Sample Size for a Specified Half-Width of a Tolerance Interval for a Normal Distribution

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Description

Compute the sample size necessary to achieve a specified half-width of a tolerance interval for a normal distribution, given the estimated standard deviation, coverage, and confidence level.

Usage

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  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 = 1e-07, maxiter = 1000)

Arguments

half.width

numeric vector of (positive) half-widths. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.

sigma.hat

numeric vector specifying the value(s) of the estimated standard deviation(s). The default value is sigma.hat=1.

coverage

numeric vector of values between 0 and 1 indicating the desired coverage of the tolerance interval. The default value is coverage=0.95.

cov.type

character string specifying the coverage type for the tolerance interval. The possible values are "content" (β-content; the default), and "expectation" (β-expectation).

conf.level

numeric vector of values between 0 and 1 indicating the confidence level of the prediction interval. The default value is conf.level=0.95.

method

character string specifying the method for constructing the tolerance interval. The possible values are "exact" (the default) and "wald.wolfowitz" (the Wald-Wolfowitz approximation).

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 round.up=TRUE.

n.max

positive integer greater than 1 specifying the maximum possible sample size. The default value is n.max=5000.

tol

numeric scalar indicating the tolerance to use in the uniroot search algorithm. The default value is tol=1e-7.

maxiter

positive integer indicating the maximum number of iterations to use in the uniroot search algorithm. The default value is maxiter=1000.

Details

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 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} ∑_{i=1}^n x_i \;\;\;\;\;\; (2)

s denotes the sample standard deviation:

s^2 = \frac{1}{n-1} ∑_{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)

The function tolIntNormN uses the uniroot search algorithm to determine the sample size for specified values of the half-width, sample standard deviation, coverage, and confidence level. Note that unlike a confidence interval, the half-width of a tolerance interval does not approach 0 as the sample size increases.

Value

numeric vector of sample sizes.

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, tolIntNormHalfWidth, plotTolIntNormDesign, Normal.

Examples

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  # 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 half-width:

  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 half-width:

  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 half-width:

  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 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 required sample sizes for 
  # various half-widths on the log-scale.

  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 half-width 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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