AOQ2 Function :: SAS/QC(R) 12.1 User's Guide

Syntax

AOQ2(replacement, $N,a_{1},r_{1},a_{2},n_{1},n_{2},p$ )

where

replacement	has the value 'REP' or 'NOREP', respectively, depending on whether nonconforming items are replaced with conforming items.
N	is the lot size, where $N\geq 2$ .
$a_{1}$	is the acceptance number for the first sample, where $a_{1}\geq 0$ .
$r_{1}$	is the rejection number for the first sample, where $r_{1}>a_{1}+1$ .
$a_{2}$	is the acceptance number for the second sample, where $a_{2}\geq a_{1}$ .
$n_{1}$	is the size of the first sample, where $n_{1}\geq 1$ and $n_{1}+n_{2}\leq N$ .
$n_{2}$	is the size of the second sample, where $n_{2}\geq 1$ and $n_{1}+n_{2}\leq N$ .
p	is the proportion of nonconforming items produced by the process, where 0 < p < 1.

Description

The AOQ2 function returns the average outgoing quality for a Type B double-sampling plan in which nonconforming items are replaced with conforming items (replacement is 'REP') or not replaced (replacement is 'NOREP'). For details on Type B double-sampling plans, see Types of Sampling Plans.

For replacement, the average outgoing quality is

$\mr {AOQ}=\frac{pP_{a_{1}}(N-n_{1}) + pP_{a_{2}}(N-n_{1}-n_{2}) }{N}$

and for no replacement, the average outgoing quality is

$\mr {AOQ}=\frac{ pP_{a_{1}}(N-n_{1}) }{N-n_{1}p } + \frac{ pP_{a_{2}}(N-n_{1}-n_{2}) }{N-n_{1}p-n_{2}p }$

where, in both situations,

$\displaystyle P_{a_{1}}$	$\displaystyle =$	$\displaystyle \sum _{d=0}^{a_{1}} f(d\|n)$
$\displaystyle$	$\displaystyle =$	$\displaystyle \mbox{probability of acceptance for first sample}$
$\displaystyle P_{a_{2}}$	$\displaystyle =$	$\displaystyle \sum _{d=a_{1}+1}^{r_{1}-1} f(d\|n_{1})F(a_{2}-d\|n_{2})$
$\displaystyle$	$\displaystyle =$	$\displaystyle \mbox{probability of acceptance for second sample}$

and

$\displaystyle f(d\|n)$	$\displaystyle =$	$\displaystyle (\stackrel{n}{_ d})p^{d}(1-p)^{n-d}$
$\displaystyle$	$\displaystyle =$	$\displaystyle \mbox{binomial probability that the number of nonconforming items }$
$\displaystyle$	$\displaystyle$	$\displaystyle \mbox{in a sample of size \Mathtext{n} is exactly \Mathtext{d}}$
$\displaystyle F(a\|n)$	$\displaystyle =$	$\displaystyle \sum _{d=0}^{a}f(d\|n)$
$\displaystyle$	$\displaystyle =$	$\displaystyle \mbox{probability that the number of nonconforming items is less}$
$\displaystyle$	$\displaystyle$	$\displaystyle \mbox{than or equal to \Mathtext{a}}$

Examples

The first set of statements results in a value of 0.0148099904. The second set of statements results in a value of 0.0144743043.

data;
   aoq=aoq2('norep',120,0,2,1,13,13,0.18);
   put aoq;
run;

data;
   aoq=aoq2('rep',120,0,2,1,13,13,0.18);
   put aoq;
run;