W. John Braun

University of Western Ontario

Journal of Statistics Education v.8, n.2 (2000)

Copyright (c) 2000 by W. John Braun, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.

**Key Words:** Acceptance sampling;
Binomial nomograph; Single-sampling plan.

A simple procedure is presented for obtaining the
sample size and acceptance number for a single sample
acceptance sampling plan, given the probability of lot
acceptance for lots having proportion defective equal to
*p*_{1}, and the probability of lot rejection
for lots having proportion defective equal to
*p*_{2}. The procedure gives a practical
illustration of the use of the normal approximation to the
binomial distribution that is appropriate for courses on
statistical quality control as well as on introductory
statistics.

1 The most basic acceptance sampling plan considered in
courses on statistical quality control can be described as
follows. A large lot of items is to be inspected in order
to ascertain its quality. A random sample of *n*
items is selected, and *D*, the number of defectives
(or nonconforming items) in the sample is counted. If
*D* exceeds *c*, the acceptance number, then the
lot is rejected. Otherwise, it is accepted. This is the
so-called single-sampling plan.

2 Because of the simplicity and practicality of such
plans, they are also appropriate for discussion or
exercises in introductory courses on statistics, especially
those designed for mathematics or statistics majors. They
are useful as examples of binomial and hypergeometric
models. In addition, the designing of such plans (that is,
deciding upon *n* and *c*) provides nice
nontrivial examples of the use of the normal approximation
to the binomial distribution, highlighting the importance
of the continuity correction, which is often one of the
more difficult topics to motivate in an introductory
course. Strangely, the approach taken toward designing
such plans described in popular quality control textbooks
avoids mention of the normal approximation to the binomial,
even if the approximation is described in the 'statistical
background' chapter. Instead, a 'black-box' method based on
something called a binomial nomograph is described (see,
e.g., Montgomery 1996), and an
opportunity to demonstrate the normal approximation in a
practical setting is missed.

3 Such plans are usually designed (that is, *n* and
*c* are chosen) to satisfy the competing interests
of the lot producer and the lot consumer. The lot producer
would like the probability of lot acceptance *p*_{1}) is low. The consumer requires the
probability of lot acceptance ()
to be low when the proportion of nonconforming units
(*p*_{2}) is high. In the quality control
textbook by Montgomery (1996, p.
620), it is stated that *n* and *c* should be
taken to satisfy

Montgomery goes on to say that
the nonlinear equations (1) and
(2) have no simple, direct solution. A binomial
nomograph is then exhibited for use in obtaining solutions
to these equations. The nomograph is a nonregular grid
for which a relatively simple, but apparently magical, set
of rules can be followed to obtain *n* and *c*,
for given ,
,
*p*_{1}, and *p*_{2}. An
equivalently magical procedure is provided by Mitra (1998, p. 438-441), in which case a
table of Grubbs (1949) has been used
to obtain sampling plans.

Figure 1 (123.2K jpg)

Figure 1. A Binomial Nomograph from Montgomery (1996, p. 620) (used with permission).

4 The goal of the present note is to make some remarks
about the above equations and procedure and to present an
alternative, simpler procedure for designing
single-sampling plans. The idea is that the normal
approximation to the binomial distribution leads to
approximations for *n* and *c*, given
and .
A key motivation is to replace the black-box nature of the
above nomograph procedure with a procedure that can be
relatively easily understood. Such a procedure could be
demonstrated in either an introductory statistics course or
in a quality control course.

5 The first thing to observe is that the nonlinear
equations will usually not have an integer-valued
solution. Thus, the nomograph will not usually provide a
true solution, but it will yield an approximate solution.
This will be accomplished by choosing the nearest grid
point on the nomograph to the real-valued solution that the
nomograph provides. One problem with this technique is
that the resulting sampling design may sometimes result in
probabilities of acceptance that are too low at
*p*_{1} and/or too high at
*p*_{2}. What is really sought is a sampling
design that satisfies (or comes close to satisfying) the
inequalities

Usually, one would want to use the smallest value of
*n* satisfying both inequalities. Using the normal
approximation to the binomial distribution with the
continuity correction, we have

(5) |

where *Z* is a standard normal random variable.
Thus, inequality (3) implies

(6) |

where , and inequality (4) implies

Thus, andA quadratic inequality in can then be obtained by subtracting the first inequality from the second. The relevant solution satisfies

One possible value of *n* to try is the smallest
integer satisfying the above inequality. The value of
*c* may then be chosen as the smallest integer
satisfying (8). However, the
previously chosen value of *n* may not satisfy (9) for this particular value of
*c*, so the value of *n* may need to be revised
accordingly. This time, (9) may be
viewed as a quadratic inequality in ,
and the relevant solution set is given by

The value of *n* should then be taken as the
smallest integer satisfying (11). In some circumstances, one
may wish to revise *c*, using the newly revised value
of *n* and inequality (8),
but this is usually not necessary.

6 Table 1 gives an indication of
the quality of the sampling plans obtained using the normal
approximation (with continuity correction) for some
typical situations. The nominal values of ,
,
and *p*_{1} are fixed at .05, .1, and .01,
respectively. The table provides the sampling plans for
the tabulated values of *p*_{2}, and also
gives the true binomial probabilities
(at *p*_{1}) and
(at *p*_{2}). These are listed in the fourth
and fifth columns, respectively.

**Table 1. **Some Continuity Corrected
Single-Sampling Plans for Various Values of
*p*_{2}, With *p*_{1} =
0.01,*p*_{1})
and Acceptance (at *p*_{2})

p_{2} |
n |
c |
error in | error in | ||

0.020 | 1184 | 17 | 0.0561 | 0.0952 | 0.122 | 0 |

0.025 | 620 | 10 | 0.0505 | 0.0933 | 0.011 | 0 |

0.030 | 395 | 7 | 0.0473 | 0.0929 | 0 | 0 |

0.035 | 268 | 5 | 0.0542 | 0.0905 | 0.084 | 0 |

0.040 | 202 | 4 | 0.0536 | 0.0906 | 0.071 | 0 |

0.045 | 179 | 4 | 0.0349 | 0.0914 | 0 | 0 |

0.050 | 135 | 3 | 0.0474 | 0.0901 | 0 | 0 |

0.060 | 90 | 2 | 0.0619 | 0.0880 | 0.239 | 0 |

0.070 | 77 | 2 | 0.0424 | 0.0875 | 0 | 0 |

0.080 | 67 | 2 | 0.0298 | 0.0882 | 0 | 0 |

0.090 | 60 | 2 | 0.0224 | 0.0846 | 0 | 0 |

0.100 | 40 | 1 | 0.0607 | 0.0805 | 0.215 | 0 |

0.120 | 33 | 1 | 0.0430 | 0.0810 | 0 | 0 |

0.150 | 26 | 1 | 0.0277 | 0.0817 | 0 | 0 |

NOTE: The relative error in the constraints (3) and (4) is indicated in the last two columns.

7 It should be noted that, because the normal approximation is used, the required inequalities are sometimes mildly violated, especially (3); however, the violations are usually no worse than those for the Grubbs' table or the nomograph. The sixth and seventh columns of Table 1 indicates the relative size of these errors. That is,

8 The method is surprisingly accurate even for cases
where *n* turns out to have a small value. This is
consistent with observations made by Kupper and Hafner (1989) about finding
sample sizes for hypothesis tests when both test size and
power at a particular alternative are specified.

9 In practice, one could check
for both values of *p* to ensure that the sampling
plan is satisfactory. If it is not, one could experiment
with slightly larger values of *n* (together with the
corresponding *c* values) to obtain plans that
conform more closely to the nominal values of

10 It is also possible to try to correct the
approximation using Cornish-Fisher expansions (e.g.,

(12) |

(13) |

Then, a sampling plan can be obtained by solving another
quadratic inequality for .
The resulting plans obey inequalities (3) and (4) more often than the
uncorrected plans. Table 2 lists the
corrected plans that correspond to the ones in Table 1, together with the actual
probabilities of rejection at *p*_{1} and
acceptance at *p*_{2}. Although not listed in
the table, there are some plans found by this procedure
that violate inequality (3).

**Table 2. **Cornish-Fisher Corrected
Single-Sampling Plans for Various Values of
*p*_{2}, With *p*_{1} = 0.01,
= 0.05 (Nominal), and
= 0.1 (Nominal), Together With Actual
Probabilities of Lot Rejection (at *p*_{1})
and Acceptance (at *p*_{2})

p_{2} |
n |
c |
error in | error in | ||

0.020 | 1236 | 18 | 0.0466 | 0.0989 | 0 | 0 |

0.025 | 615 | 10 | 0.0483 | 0.0985 | 0 | 0 |

0.030 | 391 | 7 | 0.0451 | 0.0985 | 0 | 0 |

0.035 | 300 | 6 | 0.0328 | 0.0976 | 0 | 0 |

0.040 | 231 | 5 | 0.0298 | 0.0972 | 0 | 0 |

0.045 | 177 | 4 | 0.0335 | 0.0964 | 0 | 0 |

0.050 | 133 | 3 | 0.0453 | 0.0961 | 0 | 0 |

0.060 | 110 | 3 | 0.0250 | 0.0980 | 0 | 0 |

0.070 | 75 | 2 | 0.0397 | 0.0968 | 0 | 0 |

0.080 | 66 | 2 | 0.0287 | 0.0935 | 0 | 0 |

0.090 | 58 | 2 | 0.0205 | 0.0965 | 0 | 0 |

0.100 | 52 | 2 | 0.0154 | 0.0966 | 0 | 0 |

0.120 | 43 | 2 | 0.0092 | 0.0970 | 0 | 0 |

0.150 | 25 | 1 | 0.0258 | 0.0931 | 0 | 0 |

NOTE: The relative error in the constraints (3) and (4) is indicated in the last two columns.

11 One might argue that when using the Cornish-Fisher correction, the simplicity and directness of the method are sacrificed. For most practical purposes, the normal approximation is probably adequate, and it is certainly easier for an undergraduate student to understand. On the other hand, it might not hurt for a senior undergraduate to see that there are relatively simple ways to improve on the normal approximation.

12 The continuity correction itself seems to be necessary in order to provide accurate results. If the correction is ignored, the above inequalities are violated fairly often, and sometimes by a substantial margin, as can be seen from Table 3. That table corresponds exactly to Table 1, except that

is used in place of (8) and

is used in place of (11).
Inequality (10) is still used to
obtain the initial estimate of *n*.

**Table 3. **Uncorrected
Single-Sampling Plans for Various Values of
*p*_{2}, With *p*_{1} = 0.01,
= 0.05 (Nominal), and
= 0.1 (Nominal), Together With Actual
Probabilities of Lot Rejection (at *p*_{1})
and Acceptance (at *p*_{2})

p_{2} |
n |
c |
error in | error in | ||

0.020 | 1213 | 18 | 0.0400 | 0.115 | 0 | 0.155 |

0.025 | 596 | 10 | 0.0401 | 0.120 | 0 | 0.203 |

0.030 | 375 | 7 | 0.0368 | 0.123 | 0 | 0.240 |

0.035 | 286 | 6 | 0.0262 | 0.125 | 0 | 0.251 |

0.040 | 218 | 5 | 0.0233 | 0.128 | 0 | 0.286 |

0.045 | 165 | 4 | 0.0258 | 0.131 | 0 | 0.317 |

0.050 | 148 | 4 | 0.0170 | 0.132 | 0 | 0.329 |

0.060 | 101 | 3 | 0.0189 | 0.137 | 0 | 0.378 |

0.070 | 87 | 3 | 0.0115 | 0.133 | 0 | 0.339 |

0.080 | 59 | 2 | 0.0214 | 0.139 | 0 | 0.392 |

0.090 | 52 | 2 | 0.0153 | 0.141 | 0 | 0.417 |

0.100 | 47 | 2 | 0.0116 | 0.138 | 0 | 0.383 |

0.120 | 39 | 2 | 0.0069 | 0.137 | 0 | 0.374 |

0.150 | 21 | 1 | 0.0185 | 0.155 | 0 | 0.550 |

NOTE: The relative error in the constraints (3) and (4) is indicated in the last two columns.

13 It should be noted that the nomograph is unable to
provide sampling plans outside a certain range. For
example, if
= .001,
= .1, *p*_{1} = .01, and *p*_{2}
= .02, then the nomograph cannot be used to obtain a plan,
but the normal approximation method gives the sampling
plan: *n* = 2416 and *c* = 39 and

Also, the nomograph will not yield any sampling plans
when *p* < .01. The Grubbs' table will not
provide any sampling plans where *c* exceeds 15. The
normal approximation is much more widely applicable.

14 Finally, there is the important pedagogical value of the above approach. Not only is this a relatively simple way to replace a black-box (or striped-box) solution, but it is also a useful application of the normal approximation to the binomial distribution.

The helpful comments and suggestions of three anonymous referees have led to a substantial improvement in the paper and are gratefully acknowledged. This work was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) and was completed during a visit to the Centre for Mathematics and Its Applications at the Australian National University in Canberra, Australia.

Hall, P. (1992), The Bootstrap and Edgeworth Expansion, New York: Springer-Verlag.

Kupper, L. L., and Hafner, K. B. (1989), "How Appropriate Are Popular Sample Size Formulas?" The American Statistician, 43, 101-105.

Mitra, A. (1998), Fundamentals of Quality Control and Improvement (2nd ed.), Upper Saddle River, New Jersey: Prentice Hall.

Montgomery, D. C. (1996), Introduction to Statistical Quality Control (3rd ed.), New York: Wiley.

W. John Braun

Department of Statistical and Actuarial Sciences

Western Science Centre

University of Western Ontario

London, Ontario, Canada N6A 5B7

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