Description Usage Arguments Details Value Note Author(s) References See Also Examples

Construct a *β*-content or *β*-expectation tolerance
interval for a normal distribution.

1 2 | ```
tolIntNorm(x, coverage = 0.95, cov.type = "content",
ti.type = "two-sided", conf.level = 0.95, method = "exact")
``` |

`x` |
numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a normal (Gaussian) distribution
(i.e., |

`coverage` |
a scalar 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 |

`ti.type` |
character string indicating what kind of tolerance interval to compute.
The possible values are |

`conf.level` |
a scalar between 0 and 1 indicating the confidence level associated with the tolerance
interval. The default value is |

`method` |
for the case of a two-sided tolerance interval, a character string specifying the method for
constructing the tolerance interval. This argument is ignored if |

If `x`

contains any missing (`NA`

), undefined (`NaN`

) or
infinite (`Inf`

, `-Inf`

) values, they will be removed prior to
performing the estimation.

A tolerance interval for some population is an interval on the real line constructed so as to
contain *100 β \%* of the population (i.e., *100 β \%* of all
future observations), where *0 < β < 1*. The quantity *100 β \%* is called
the *coverage*.

There are two kinds of tolerance intervals (Guttman, 1970):

A

*β*-content tolerance interval with confidence level*100(1-α)\%*is constructed so that it contains at least*100 β \%*of the population (i.e., the coverage is at least*100 β \%*) with probability*100(1-α)\%*, where*0 < α < 1*. The quantity*100(1-α)\%*is called the confidence level or confidence coefficient associated with the tolerance interval.A

*β*-expectation tolerance interval is constructed so that the*average*coverage of the interval is*100 β \%*.

**Note:** A *β*-expectation tolerance interval with coverage *100 β \%* is
equivalent to a prediction interval for one future observation with associated confidence level
*100 β \%*. Note that there is no explicit confidence level associated with a
*β*-expectation tolerance interval. If a *β*-expectation tolerance interval is
treated as a *β*-content tolerance interval, the confidence level associated with this
tolerance interval is usually around 50% (e.g., Guttman, 1970, Table 4.2, p.76).

For a normal distribution, the form of a two-sided *100(1-α)\%* tolerance
interval is:

*[\bar{x} - Ks, \, \bar{x} + Ks]*

where *\bar{x}* denotes the sample
mean, *s* denotes the sample standard deviation, and *K* denotes a constant
that depends on the sample size *n*, the coverage, and, for a *β*-content
tolerance interval (but not a *β*-expectation tolerance interval), the
confidence level.

Similarly, the form of a one-sided lower tolerance interval is:

*[\bar{x} - Ks, \, ∞]*

and the form of a one-sided upper tolerance interval is:

*[-∞, \, \bar{x} + Ks]*

but *K* differs for one-sided versus two-sided tolerance intervals.
The derivation of the constant *K* is explained in the help file for
`tolIntNormK`

.

If `x`

is a numeric vector, `tolIntNorm`

returns a list of class
`"estimate"`

containing the estimated parameters, a component called
`interval`

containing the tolerance interval information, and other
information. See `estimate.object`

for details.

If `x`

is the result of calling an estimation function, `tolIntNorm`

returns a list whose class is the same as `x`

. The list contains the same
components as `x`

. If `x`

already has a component called
`interval`

, this component is replaced with the tolerance interval
information.

Tolerance intervals have long been applied to quality control and life testing problems (Hahn, 1970b,c; Hahn and Meeker, 1991; Krishnamoorthy and Mathew, 2009). References that discuss tolerance intervals in the context of environmental monitoring include: Berthouex and Brown (2002, Chapter 21), Gibbons et al. (2009), Millard and Neerchal (2001, Chapter 6), Singh et al. (2010b), and USEPA (2009).

Steven P. Millard ([email protected])

Berthouex, P.M., and L.C. Brown. (2002). *Statistics for Environmental Engineers*.
Lewis Publishers, Boca Raton.

Draper, N., and H. Smith. (1998). *Applied Regression Analysis*. Third Edition.
John Wiley and Sons, New York.

Ellison, B.E. (1964). On Two-Sided Tolerance Intervals for a Normal Distribution.
*Annals of Mathematical Statistics* **35**, 762-772.

Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009).
*Statistical Methods for Groundwater Monitoring*, Second Edition.
John Wiley & Sons, Hoboken.

Guttman, I. (1970). *Statistical Tolerance Regions: Classical and Bayesian*.
Hafner Publishing Co., Darien, CT.

Hahn, G.J. (1970b). Statistical Intervals for a Normal Population, Part I: Tables, Examples
and Applications. *Journal of Quality Technology* **2**(3), 115-125.

Hahn, G.J. (1970c). Statistical Intervals for a Normal Population, Part II: Formulas, Assumptions,
Some Derivations. *Journal of Quality Technology* **2**(4), 195-206.

Hahn, G.J., and W.Q. Meeker. (1991). *Statistical Intervals: A Guide for Practitioners*.
John Wiley and Sons, New York.

Krishnamoorthy K., and T. Mathew. (2009).
*Statistical Tolerance Regions: Theory, Applications, and Computation*.
John Wiley and Sons, Hoboken.

Millard, S.P., and N.K. Neerchal. (2001). *Environmental Statistics with S-PLUS*.
CRC Press, Boca Raton.

Odeh, R.E., and D.B. Owen. (1980). *Tables for Normal Tolerance Limits, Sampling Plans,
and Screening*. Marcel Dekker, New York.

Owen, D.B. (1962). *Handbook of Statistical Tables*. Addison-Wesley, Reading, MA.

Singh, A., R. Maichle, and N. Armbya. (2010a).
*ProUCL Version 4.1.00 User Guide (Draft)*. EPA/600/R-07/041, May 2010.
Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.

Singh, A., N. Armbya, and A. Singh. (2010b).
*ProUCL Version 4.1.00 Technical Guide (Draft)*. EPA/600/R-07/041, May 2010.
Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.

USEPA. (2009). *Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance*.
EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division.
U.S. Environmental Protection Agency, Washington, D.C.

USEPA. (2010). *Errata Sheet - March 2009 Unified Guidance*.
EPA 530/R-09-007a, August 9, 2010. Office of Resource Conservation and Recovery, Program Information and Implementation Division.
U.S. Environmental Protection Agency, Washington, D.C.

Wald, A., and J. Wolfowitz. (1946). Tolerance Limits for a Normal Distribution.
*Annals of Mathematical Statistics* **17**, 208-215.

`tolIntNormK`

, `tolIntLnorm`

, `Normal`

,
`estimate.object`

, `enorm`

, `eqnorm`

,
`predIntNorm`

, Tolerance Intervals,
Estimating Distribution Parameters, Estimating Distribution Quantiles.

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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | ```
# Generate 20 observations from a normal distribution with parameters
# mean=10 and sd=2, then create a tolerance interval.
# (Note: the call to set.seed simply allows you to reproduce this
# example.)
set.seed(250)
dat <- rnorm(20, mean = 10, sd = 2)
tolIntNorm(dat)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Normal
#
#Estimated Parameter(s): mean = 9.861160
# sd = 1.180226
#
#Estimation Method: mvue
#
#Data: dat
#
#Sample Size: 20
#
#Tolerance Interval Coverage: 95%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact
#
#Tolerance Interval Type: two-sided
#
#Confidence Level: 95%
#
#Tolerance Interval: LTL = 6.603328
# UTL = 13.118993
#----------
# Clean up
rm(dat)
#--------------------------------------------------------------------
# Example 17-3 of USEPA (2009, p. 17-17) shows how to construct a
# beta-content 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 construct the tolerance interval; note however that it is
# easier to call the function tolIntLnorm instead using the
# original data.
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
tol.int.list <- with(EPA.09.Ex.17.3.chrysene.df,
tolIntNorm(log(Chrysene.ppb[Well.type == "Background"]),
ti.type = "upper", coverage = 0.95, conf.level = 0.95))
tol.int.list
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Normal
#
#Estimated Parameter(s): mean = 2.5085773
# sd = 0.6279479
#
#Estimation Method: mvue
#
#Data: log(Chrysene.ppb[Well.type == "Background"])
#
#Sample Size: 8
#
#Tolerance Interval Coverage: 95%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact
#
#Tolerance Interval Type: upper
#
#Confidence Level: 95%
#
#Tolerance Interval: LTL = -Inf
# UTL = 4.510032
# Compute the upper tolerance interaval on the original scale
# by exponentiating the upper tolerance limit:
exp(tol.int.list$interval$limits["UTL"])
# UTL
#90.9247
#----------
# Clean up
rm(tol.int.list)
``` |

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