tolerance (version 3.0.0)

diffnormtol.int: 1-Sided Tolerance Limits for the Distribution of the Difference Between Two Independent Normal Random Variables

Description

Provides 1-sided tolerance limits for the difference between two independent normal random variables. If the ratio of the variances is known, then an exact calculation is performed. Otherwise, approximation methods are implemented.

Usage

diffnormtol.int(x1, x2, var.ratio = NULL, alpha = 0.05, 
                P = 0.99, method = c("HALL", "GK", "RG"))

Value

diffnormtol.int returns a data frame with items:

alpha

The specified significance level.

P

The proportion of the population covered by this tolerance interval.

diff.bar

The difference between the sample means.

1-sided.lower

The 1-sided lower tolerance bound.

1-sided.upper

The 1-sided upper tolerance bound.

Arguments

x1

A vector of sample data which is distributed according to a normal distribution (sample 1).

x2

Another vector of sample data which is distributed according to a normal distribution (sample 2). It can be of a different sample size than the sample specified by x1.

var.ratio

A specified, known value of the variance ratio (i.e., the ratio of the variance for population 1 to the variance of population 2). If NULL, then the variance ratio is estimated according to one of the three methods specified in the method argument.

alpha

The level chosen such that 1-alpha is the confidence level.

P

The proportion of the population to be covered by the tolerance limits.

method

The method for estimating the variance ratio. This only needs to be specified in the case when var.ratio is not NULL. "HALL" is Hall's method, which takes a bias-corrected version of the ratio between the sample variance for sample 1 to the sample variance for sample 2. "GK" is the Guo-Krishnamoorthy method, which first calculates a bias-corrected version of the ratio between the sample variance for sample 2 to the sample variance for sample 1. The resulting limit is then compared to the limit from Hall's method and the most conservative limit is chosen. "RG" is the Reiser-Guttman method, which is a biased version of the variance ratio that is calculated by taking the sample variance for sample 1 to the sample variance for sample 2. Typically, Hall's method or the Guo-Krishnamoorthy method are preferred to the Reiser-Guttman method.

Details

Satterthwaite's approximation for the degrees of freedom is used when the variance ratio is unknown.

References

Guo, H. and Krishnamoorthy, K. (2004), New Approximate Inferential Methods for the Reliability Parameter in a Stress-Strength Model: The Normal Case, Communications in Statistics - Theory and Methods, 33, 1715--1731.

Hall, I. J. (1984), Approximate One-Sided Tolerance Limits for the Difference or Sum of Two Independent Normal Variates, Journal of Quality Technology, 16, 15--19.

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

Reiser, B. J. and Guttman, I. (1986), Statistical Inference for Pr(Y < X): The Normal Case, Technometrics, 28, 253--257.

See Also

Normal, K.factor, normtol.int

Examples

## 90%/99% tolerance limits for the difference between two
## simulated normal data sets.  This data is taken from
## Krishnamoorthy and Mathew (2009).  Note that there is a
## calculational error in their example, which yields different
## results with the output below. 

x1 <- c(10.166, 5.889, 8.258, 7.303, 8.757)
x2 <- c(-0.204, 2.578, 1.182, 1.892, 0.786, -0.517, 1.156,
        0.980, 0.323, 0.437, 0.397, 0.050, 0.812, 0.720)

diffnormtol.int(x1, x2, alpha = 0.10, P = 0.99, method = "HALL")
diffnormtol.int(x1, x2, alpha = 0.10, P = 0.99, method = "GK")
diffnormtol.int(x1, x2, alpha = 0.10, P = 0.99, method = "RG")
diffnormtol.int(x1, x2, var.ratio = 3.8, alpha = 0.10, P = 0.99)