PROBACC2 Function :: SAS/QC(R) 12.1 User's Guide

Syntax

PROBACC2

where

$a_{1}$	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 $a_{2} > a_{1}$ .
$n_{1}$	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 $n_{1}+n_{2}\leq N$ .
D	is the number of nonconforming items in the lot, where $0\leq D\leq N$ .
N	is the lot size, where $N\geq 2$ .
p	is the proportion of nonconforming items produced by the process, where 0 < p < 1.

Description

The PROBACC2 function returns the acceptance probability for a double-sampling plan of Type A if you specify the parameters D and N, and it returns the acceptance probability for a double-sampling plan of Type B if you specify the parameter p. For details on Type A and Type B double-sampling plans, see Types of Sampling Plans.

For either type of sampling plan, the acceptance probability is calculated as

$P_{a_1} + P_{a_2}$

where

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

and

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

These probabilities are determined from either the hypergeometric distribution (Type A sampling) or the binomial distribution (Type B sampling).

Examples

The first set of statements results in a value of 0.2396723824. The second set of statements results in a value of 0.0921738126.

data;
   prob=probacc2(1,4,3,50,100,10,200);
   put prob;
run;

data;
   prob=probacc2(0,2,1,13,13,0.18);
   put prob;
run;