Sampling Plans

We will first consider a single sampling plan in which accept/reject decisions are based on the results of a single sample of n items from the lot of n items. Each of the n sample items is inspected and categorized as either acceptance or defective. Such a plan is known as Sampling by Attributes. (We will not discuss Sampling by Variables in this unit. The interested riders may refer to the Further Reading Section given at the end of the block.) If the number of defective items in the sample exceeds a pre-specified cut-off level, c, thee entire batch is rejected. (Depending on costs, a rejected lot may be sapped, 100 per cent inspected or returned back to the manufacturer.) Since a finding of c or fewer defective items in the sample implies accepting the batch, often referred to as the acceptance lever. A sampling  plan is specified by the values of n aid c. 

The sampling plan is supposed to separate good lots from bad lots. As mentioned earlier there are bound to be sampling errors. We will now study the probabilities of such error graphically, using an Operating Characteristic Curve. 

The Operating Characteristic Curve  

It is useful to have a picture that allows us to compare sampling plans as to how they will react to different lots with unknown, varying fraction defective. Such a comparison is provided by the Operating Characteristic Curve (OCC) which displays the probability of accepting a lot with any fraction defective. 

  Operating Characteristic  Curve

Figure shows OCC for singlet sampling plans A and B with n = 35, c= 1 and n = 150, c = 6, respectively. For example, suppose that a lot with F = 10 percent defectives is considered to be a bad lot and a lot with f = 2' percent defectives is considered to be a good lot. From Figure 9.6, it is clear that sampling plan A would 14 percent chance of accepting a bad lot. The same unfortunate error can occur with the sampling plan B, with larger sample size also, but the probability of error is much smaller. In fact, it is only 1 percent. The sampling plan B is also better at not rejecting goads lots (f = 2 percent). Sampling plan A has 1 6 percent chance of rejecting a good lot whereas plan B h'as only 3 percent chance of rejecting a good lot. 

It is not surprising that a larger sample does a better job of discriminating between good and bad lots. It has more information. However, the price for increased accuracy is higher inspection costs. The design of a sampling plan$has  to optimally trade off cost with discrimination. 

The values of subordinates of the Operating Characteristics Curve are determined from the Poisson Distribution. The actual details can be found in the advanced texts listed in the reference.  

At  this moment, pause for a while and check for yourself whether you have  understood OCC. Do the following Activity.