tolIntNormHalfWidth {EnvStats} | R Documentation |

## Half-Width of a Tolerance Interval for a Normal Distribution

### Description

Compute the half-width of a tolerance interval for a normal distribution.

### Usage

```
tolIntNormHalfWidth(n, sigma.hat = 1, coverage = 0.95, cov.type = "content",
conf.level = 0.95, method = "wald.wolfowitz")
```

### Arguments

`n` |
numeric vector of positive integers greater than 1 indicating the sample size upon
which the prediction interval is based.
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 |

### Details

If the arguments `n`

, `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 two-sided tolerance interval based on the sample size, the
observed sample mean and sample standard deviation, and specified confidence level
and coverage. Specifically, the two-sided tolerance interval is given by:

`[\bar{x} - Ks, \bar{x} + Ks] \;\;\;\;\;\; (1)`

where `\bar{x}`

denotes the sample mean:

`\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \;\;\;\;\;\; (2)`

`s`

denotes the sample standard deviation:

`s^2 = \frac{1}{n-1} \sum_{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 half-width of the tolerance interval is
given by:

`HW = Ks \;\;\;\;\;\; (4)`

### Value

numeric vector of half-widths.

### Note

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 half-width
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 normally-distributed observations.

### Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

### References

See the help file for `tolIntNorm`

.

### See Also

`tolIntNorm`

, `tolIntNormK`

,
`tolIntNormN`

, `plotTolIntNormDesign`

,
`Normal`

.

### Examples

```
# Look at how the half-width of 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
round(tolIntNormHalfWidth(n = 10, coverage = seq(0.5, 0.9, by = 0.1)), 2)
#[1] 1.17 1.45 1.79 2.21 2.84
#----------
# Look at how the half-width of a tolerance interval decreases with
# increasing sample size:
2:5
#[1] 2 3 4 5
round(tolIntNormHalfWidth(n = 2:5), 2)
#[1] 37.67 9.92 6.37 5.08
#----------
# Look at how the half-width of a tolerance interval increases with
# increasing estimated standard deviation for a fixed sample size:
seq(0.5, 2, by = 0.5)
#[1] 0.5 1.0 1.5 2.0
round(tolIntNormHalfWidth(n = 10, sigma.hat = seq(0.5, 2, by = 0.5)), 2)
#[1] 1.69 3.38 5.07 6.76
#----------
# Look at how the half-width of a tolerance interval increases with
# increasing confidence level for a fixed sample size:
seq(0.5, 0.9, by = 0.1)
#[1] 0.5 0.6 0.7 0.8 0.9
round(tolIntNormHalfWidth(n = 5, conf = seq(0.5, 0.9, by = 0.1)), 2)
#[1] 2.34 2.58 2.89 3.33 4.15
#==========
# 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 estimate the
# standard deviation based on the two background wells. We will use this
# estimate of standard deviation to compute the half-widths of
# future tolerance intervals on the log-scale for various sample sizes.
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
tolIntNormHalfWidth(n = c(4, 8, 16), sigma.hat = sigma.hat)
#[1] 3.999681 2.343160 1.822759
#==========
# Clean up
#---------
rm(summary.stats, sigma.hat)
```

*EnvStats*version 2.8.1 Index]