Bill Smith buys a gross of screws for a do-it-yourself project. When he begins to use them, he finds that one of the first few he takes from the box has no slot in the head. Another has a defective thread. As many others might, Bill thinks, "I’ll look at a few more, and if any others are defective, I’ll take them back and complain." Here is a common situation­for Bill Smith, read A. N. Other; for screws, read drawing pins, balloons, matches; for ‘no slot’ or ‘damaged thread’, read ‘loose head’, ‘hole in neck’, ‘broken matchstick’. In general, we have a batch of discrete items, from which we have some data about the incidence of an attribute, the data being based on a sample from the batch. On the evidence of the sample, is it reasonable to reject the batch, or complain about it?

In real life, we recognise that it would be unreasonable to expect the manufacturers to ensure that every screw, drawing pin, balloon, match (or even chocolate drop) is perfect, and once this fact of life is recognised, how many defective items do we need to convince ourselves (and the supplier!) that our batch is below an acceptable standard?

These are domestic examples of a very common industrial problem, that of assessing the acceptability of a batch of items without examining every one of them. In these industrial situations, the items may be engineering components, packages of foodstuffs or domestic goods, plastic mouldings, etc. They may be assembled in very large batches comprising many thousands of items, and the decision whether to accept the batches will then have important consequences. These apply both to the supplier, who may find himself having to replace or rectify rejected batches, and the consumer, who may either find he has accepted batches which subsequently prove faulty, or face the disruption caused by rejecting batches of urgently needed items.

Before inspection of batches by sampling can be considered, it has to be acknowledged that some proportion of defective items is tolerable. Such items may, for example, still perform their function adequately, but have some aesthetic flow (discoloured flecks in sheets of office stationery, for example). In other cases the defect may be rectifiable, but causes some nuisance­tight threads in nuts and bolts can be eased with a die, but this will disrupt a production line. Yet again, a small proportion of totally unusable items (like slotless screws) may be tolerable, but a larger proportion would not be. Obviously, if no defective items whatever are permissible, the batches must be fully inspected or tested, and any defective items eliminated. This inspection should preferably be carried out by the supplier before despatch!

For less critical defects, there needs to be agreement between the supplier (or producer) and the consumer on many matters of definition­for example, what constitutes a defect or defective item; what proportion of defective items is to be regarded as reasonable; what risks of erroneous decisions can be accepted? Regarding the latter point, some risks are inevitable when batches are judged by samples. Thus a good batch may be rejected on the basis of a "pessimistic" sample, whilst a bad batch may be accepted if the sample is "optimistic".

In the next article in this series, we shall look more closely at some of these matters of definition, but for the present let us consider a simple example and evaluate some of the risks involved. Suppose that a batch of 100 items is to be inspected, and that a random sample (more about random sampling next time!) of twenty items to be drawn for inspection. If no defectives, or just one, are found in the sample, the batch is accepted; if two or more defectives turn up in the sample, the batch is rejected (perhaps implying that it is returned to the supplier, or that it is to be fully inspected, or perhaps retained for use but with some financial refund for the inconvenience it may cause).

If the batch contains no defectives or just one, then obviously the sample can only lead to an "accept" decision. However, if two defectives exist in the batch, it is just possible that they may both appear in the sample. On the assumption of random sampling, we can evaluate the probability of this occurrence. It occurs when, from a "population" of N = 100, comprising A = 98 good items and B = 2 bad ones, our sample of n 20 items contains a = 18 good items and b = 2 bad ones. From first principles, it can be deduced that this is equal to

(no. of ways of choosing a from A)*(no. of ways of choosing b from B)/(no. of ways of choosing n from N)

For N = 100, n = 20, A = 98, a = 18, B = 2, b = 2, we find that this becomes (cancelling factorial terms where possible)

Since 0! = 1 this finally reduces to 19*20/(100*99) = 19/495

i.e. 0.3838…

This result may be interpreted as meaning that if a succession of batches (of 100 items each), every one containing just two defectives, then in the long run about 4% of such batches would be rejected, and about 96 % (of batches) accepted. This may appear a reasonable risk to the producer.

Supposing that 10% of the items are defective, however. We then have A = 90, B = 10, whilst N, n, a, b remain as before.

From the above expression, we now find the probability of rejection is 0.31817 (actually 101355025/318555566; no further cancellation is possible). This seems much less satisfactory - it implies that, of batches containing 10% defective, twice as many will be accepted as rejected.

There is a dilemma here. If the decision rule is tightened so that rejection occurs if just one defective occurs in the sample, there is a much greater risk of rejecting satisfactory batches. Better discrimination between good and bad quality batches would result from taking larger samples, but this may be expensive or time consuming.

In the next article, as well as resolving some matters of definition, we shall look at more general, though approximate, methods of assessing risks by using the Binomial or Poisson distributions as models for the behaviour of samples in these situations.

Abergavenny

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