greater than 1.
tolIntNormHalfWidth(n = 1e6, sigma.hat = sigma.hat)
#[1] 1.232095
#==========
# Clean up
#
rm(summary.stats, sigma.hat)

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