Calculate k factor for basis values (\(kB\), \(kA\)) with normal distribution

The factors returned by this function are used when calculating basis values (one-sided confidence bounds) when the data are normally distributed. The basis value will be equal to \(\bar{x} - k s\), where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation and \(k\) is the result of this function. This function is internally used by basis_normal() when computing basis values.

k_factor_normal(n, p = 0.9, conf = 0.95)

Arguments

n

the number of observations (i.e. coupons)

p

the desired content of the tolerance bound. Should be 0.90 for B-Basis and 0.99 for A-Basis

conf

confidence level. Should be 0.95 for both A- and B-Basis

Value

the calculated factor

Details

This function calculates the k factors used when determining A- and B-Basis values for normally distributed data. To get \(kB\), set the content of the tolerance bound to p = 0.90 and the confidence level to conf = 0.95. To get \(kA\), set p = 0.99 and conf = 0.95. While other tolerance bound contents and confidence levels may be computed, they are infrequently needed in practice.

The k-factor is calculated using equation 2.2.3 of Krishnamoorthy and Mathew (2008).

This function has been validated against the \(kB\) tables in CMH-17-1G for each value of \(n\) from \(n = 2\) to \(n = 95\). It has been validated against the \(kA\) tables in CMH-17-1G for each value of \(n\) from \(n = 2\) to \(n = 75\). Larger values of \(n\) also match the tables in CMH-17-1G, but R emits warnings that "full precision may not have been achieved." When validating the results of this function against the tables in CMH-17-1G, the maximum allowable difference between the two is 0.002. The tables in CMH-17-1G give values to three decimal places.

For more information about tolerance bounds in general, see Meeker, et. al. (2017).

References

K. Krishnamoorthy and T. Mathew, Statistical Tolerance Regions: Theory, Applications, and Computation. Hoboken: John Wiley & Sons, 2008.

W. Meeker, G. Hahn, and L. Escobar, Statistical Intervals: A Guide for Practitioners and Researchers, Second Edition. Hoboken: John Wiley & Sons, 2017.

“Composite Materials Handbook, Volume 1. Polymer Matrix Composites Guideline for Characterization of Structural Materials,” SAE International, CMH-17-1G, Mar. 2012.

See also

basis_normal()

Examples

kb <- k_factor_normal(n = 10, p = 0.9, conf = 0.95)
print(kb)
#> [1] 2.35464

## [1] 2.35464

# This can be used to caclulate the B-Basis if
# the sample mean and sample standard deviation
# is known, and data is assumed to be normally
# distributed

sample_mean <- 90
sample_sd <- 5.2
print("B-Basis:")
#> [1] "B-Basis:"
print(sample_mean - sample_sd * kb)
#> [1] 77.75587

## [1] B-Basis:
## [1] 77.75587