Process or Product Monitoring and Control
6.2. Test Product for Acceptability: Lot Acceptance Sampling
|Double Sampling Plans|
|How double sampling plans work||
Double and multiple sampling plans were invented to give a
questionable lot another chance. For example, if in double sampling
the results of the first sample are not conclusive with regard to
accepting or rejecting, a second sample is taken. Application of
double sampling requires that a first sample of size
n1 is taken at random from the (large) lot. The
number of defectives is then counted and compared to the first
sample's acceptance number a1 and rejection number
r1. Denote the number of defectives in sample 1
by d1 and in sample 2 by d2,
If d1 r1, the lot is rejected.
If a1 < d1 < r1, a second sample is taken.
If D2 r2, the lot is rejected.
|Design of a Double Sampling Plan|
|Design of a double sampling plan||
The parameters required to construct the OC curve are similar to the
single sample case. The two points of interest are
where p1 is the lot fraction defective for plan 1
and p2 is the lot fraction defective for plan 2.
As far as the respective sample sizes are concerned, the second
sample size must be equal to, or an even multiple of, the first
There exist a variety of tables that assist the user in constructing double and multiple sampling plans. The index to these tables is the p2/p1 ratio, where p2 > p1. One set of tables, taken from the Army Chemical Corps Engineering Agency for = .05 and = .10, is given below:
|Example of a double sampling plan||
We wish to construct a double sampling plan according to
The left holds constant at 0.05 (P = 0.95 = 1 - ) and the right holds constant at 0.10. (P = 0.10). Then holding constant we find pn1 = 1.16 so n1 = 1.16/p1 = 116. And, holding constant we find pn1 = 5.39, so n1 = 5.39/p2 = 108. Thus the desired sampling plan is
The first plan needs less samples if the number of defectives in sample 1 is greater than 2, while the second plan needs less samples if the number of defectives in sample 1 is less than 2.
|ASN Curve for a Double Sampling Plan|
|Construction of the ASN curve||
Since when using a double sampling plan the sample size depends on
whether or not a second sample is required, an important
consideration for this kind of sampling is the Average Sample Number
curve. This curve plots the ASN versus p', the true
fraction defective in an incoming lot.
We will illustrate how to calculate the ASN curve with an example. Consider a double-sampling plan n1 = 50, c1= 2, n2 = 100, c2 = 6, where n1 is the sample size for plan 1, with accept number c1, and n2, c2, are the sample size and accept number, respectively, for plan 2.
Let p' = .06. Then the probability of acceptance on the first sample, which is the chance of getting two or less defectives, is .416 (using binomial tables). The probability of rejection on the second sample, which is the chance of getting more than six defectives, is (1-.971) = .029. The probability of making a decision on the first sample is .445, equal to the sum of .416 and .029. With complete inspection of the second sample, the average size sample is equal to the size of the first sample times the probability that there will be only one sample plus the size of the combined samples times the probability that a second sample will be necessary. For the sampling plan under consideration, the ASN with complete inspection of the second sample for a p' of .06 is
|The ASN curve for a double sampling plan|