Functions

Evaluating Single-Sampling Plans

Subsections:

You can use the Base SAS functions PROBBNML and PROBHYPR to evaluate single-sampling plans. Measures of the performance of single-sampling plans include

  • the probability of acceptance $P_ a$

  • the average sample number ASN

  • the average outgoing quality AOQ

  • the average total inspection ATI

Probability of Acceptance

Since $P_ a$ is the probability of finding c or fewer defectives in the sample, you can calculate the acceptance probability using the function PROBHYPR$(N,D,n,c)$ for Type A sampling and the function PROBBNML$(p,n,c)$ for Type B sampling.

For example, the following statements calculate $P_ a$ for the plan n = 20, c = 1 when sampling from a single lot of size N = 120 that contains D = 22 nonconforming items, resulting in a value of 0.0762970752: 

data;
   prob=probhypr(120,22,20,1);
   put prob;
run;

Similarly, the following statements calculate $P_ a$ for the plan n = 20, c = 1 when sampling from a series of lots for which the proportion of nonconforming items is p = 0.18, resulting in a value of 0.1018322793: 

data;
   prob=probbnml(0.18,20,1);
   put prob;
run;

Other Measures of Performance

The measures ASN, AOQ, and ATI are meaningful only for Type B sampling and can be calculated using the PROBBNML function. For reference, the following equations are provided.

Average sample number: Following the notation of Schilling (1982), let $F(c|n)$ denote the probability of finding c or fewer nonconforming items in a sample of size n. Note that $F(c|n)$ is equivalent to PROBBNML$(p,n,c)$. Then, depending on the mode of inspection, the average sample number can be expressed as shown in the following table:

Mode of Inspection

ASN

Full

n

Semicurtailed

$nF(c|n) + \displaystyle \frac{ (c+1)(1-F(c+1|n+1)) }{p} $

Fully curtailed

$\displaystyle \frac{(n-c)F(c|n+1)}{1-p} + \displaystyle \frac{(c+1)(1-F(c+1|n+1))}{p} $

Average outgoing quality can be expressed as

\[ \mbox{AOQ}=\frac{p(N-n)F(c|n)}{N} \]

if the nonconforming items found are replaced with conforming items, and as

\[ \mbox{AOQ}=\frac{p(N-n)F(c|n)}{N-np} \]

if the nonconforming items found are not replaced.

Average total inspection can be expressed as

\[ \mbox{ATI}=n+(1-F(c|n))(N-n) \]