In this example, we have a lot of 400 items. We take a sample of 100 items from the lot. We accept the entire lot as being good if the sample has two or less defective items.

We use the hypergeometric distribution (@PHG) to determine the exact producer risk (probability of rejecting a good lot) and the exact consumer risk (probability of accepting a bad lot). In the days before computers were widely available, statisticians had to rely on published tables of the probability distributions to compute probabilities such as these. Because the hypergeometric distribution is specified by four parameters, it would have been unrealistic to carry around hypergeometric tables that covered all possible scenarios. Instead, statisticians routinely used distributions of fewer parameters to approximate the hypergeometric. So, in deference to the good old days, we make use of the binomial, Poisson, and normal approximations to the hypergeometric to compute these same risk probabilities. The interested reader can compare the accuracy of the various approximations.

MODEL:

 

! Acceptance sampling: taking one or more samples

  at random from a lot, inspecting each of the

  items in the sample(s), and deciding on the basis

  of inspection results whether to accept or reject

  the entire lot. (See Schroeder, Oper. Mgt.) This

  Acceptance Sampling model illustrates the effect

  of choice of distribution.;

 

! From a lot of 400 items;

 LOTSIZE = 400;

! We take a sample of size 100;

 SAMPSIZE = 100;

! Producer considers the lot good if

   the lot fraction defective is .0075 or less;

 FGOOD = .0075;

! Consumer considers the lot bad if

   the lot fraction defective is .025 or more;

 FBAD = .025;

! We accept the lot if sample contains 2 or less;

 ACCEPTAT = 2;

 

! The model;

! What is producer risk of rejecting a good lot;

  ! Using the (exact) hypergeometric distribution;

 PGOODH = 1 - @PHG( LOTSIZE, LOTSIZE * FGOOD,

  SAMPSIZE, ACCEPTAT);

  ! Using binomial approx. to the hypergeometric;

 PGOODB = 1 - @PBN( FGOOD, SAMPSIZE, ACCEPTAT);

  ! Using the Poisson approx. to the binomial;

 PGOODP = 1 - @PPS( FGOOD * SAMPSIZE, ACCEPTAT);

  ! Using Normal approximation;

 PGOODN =

  1 - @PSN( (ACCEPTAT + .5 - MUG) / SIGMAG);

  ! where;

 MUG = SAMPSIZE * FGOOD;

 SIGMAG = ( MUG * ( 1 - FGOOD)) ^ .5;

 

! What is the consumer risk of accepting a bad lot;

  ! Using the hypergeometric;

 PBADH = @PHG( LOTSIZE, LOTSIZE * FBAD,

  SAMPSIZE, ACCEPTAT);

  ! Binomial;

 PBADB = @PBN( FBAD, SAMPSIZE, ACCEPTAT);

  ! Poisson;

 PBADP = @PPS( FBAD * SAMPSIZE, ACCEPTAT);

  ! Using Normal approximation;

 PBADN = @PSN( ( ACCEPTAT + .5 - MUB) / SIGMAB);

 ! where;

 MUB = SAMPSIZE * FBAD;

 SIGMAB = ( MUB * ( 1 - FBAD)) ^ .5;

 

END

Model: SAMPLE