February 2012 

NASA/TM-2012-217335/Volume I 

NESC-RP-08-00448 

 

Probabilistic Requirements (Partial) Verification 

Methods Best Practices Improvement 

Variables Acceptance Sampling Calculators: Derivations 

and Verification of Plans 

Kenneth L. Johnson/NESC 

Langley Research Center, Hampton, Virginia 

K. Preston White, Jr. 

University of Virginia, Charlottesville, Virginia 
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February 2012 

NASA/TM-2012-217335/Volume I 

NESC-RP-08-00448 

 

Probabilistic Requirements (Partial) Verification 

Methods Best Practices Improvement 

Variables Acceptance Sampling Calculators: Derivations 

and Verification of Plans 

Kenneth L. Johnson/NESC 

Langley Research Center, Hampton, Virginia 

K. Preston White, Jr. 

University of Virginia, Charlottesville, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

The use of trademarks or names of manufacturers in the report is for accurate reporting and does not 

constitute an official endorsement, either expressed or implied, of such products or manufacturers by the 

National Aeronautics and Space Administration. 
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Volume I

Probabilistic Requirements (Partial) Verification Methods Best 

Practices Improvement

Variables Acceptance Sampling Calculators:

Derivations and Verification of Plans

October 27, 2011
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Approval and Document Revision History 

NOTE: This document was approved at the October 27, 2011, NRB. This document was 

submitted to the NESC Director on January 30, 2012, for configuration control.

Approved: Original Signature on File 1/30/12

NESC Director Date

Version Description of Revision Office of Primary 

Responsibility

Effective Date

1.0 Initial Release Kenneth L. Johnson, Statistics 

and Trending, Marshall Space 

Flight Center

10/27/11
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Table of Contents

Volume I: Technical Assessment Report

1.0 Notification and Authorization.........................................................................................5

2.0 Signature Page....................................................................................................................6

3.0 Team List............................................................................................................................7

4.0 Abstract...............................................................................................................................8

5.0 Introduction........................................................................................................................9

6.0 Probabilistic Requirements and Limit Standards........................................................10

7.0 Sampling Plans and Operating Characteristics............................................................11

8.0 Hypothesis Testing and Acceptance Limits...................................................................12

9.0 General Procedure for Developing Variables Acceptance Sampling Plans...............13

10.0 Methods Implemented.....................................................................................................15

10.1 Exponential ........................................................................................................... 15

10.2 Normal (-known)................................................................................................ 16

10.3 Normal (-unknown)............................................................................................ 18

10.4 Gamma.................................................................................................................. 19

10.5 Weibull.................................................................................................................. 20

10.6 Inverse Gaussian ................................................................................................... 21

10.7 Poisson .................................................................................................................. 23

11.0 Conclusions.......................................................................................................................24

12.0 Acronyms List ..................................................................................................................25

13.0 References.........................................................................................................................25

14.0 Appendix...........................................................................................................................26

List of Figures

Figure 9.0-1. Distribution Functions and Critical Values for X Under the Null and 

Alternate Hypotheses......................................................................................... 14

Figure 9.0-2. Sampling Distributions for the Estimator 

ˆ



Under the Null and Alternate 

Hypotheses......................................................................................................... 14

Figure 10.0-1. The Individual Terms of the Inverse-Gaussian Distribution, the Ratio 

R = F2(x;,)/F1(x;,), and the Sum p = F1(x;,) + F2(x;,), as Functions of 
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the Lower-Limit-to-Mean Ratio xmin/ for the Shape-to-Mean Ratio 

/ = 10.............................................................................................................. 22

Figure 10.0-2. The Ratio of the Terms of the Inverse-Gaussian Distribution, 

R = F2(x;,)/F1(x;,), and the Corresponding Failure Probability, p, as 

Functions of the Lower-Limit-to-Mean Ratio xmin/, Parameterized on the 

Shape-to-Mean Ratio for / = 10, 20, 100....................................................... 23
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Volume I: Technical Assessment Report

1.0 Notification and Authorization

The NASA Engineering and Safety Center (NESC) was requested to improve on the Best 

Practices document produced for the NESC assessment 07-001-E, Verification of Probabilistic 

Requirements for the Constellation Program (CxP), by giving a recommended procedure for 

using acceptance sampling by variables techniques. This recommended procedure would be 

used as an alternative to the potentially resource-intensive acceptance sampling by attributes 

method given in the document.

An NESC out-of-board activity was approved on January 9, 2008. Mr. Kenneth Johnson at 

Marshall Space Flight Center (MSFC) was selected to lead this assessment. The assessment plan 

was approved at the NESC Review Board (NRB) on March 20, 2008. 
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2.0 Signature Page 

Submitted by:

Team Signature Page on File – 2-2-12

Kenneth L. Johnson Date

Significant Contributors:

K. Preston White, Jr. Date

Signatories declare the findings and observations compiled in the report are factually based from 

data extracted from Program/Project documents, contractor reports, and open literature, and/or 

generated from independently conducted tests, analyses, and inspections. 
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3.0 Team List

Name Discipline Organization/Location

Core Team

Kenneth Johnson Statistics and Trending MSFC

K. Preston White, Jr. Department of Systems and Information 

Engineering University of Virginia

Chris Johansen MTSO LaRC

Administrative Support

Christina Williams Technical Writer LaRC/ATK
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4.0 Abstract

Acceptance sampling is a method for verifying quality or performance requirements using 

sample data. Acceptance sampling by variables (ASV) is an alternative to acceptance sampling 

by attributes (ASA). In many instances, ASV requires significantly smaller samples than ASA. 

In an effort to make ASV more widely accessible, the team developed calculators for existing 

sampling plans identified in the open literature. For this paper, the literature on ASV was 

consolidated by providing a unified exposition of the approach used to develop such plans. From 

within this framework, the derivation of plans for exponential, normal, gamma, Weibull, inverseGaussian, and Poisson random variables were reviewed and verified. Verification unexpectedly 

surfaced a flaw in a published plan for inverse-Gaussian variables. A companion paper 

confirmed these results empirically and addressed the practical application of ASV when 

distributional forms were themselves uncertain.
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5.0 Introduction

One of the oldest problems in quality engineering is to assess the acceptability of items that a 

customer receives from a producer. Acceptance sampling is an alternative to 100-percent

inspection applied when inspection is destructive, or when the time and/or cost of 100-percent 

inspection are unwarranted or prohibitive. The customer decides the disposition of an incoming 

lot based on a standard specifying the minimum proportion of nonconforming items in the 

sample. The decision can be to accept or reject the entire lot, or to continue sampling. 

Acceptance sampling also has been adapted to problems not identified with procurement. 

Smith et al. (2003) advocated the application of acceptance sampling in the context of water 

quality assessment. Bayard et al. (2007) applied the underlying concept to the assessment of 

future aircraft runway-incursion controls. White, et al. (2009) showed that acceptance sampling 

can be applied to any sampling experiment, including those commonly employed to verify 

design requirements using simulation and Monte Carlo methods.

ASA determines the acceptability of a lot based on a count of the number of nonconforming 

items relative to the size of a random sample drawn from this lot. The inspection variable is 

binary (pass/fail) and therefore the count is necessarily a binomial random variable. ASA can be 

used with categorical outputs, or with outputs measured on a continuous or discrete scale, by 

reference to a required limiting value. Conceptually simple, easily applied, and universally 

applicable, ASA is the first choice for sampling inspection.

ASV is an alternate approach, which in many instances prescribes significantly smaller samples 

than ASA. ASV requires that the inspection variable is measured on a continuous or discrete 

scale, that the distribution of this variable is known a priori and stable, and that a plan exists for 

this particular distribution. While far more restrictive in its assumptions, ASV may be 

considered when the assumptions are appropriate and the larger samples required by ASA are 

unavailable.

White, et al. (2009) describes the development of ASA plans and provides an example 

illustrating the application of ASV to a Level 2 requirement from the Constellation Program

(CxP). The first objective of this paper was to consolidate the scholarly literature on ASV and 

provide a readable tutorial on the concept and general approach used to develop ASV plans for 

alternative parent distributions. While ASV does employ a consistent procedure for determining 

sampling plans, this commonality was obscured by different presentation styles and the vastly 

different notation adopted by researchers in the field. This paper presents a unified presentation 

and a consistent notation. An example is provided in Appendix A.

The second objective was to validate the mathematical derivations. Validation was successful in 

all but one case. A subtle flaw in the published plan for inverse-Gaussian variables was 

discovered that rendered it unusable, at least in its present form. A companion paper confirmed
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these results and addressed the practical application of ASV when distributional forms were 

themselves uncertain.

6.0 Probabilistic Requirements and Limit Standards

In uncertain environments, requirements verification seeks to determine whether a measureable 

quantity is conforming or nonconforming, i.e., to determine whether or not the parent population 

from which a sample is drawn achieves a specified level of quality or performance. It is 

important to remember that this is a simplified and incomplete view of verification. A true 

requirement verification using these methods alone would be incomplete—a requirement can be 

judged as verified when a full understanding of the assumptions is clearly presented and 

understood. NASA-STD-7009, Standard for Models and Simulations (2008), offers a 

framework for doing much of this. The discussion on epistemic versus aleatory uncertainties in 

the Probabilistic Risk Assessment Procedures Guide for NASA Managers and Practitioners

Version 1.1 (2002)1is also recommended. It is important to recognize that this verification 

process does not generally apply at all to the ―Probable Risk Assessment-type‖ requirements that 

directly address loss of crew or vehicle.

The remainder of the document will address this partial verification. The methods outlined in 

this paper were meant to improve the rigor of the portion of the requirements verification process 

involving the raw exercise of a simulation model and the comparison of its output to the stated 

quantified requirement threshold. These methods do not address the critical wider issue of 

analysis method assumptions.

A probabilistic requirement can be stated probabilistically as a (I, , ) limit standard

(White, et al., 2009), where:

(1) I is the performance indicator. The measured quantity may be inherently categorical or 

qualitative in nature and performance indicated by occurrence or nonoccurrence of some 

event. Alternately, the measured quantity may be inherently quantitative and 

performance is indicated by success or failure in achieving a limit or tolerance. 

(2)  is the minimum reliability for the population. This is the minimum, acceptable, longrun proportion of observations on which the population achieves the desired performance. 

If p is the failure probability for the population, then the requirement is p < 1  .

 

1

Probabilistic Risk Assessment Procedures Guide for NASA Managers and Practitioners Version 1.1 has been 

replaced by NASA/SP‐2011‐3421 Probabilistic Risk Assessment Procedures Guide for NASA Managers and 

Practitioners second edition.
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 is the maximum acceptable consumer’s risk. This is the probability of incorrectly 

accepting a nonconforming population as the result of sampling instead of 100-percent

inspection.

Formally, a measurable quantity is represented as the random variable X with a probability 

distribution of F(x;) and density is f(x;), where  is an unknown distribution parameter. Let 

{Xi; i = 1, ..., n} be a random sample with observed values {xi; i = 1, ..., n}. The verification 

problem is to determine whether the population as a whole conforms to a specified limit 

standard, based on the statistics of the sample observed. 

7.0 Sampling Plans and Operating Characteristics

A sampling plan is the pair (n,), where n is the minimum sample size, i.e., the minimum 

number of observations required to verify statistically the requirement imposed by the standard. 

 is a constant factor which is used to assess whether or not the population is conforming. The 

interpretation of  depends on whether the characteristic is continuous or discrete, as discussed 

in Section 8.0. 

For a given distribution, every sampling plan has a unique operating characteristic (OC). The 

OC is a function that defines the probability of accepting a population, Pa, for every value of the 

failure probability, p, i.e., the OC is the function Pa(p) where p[0,1]. A sampling plan is 

derived by first defining two operating points, (p0,1  ) and (p1,), where p0 < p1 and  and 

are small probabilities. It is then required that OC must (minimally) satisfy the following 

inequalities:

Pa(p0) > 1   (Eq. 1)

Pa(p1) < . (Eq. 2)

Alternately, it is sometimes more convenient to express these in terms of the rejection 

probabilities as the power conditions Pr(p0) = 1  Pa(p0) <  and Pr(p1) = 1  Pa(p1) > 1  ). 

Note that when p1 = 1  and  is the consumer’s risk, inequality (Eq. 2) enforces the limit 

standard. Under this plan, a population with failure probability p1 as conforming to small 

probability  is noted. Inequality (Eq. 1) captures the competing good. Under this plan, a 

population with probability p0 as conforming to high probability 1   is noted. 

The probability of incorrectly rejecting a conforming system, , was called the producer’s risk.

When given as percentages, 100-percent x (1  p0) was called the acceptable quality level and 

100-percent x (1  p1) was called the lot tolerance percent defective. Note that the limit standard 

does not specify the (p0,1  ) operating point. It was common to develop a range of plans with 

the required consumer’s risk  and differing. Plans with larger sample sizes n then have 

smaller corresponding. 
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8.0 Hypothesis Testing and Acceptance Limits

The underlying problem was framed as a hypothesis test, which was intended to enforce both 

significance and power requirements. The null and alternate hypotheses are:

H0: p = p0 and H1: p = p1 > p0

Under H0, the population was accepted as conforming and under H1 the population was rejected 

as nonconforming. Inequality (Eq. 1) establishes the significance of test as  and inequality 

(Eq. 2) establishes the power of the test as 1  , where  and  are the probabilities of a Type I 

error (the producer’s risk or the risk of rejecting a population that should have been accepted), 

and a Type II error (the consumer’s risk, or the risk of accepting a population that should have 

been rejected), respectively.

The sample data were used to choose between the null and alternate hypotheses. To accomplish 

this, the critical value of an appropriate test statistic needed to be determined. Denote the test 

statistic as the acceptance limit A(n,), where the arguments are the parameters of the sampling 

plan. For a continuous quantity,  = k is the acceptance limit and typically has the form:

ˆ

A n k n k n ( , ) ( ) ( )     ˆ

where the plus is used with an upper bound and the minus with a lower bound. It follows that:

ˆ

( , ) ( )

ˆ( )

A n n k

n

 







(Eq. 3)

where 



ˆ

and 



ˆ

are estimators for the unknown parameter and the standard deviation of the 

population. 

For I a lower specification limit with value xmin, the desired performance on the i

th observation is 

achieved if and only if xi > xmin. For the sample as a whole, the null hypothesis is rejected if

min A n k x ( , )   . That is, the acceptance limit is smaller than the specified limit.

For I an upper specification limit with value xmax, the desired performance on the i

th observation 

is achieved if and only if xi < xmax. For the sample as a whole, the null hypothesis is rejected if

max A n k x ( , )  

. That is, the acceptance limit is larger than the specified limit.

The effect in either case is to move the critical point of the acceptance limit away from the 

estimate of the unknown quantity in the direction of the specified limit by k standard deviations. 

Intuitively, this hedge is intended to compensate for error in the estimated mean for a sample of 

n observations in a statistically exact way.

Note that in this paper the third case will not be addressed, where two limits are specified. In 

this case, I is a tolerance interval, xmax > xi > xmin. The derivation of sampling plans for tolerance 
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is modestly more complicated than that described in Section 8.0. The reader is referred to the 

literature on tolerance intervals.

For a discrete population characteristic, the acceptance limit has the form:

A n c Y n c ( , ) ( )  

where Y is some function of X with nonnegative integer values. It follows that:

c Y n A n c   ( ) ( , ).

(Eq. 4)

For a lower specification limit on Y, the acceptance number, c, is the minimum acceptable value 

of Y(n) and the rejection criteria is A(n,c) < 0. For an upper specification limit on Y, c is the 

maximum acceptable value of Y(n) and the rejection criteria is A(n,c) > 0.

9.0 General Procedure for Developing Variables Acceptance 

Sampling Plans

For a continuous distribution with lower specification limit, the procedure comprises 4 steps: 

(1) Determine the value of 0 (typically the mean) for the null probability distribution with 

failure probability p0 = Pr[X < xmin] = F(xmin;0); determine the value of 1 for the 

alternate distribution with failure probability p1 = Pr[X < xmin.] = F(xmin;1). This is 

accomplished using the inverse distribution. As shown in Figure 9.0-1, the requirement 

that p0 < p1 implies that 0 > 1.

(2) Determine the minimum value of n from the sampling distributions for 0 and 1 such 

that both inequalities (Eq. 1) and (Eq. 2) are satisfied.

(3) Determine the maximum acceptance limit, A0, from the null sampling distribution with

0 0

ˆ ˆ

       Pr[ ] ( ; , ) A F n

; determine the minimum acceptance limit, A1, from the 

alternate sampling distribution with 

1 1        Pr[ ] ( ; , )

ˆ ˆ

A F n

; as illustrated in 

Figure 9.0-2.

(4) Determine the factors k0 and k1 corresponding to A0 and A1, respectively, from 

equation (3). In general, these factors are not equal. A conservative choice is to use the 

larger of the two values as the factor k for the sampling plan. Alternately, k is sometimes 

taken as the average of these two values. 

For I, an upper specification limit, the failure probability of the population is 

p = Pr[X > xmax] = 1  F(x;). The procedure is identical with this single exception. For a 

discrete distribution, instead of the factors k0 and k1, at Step 4 the constant c is determined.
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Figure 9.0-1. Distribution Functions and Critical Values for X Under the Null and Alternate 

Hypotheses

Figure 9.0-2. Sampling Distributions for the Estimator 

ˆ



Under the Null and Alternate 

Hypotheses

If the random variable X and the unknown parameter  can be standardized, it is generally more 

convenient to use the standardized distribution and standardized sampling distribution to derive 
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sampling plans. This convenience is illustrated in Section 10 for exponential and normal random 

variables. 

10.0 Methods Implemented

A literature search found derivations of (n,k) sampling plans for exponential, normal, gamma, 

Weibull, and inverse-Gaussian distributions. These derivations follow the general procedure 

given in Section 9.0. A derivation was also found for the (n,c) sampling plan for the Poisson 

distribution. In this section, these derivations are outlined with reference to lower specification 

limits. The corresponding derivations for upper specification limits are easily deduced.

10.1 Exponential

Guenther (1977) considers the exponential random variable X with unknown mean , supports

x[0,), and distribution function:

/

( ; ) 1 . x F x e 



 

Note that exponential and chi-squared distributions are both special cases of the gamma 

distribution, with exponential () = chi-squared(2) = gamma(1,).

Denote a random variable distributed chi-squared with  degrees of freedom as Y X can be 

standardized as:

2

2

Y X





so that 

Pr( ) Pr( )

p v p 2; p X x Y y    

and

2; min

2

p

y x





(Eq. 5)

where p is either p0 or p1. Similarly, the estimator for the mean  is sample mean

X n( )

, which 

can be standardized as:

2

2

( ).

n

n

Y X n





(Eq. 6)

Substituting equation (5) into equation (6) yields:

;

2 ;

min

2

( ) ( ) n p

n p

n ny

y x n x n

 x

 

where p is either p0 or p1. Inequalities (Eq. 1) and (Eq. 2) imply:
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0

2;

2 ; 2 ;

min

2

( ) ( ) p

n p n

n y

y x n x n y

x





   

1

1

2;

2 ; 2 ;(1 )

min

2

( ) ( ) . p

n p n

n y

y x n x n y

x





   

 

Together, these in turn imply that:

0 1

2 ; 2 ;(1 )

2; min 2;

n ( ) n

p p

y x n y

y x y

 

  

or

1

0

2; 2 ;(1 )

2; 2 ;

.

p n

p n

y y

y y







 (Eq. 7)

The minimum required sample size is the smallest value of the integer n satisfying inequality 

(Eq. 7), which can be determined by table lookup.

With n known, the k-factor can be computed. The estimator for both the mean and standard 

deviation is the sample mean,

X n( )

; therefore, the acceptance limit is:

2

( , ) ( ) ' ( ) (1 ) ( ) (1 ') . ˆ ˆ

2

A n k n k n k X n k Y n

n



        

It is customary to let k = 1  k’ and use the k-factors and

0

2;

0

2 ;

p

n

ny

k

y 

 

1

2;

1

2 ;(1 )

.

p

n

ny

k

y 

 

10.2 Normal (-known)

The derivation of sampling plans for measureable quantities distributed N(,

2

) is 

widely published (Bowker and Goode, 1952; Guenther, 1977; Kao, 1971; Lieberman and 

Resnikoff, 1955; and Montgomery, 2005; among others) and the basis for standards 

Military Standard (MIL-STD)-414 and American National Standards Institute/American Society 

of Quality Control Z1.9, and International Organization for Standardization 2859. Consider the 
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normal random variable X with unknown mean , known standard deviation , supports

x(,), and density function:

2 2 ( ) /(2 )

2

1

( ; , ) .

2

x

f x e    



 



Denote a random variable distributed a standard normal N(0,1) as Z. X can be standardized as:

X

Z









such that 

Pr( ) Pr( )

p p

p X x Z z    

is:

( )

.

p

A x n

z







(Eq. 8)

The mean estimator 

X n( )

is distributed N(,

2

/n), which can be standardized as:

( ) ( ) X n Z n

n









(Eq. 9)

where 

Z n( )

also is a standard normal N(0,1) deviate. 

The acceptance criterion is

min A n k x n k x ( , ) ( )     

, so that the population is rejected if:

min ( )

.

x n x k







Inequalities (Eq. 1) and (Eq. 2) imply:

 

min min

min

( ) ( ) Pr Pr

( ) Pr

Pr

Pr .

p

p

X n x X n x k k n

n

X n x

k n

n n

Z z n k n

Z n k z

 

 

 

     

          

   

      

      

      

The acceptance-limit criteria (expressed as power requirements) from inequalities (Eq. 1) and 

(Eq. 2) then imply:

0

( )

p

n k z z   
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1

(1 ) 1 ( ) .p

n k z z z z          

Together, these imply that:

0

1

0

1

 

p

p

z

k k z

n

z

k z

n





 



 

(Eq. 10)

and therefore

1 0

2

.

p p

z z

n

z z

 

  

     

 

(Eq. 11)

The minimum required sample size is the smallest integer value of n satisfying inequality 

(Eq. 11). With n known, the k-factors can be computed as from equations (Eq. 10). 

10.3 Normal (-unknown)

The more usual case is where  is unknown and must be estimated from the sample data. While 

an exact approach is available using a non-central t-distribution (see Guenther, 1977, among 

others). For modest sample sizes, excellent approximations for n and k are achieved applying a 

result by Cramér (1945). Let

X n( )

and S(n) be sample mean and standard deviation, 

respectively. For n sufficiently large, the distribution of the acceptance limit 

A n k X n kS n ( , ) ( ) ( )  

is asymptotically normally distributed N(A,

2

A), with mean

A      k

and variance:

2

2

3 4 ( , , ). A

e k

n



   

The limiting variance 

2

/n is weighted by the expansion factor e(k,3,4) (Takagi, 1972). For a 

non-Normal random variable X with mean  , standard deviation , skew 3, and kurtosis 4,

 

2

3 4 4 3 ( , , ) 1 1

4

k

e k k    

 

       

(Eq. 12)
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where the plus is used with an upper bound and the minus with a lower bound. For a Normal

random variable X, the skew is 3 = 0 and the kurtosis is 4 = 3. The expansion factor therefore 

reduces to (Wallis, 1947):

2

( ,0,3) 1 .

2

k

e k       

The derivation of n and k then proceeds as in the -known case, resulting in:

1 0

2

( ,0,3)

p p

z z

n e k

z z

 

  

     

 

and

1 0

2

.

p p

z z z z

k

z z

 

 

  

       

The minimum required sample size is the smallest integer value of n satisfying the inequality;

with n known, the k-factors can be computed.

10.4 Gamma

Takagi (1972) considers the gamma random variable X with unknown location  < x <  and 

estimated shape > 0 and scale  > 0 (scale) parameters, support x[,), and density function:

 

1

( )/ ( ) /

( ; , , ) .

( )

x

x

f x e



 

 

  

 



  





The moments for this distribution are:

2 2

3

4

2

3 6 / .

  

 

 

 

 





 

(Eq. 13)

With the same normal approximation N(A,

2

A) for A(k,n), the expansion factor given in

equation (12) is computed using the gamma moments in equations (13). The derivation is 

essentially the same as that for the normal distribution, resulting in:
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1 0

2

3 4 ( , , )

p p

z z

n e k

t t

 

 

  

     

 

(Eq. 14)

and

1 0

2

.

p p

z t z t

k

z z

 

 

  

        

(Eq. 15)

Here,

X

t









is a standard gamma deviate with density:

1

( ; )

( )

x t

f t e















for which T = 0 and T = 1. 

The minimum required sample size is the smallest integer value of n satisfying the inequality; 

with n known, the k-factors can be computed.

10.5 Weibull

Takagi (1972) also considers the Weibull random variable X with unknown location  < x < 

and estimated shape > 0 and scale  > 0 (scale) parameters, support x[,), and density 

function:

 

1

( ; , , ) . x xf x e





   

  

 



        

    

The moments for this distribution are:

2 2 2

3 3

3

2 4

4

(3 1) 3 (2 1) 2 ( 1)

(4 1) 4 (3 1) 9 1) 6 (2 1) ( 1) .

T

T

T

T

b b b

b b b b b

  

  

 

 

 



            

                

(Eq. 16)

Here,
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( ) x

T









is a standard Weibull random variable with density:

1

( ; ) tf t t e



   



and mean and variance:

2 2

( 1)

(2 1) ( 1)

T

T

b

b b





  

     

where b = 1/. 

The expansion factor in (Eq. 12) is computed using the Weibull moments in (Eq. 16). The 

expressions for n and k are again given by (Eq. 14) and (Eq. 15), where it is understood that now

t is the Weibull deviate. 

10.6 Inverse Gaussian

Aminzadeh (1996) considers the inverse-Gaussian random variable X with unknown location 

parameter  < x < , estimated mean  > 0 and shape parameter  > 0, support x[0,), and 

distribution function:

1 2 F x F x F x ( ; , ) ( ; , ) ( ; , )        

where:

1

( ; , ) 1 xF x

x



  



   

           

2

2

( ; , ) 1 xF x e

x



 

  



   

           

and (

.

) is the distribution function of the standard normal distribution N(0,1). As explained 

below, the derivation of the inverse-Gaussian sampling plan appears to be flawed. 

As λ tends to infinity, the inverse-Gaussian distribution F(x;,) is asymptotically normal with 

distribution F1(x;,). On this basis, it is argued that if the shape-to-mean ratio is large 

(/ > 10), then F2(x;,) is almost zero the failure probability of p = F1(x;,). This 

relationship is used to determine the means for the null and alternative distributions as described 

in Section 9.0.
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While it is true that for F2(x;,) is almost zero when / > 10, it is also true that F1(x;,) is 

almost zero. Thus, the second term cannot simply be ignored in the calculation of p0 and p1.

This is illustrated in Figures 10.0-1 and 10.0-2 for / = 10 and lower limit xmin[0,]. 

Figure 10.0-1 shows the values of F1(x;,) and F2(x;,) as functions of the limit-to-mean ratio 

xmin/[0,1]. Also shown is the sum and ratio of these two terms, p = F1(x;,) + F2(x;,) and 

R = F2(x;,)/F1(x;,). Clearly, the contribution of F2(x;,) in the computation of p is 

nontrivial for the entire range of lower limit. For the small failure probabilities typically of 

interest in acceptance sampling, the contribution of F2(x;,) is nearly equal to that of F1(x;,). 

Figure 10.0-2 illustrates the dependency of ratio R on the shape parameter for selected values of 

/ > 10, 20, and 100. Note that as / increases and the distribution becomes increasingly 

normal in shape, the contribution F2(x;,) in the computation of p diminishes, but remains 

nontrivial, especially for small failure probabilities.

Figure 10.0-1. The Individual Terms of the Inverse-Gaussian Distribution, the Ratio 

R = F2(x;,)/F1(x;,), and the Sum p = F1(x;,) + F2(x;,), as Functions of the 

Lower-Limit-to-Mean Ratio xmin/ for the Shape-to-Mean Ratio / = 10
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Figure 10.0-2. The Ratio of the Terms of the Inverse-Gaussian Distribution, 

R = F2(x;,)/F1(x;,), and the Corresponding Failure Probability, p, as Functions of the 

Lower-Limit-to-Mean Ratio xmin/, Parameterized on the Shape-to-Mean Ratio for / = 10, 20, 100

10.7 Poisson

Guenther (1972, 1977) considers the Poisson random variable X with unknown mean , supports

x[0,1,2,...), and distribution function:

0

( ; ) ,

!

x i

i

F x e

i

 







 

The means 0 and 1 can be determined form the inverse distribution. Alternately, Guenther 

exploits the relationship between Poisson and chi-squared distributions and calculates the means 

using the expression:

min

2

2 ;1

.

2

 x p







Consider X as a discrete random variable representing the number of usable items in a lot and 

specification limit xmin as the specified minimum number of usable items in each lot. If a sample 

of n lots are obtained, the total number of usable items obtained is the sum:
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0

n

j

j

Y X



 ,

which is distributed Poisson with a mean of n, i.e., F(y; n). Because the team is working with 

discrete random variables (as is the case in attributes acceptance sampling), the rejection 

criterion is based on the minimum total number of usable items in all of the lots, y > c + 1 = d, 

instead of the distance k. Inequalities (Eq. 1) and (Eq. 2) are:

Pa(p > p0) = 1  F(yd; n0) > 1 

Pa(p > p1) = 1  F(d; n1) < .

Or, more simply, the power requirements F(d; n1) > 1   and F(d; n0) < . Again exploiting 

relationship between Poisson and chi-squared distributions, the power requirements imply:

2 2

2 ; 2 ;1

0 1

.

2 2

d d

n

   

 



 

(Eq. 14)

This expression can be solved for a (d,n) sampling plan by enumeration. The value of integer 

values of d is increased until an integer value of n is found which satisfies both inequalities

(Eq. 14).

11.0 Conclusions

The work reported in this paper is the first phase of a project intended to make ASV a practical 

alternative to ASA when appropriate variables plans are available. Published plans for 

exponential, normal, gamma, Weibull, and inverse-Gaussian random variables were discovered 

while the literature was reviewed. With the exception of normal plans, the search for off-theshelf ASV plan calculators was fruitless.

To perfect an understanding and to facilitate the future development of variable plans for a wider 

selection of distributions, a consistent notation was introduced and interpreted the procedure for 

developing plans within the common framework of hypothesis testing (following the lead of 

Guenther (1972, 1977)). The result presented here is a consolidation of the existing literature.

In reviewing the methods presented, the mathematical derivations were verified. One 

unanticipated result was the discovery of a flaw in a published plan for inverse-Gaussian random 

variables that renders this plan unusable without modification. The flaw is subtle and this plan 

has remained unchallenged for over a decade.

As reported in Volume II (Probabilistic Requirements (Partial) Verification Methods Best 

Practices Improvement), the completion of this foundational research allowed team members to 

implement plan calculators and test the accuracy of these plans empirically. Testing is necessary 

since, as this Volume has shown, the majority of variables plans are based on approximating the 
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standard deviation by employing an expansion factor to the sample standard deviation. The final 

phase of this project will address the practical application of ASV when distributional forms are 

themselves uncertain. The results of this additional research are reported in companion papers.

12.0 Acronyms List

ASA Acceptance Sampling by Attributes

ASV Acceptance Sampling by Variables

ATK Alliant Techsystems, Inc.

CxP Constellation Program

LaRC Langley Research Center

MIL-STD Military Standard

MSFC Marshall Space Flight Center

MTSO Management Technical Support Office

NESC NASA Engineering and Safety Center

NRB NESC Review Board

OC Operating Characteristic

13.0 References

1. Aminzadeh, M. S. (1996). "Inverse-Gaussian Acceptance Sampling Plans by Variables," 

Communications is Statistics--Theory and Methods 25(5)923-935.

2. Bayard, R., Cvijetic, M., Graham, J., Morales, M., and Naff, B. (2007). A Systems Approach 

to Runway Incursion Prevention, unpublished report, Department of Systems and 

Information Engineering, University of Virginia, Charlottesville, Virginia.

3. Bowker, H., and Goode H. P. (1952). Sampling Inspection by Variables. McGraw-Hill: New 

York.

4. Cramér, H. (1945). Mathematical Models of Statistics, Princeton University Press, Princeton, 

New Jersey.

5. Guenther, W. C. (1972). “Variables sampling plans for the Poisson and Binomial,‖ Statistica 

Neerlandia 26:113-120.

6. Guenther, W. C. (1977). Sampling Inspection in Statistical Quality Control, Macmillan, New 

York.

7. Kao, J. H. K. (1971). ―MIL-STD-414: Sampling Procedures and Tables for Inspection by 

Variables for Percent Defective,‖ Journal of Quality Technology 3:28–37. 

8. Lieberman, G. J., and Resnikoff, G. J. (1955). ―Sampling Plans for Inspection by Variables,‖ 

Journal of the American Statistical Association 50:457–516.
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9. Montgomery, D. C. (2005). Introduction to Statistical Quality Control, 5th ed., Wiley, 

New York, New York.

10. Smith, E. P., Zahran, A., Mahmoud, M., and Ye, K. (2003). ―Evaluation of water quality 

using acceptance sampling by variables,‖ Environmetrics 14: 373–386.

11. Takagi, K. (1972). ―On Designing Unknown-Sigma Sampling Plans on a Wide Class of NonNormal Distributions,‖ Technometrics 14(3), 669-678.

12. Wallis, W. A. (1947). ―Use of variables in Acceptance Inspection by Percent Defective,‖ in 

Selected Techniques of Statistical Analysis, McGraw-Hill, New York.

13. White, K. P., Jr., Johnson, K. L., and Creasey, R. R., Jr. (2009). ―Attribute acceptance 

sampling as a tool for verifying requirements using Monte Carlo simulation, Quality 

Engineering, 21(2)203-214). 

14.0 Appendix

Appendix A. Acceptances Sampling by Attributes
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Appendix A. Acceptances Sampling by Attributes

A producer ships a large number of bolts built to withstand 500 foot-pounds of torque without 

breaking. The shipment comes in batches called lots, with 20,000 bolts/lot. Each lot will be 

inspected to determine if the bolts satisfy the minimum torque requirement. Inspection is 

destructive (a broken bolt cannot be used), so acceptance sampling must be applied. A sample of 

n bolts from each lot will be drawn at random and each bolt in the sample will be tested to see if 

it will withstand 500 foot-pounds of torque. If more than c bolts in the lot break, then the entire 

lot will be rejected. Otherwise, the entire lot will be accepted.

The requirement is:

(1) Performance indicator. An individual bolt is conforming if X > 500 foot-pounds, where 

X is applied torque at which the bolt breaks during testing. Otherwise, the bolt is 

nonconforming. 

(2) Minimum reliability. A lot is conforming if no more than 100 bolts (0.5 percent) in that 

lot are nonconforming. That is, the minimum reliability required is  = 0.995. If p is the 

failure probability for the lot, then requirement is p < 1   = 0.005.

 Maximum acceptable consumer’s risk. To achieve the benefits of sampling, the 

consumer recognizes that occasionally a nonconforming lot will be accepted as 

conforming when it is not. The consumer requires that this happen on average no more 

than once for every twenty nonconforming received. That is, the maximum consumer’s 

risk is  = 0.05.

How large should each sample be? How many individual nonconforming bolts can be observed 

in the sample and still accept the entire lot as conforming to the requirement?

Let Y(n) be the number of bolts in a sample of n bolts that fail. Since each sampled bolt will be 

broken, the sampling in this case is sampling without replacement. The distribution of Y(n) is 

therefore hypergeometric. However, if n is less than 10 percent of the lot size, then the binomial 

distribution is a good approximation. Assume therefore that Y(n) will have a binomial 

distribution with the probability of failure p. 

Figure 1 is a screen shot of the binomial (attributes) sampling plan calculator developed during 

this assessment. A minimum reliability of 0.995 and a consumer’s risk of 0.05 have been 

entered as inputs. The third input is the reliability at which the producer’s risk is calculated. The 

results shown provide twenty (n,c) sampling plans, parameterized on producer’s risk. For 

example, a (1549,4) plan is sufficient to test the requirement with an associated producer’s risk 

of  = 0.0385. That is, samples of 1549 bolts (7.774 percent of the 20,000 bolts in a lot) will be 
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drawn at random for testing. Lots with samples which include more than four nonconforming 

bolts will be accepted. Lots with a failure probability greater than 0.005 will be accepted of 

5 percent of the time on average. Lots with failure probability less than 0.999 will be rejected 

3.85 percent on average. Also shown are the 20 corresponding OC curves, which plot the 

acceptance probability as a function of the maximum failure probability for each plan. Note, that 

by design, all of the OC curves intersect at p = 0.005 and Pa = 0.05.

Figure 1. Screen Shot of the User Interface for the Binomial Sampling Plan Calculator

Acceptance Sampling by Variables (ASV)

The ASV plan requires destruction of a considerable fraction of the bolts in each lot (but, 

obviously, far less than 100-percent inspection). Suppose that, based on past experience, it can 

be assumed that the random breaking torques have a normal distribution with mean

 = 528 and standard deviation  = 10. If this assumption is can be sustained, then a sampling 

plan based on normal random variables should reduce the waste from inspection. Let the random 

variables X1, X2,..., Xn be a random sample n such measurements, with sample mean:

0

1

( )

n

i

i

X n T

n 

 
NASA Engineering and Safety Center 

Technical Assessment Report

Document #:

NESC-RP08-00448

Version:

1.0

Title:

Probabilistic Requirements Verification Methods Best 

Practices Improvement

Page #:

29 of 30

NESC Request No.: TI-08-00448

and sample standard deviation:

 

2

0

1

( ) ( ) .

( 1)

n

i

i

S n X X n

n n 

 





How large should this sample be? How small can the sample mean be and still accept the entire 

lot as conforming to the requirement?

Figure 2 is a screen shot of the normal sampling plan calculator developed during this 

assessment. The historical mean and standard deviation have been entered as inputs. The 

required reliability, consumer’s risk, maximum reliability, and lower limit on torque also have 

been entered as inputs. 

Figure 2. Screen Shot of the User Interface for the Normal Sampling Plan Calculator

The results shown provide 20 (n,k) sampling plans, parameterized on producer’s risk. For the

same requirement, a (218, 2.283) normal plan provides comparable protection to the (1549,4) 

binomial plan examined in preceding example, with a producer’s risk of  = 0.040. Indeed, if 

the historical standard deviation is certain, then an even smaller (44, 2.283) plan will suffice. 

The significant reduction in sample size is apparent.
NASA Engineering and Safety Center 

Technical Assessment Report

Document #:

NESC-RP08-00448

Version:

1.0

Title:

Probabilistic Requirements Verification Methods Best 

Practices Improvement

Page #:

30 of 30

NESC Request No.: TI-08-00448

Also included in the results is a column which shows whether or not a normal distribution with 

given input parameters will be accepted (―Yes‖) or not (―No‖) under the corresponding plan. 

This is calculated from the inequality:

mim     k x

and the result indicates whether or not the inequality holds. A final column gives the critical 

value crit, which is the value for which the equality holds exactly (i.e., the minimum of the mean 

that will result in acceptance). 

The historical parameters for this problem were selected to illustrate an important cautionary. 

The same underlying distribution may lead to acceptance under a larger plan and rejection under 

a smaller plan. This is true in the example because the mean was chosen to be very close to the 

critical values for each of these plans. Note that for the (292, 2.7889) plan:

     k 500.12 500

and for the (256, 2.8045):

     k 499.96 500.

Depending on the application, one might question whether such a small difference (which is 

totally an artifact of the small difference in the k-value for the test) is meaningful. When this is 

the case, clearly larger samples on the whole will provide less conservative results. 
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Probabilistic Requirements (Partial) Verification Methods Best Practices 

Improvement

Variables Acceptance Sampling Calculators: Derivations and Verification 

of Plans

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Johnson, Kenneth L.; White, K. Preston, Jr.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

L-20120 NESC-RP-08-00448

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

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Unclassified - Unlimited

Subject Category 65 - Statistics and Probability

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

The NASA Engineering and Safety Center was requested to improve on the Best Practices document produced for the NESC 

assessment, Verification of Probabilistic Requirements for the Constellation Program, by giving a recommended procedure for 

using acceptance sampling by variables techniques. This recommended procedure would be used as an alternative to the 

potentially resource-intensive acceptance sampling by attributes method given in the document. This document contains the 

outcome of the assessment.

15. SUBJECT TERMS

acceptance sampling by attributes; acceptance sampling by variables; NASA Engineering and Safety Center; Constellation 

Program

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35

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(443) 757-5802

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NASA/TM-2012-217335/Volume I

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