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The Average Sample Number (ASN)

The main advantage of multiple stage sampling plans is the reduced "long run" or average sample size, required to arrive to a good decision. For now, the random variable "sample size" is a probabilistic outcome (varies with every case). Its "Expected Value," known as ASN or "Average Sample Number," depends on the value of the real parameter under test, be it the percent defective "p," reliability, or any other parameter of interest.

The ASN is obtained following the definition of Expected Value. For double sampling:

ASN = E{SN} = SNΣ SN x P{SN}

= n1 x P(n1) + (n1 + n2) x P(n1 + n2)

In the double sampling scheme, SN (sample number) can be only n1 or n1+ n2. P (n1) is the probability of drawing a "first" sample only, which occurs when arriving at a decision at the first sample (with probability 1 - P{c1 ≤ Y ≤ c2}). The probability P{n1 + n2} of having to draw a second sample, totaling a size of n1 + n2, occurs when we had an "inconclusive" outcome from the first sample (i.e., with probability: P {c1 ≤ Y ≤ c2}).

We illustrate this case using our double sampling example S (n1 = 20, n2 = 20, c1 = 14, c2 = 15, c3 = 33), described earlier. Let the true reliability "p," be p = 0.9, and let Y be the number of survivals obtained in the first sample of size n1 = 20. Then, the probability of taking "no decision" on the first sample, when p = 0.9, is P (c1 ≤ Y ≤ c2) = P (14≤ Y ≤ 15) = 0.0089 + 0.0319. This yields ASN = 20.81, barely larger than the exact n = 20 elements that would be required by a single sample plan, of fixed size n1 = 20:

ASN = n1 x P(n1) + n2 x P(n2)

= 20 x [1 - (0.0089 + 0.0319)] + 40 x (0.0089 + 0.0319)

Table 4 presents the ASN values for the double sampling plan S, described in the first section, calculated at selected values of the (reliability) parameter "p."

Table 4. Comparison of ASN for Double Sampling, Given "p"
 Reliability ASN 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 20.05 20.82 22.96 25.67 27.42 27.41 25.96 23.98

In Figure 3, we show graphically the relationship between the reliability parameter "p" and the corresponding double sample ASN.

Figure 3. ASN for Double Sampling (Click to Zoom)

In the general case of sequential tests (SPRTs), the ASN is obtained following the same principles described for the double sampling scheme. However, the stages "n" are extended to any number, not just two. We multiply each SPRT stage (the samples taken so far) times the probability of arriving to a decision at that stage, given the true parameter "p" and that we have not taken a decision earlier in the sequential test (i.e., that we have followed a path inside the SPRT "continuation region", up to the present stage):

ASN (p) = Ep{SN}

= Σ SN x Pp {Decision at Stage SN But not before}

The ASN values for sequential (SPRT) tests, however, are in general not easy to obtain. A full treatment of this topic falls beyond the scope of this START sheet. Those readers interested in pursuing it further may want to consult References 4, 5, 6, and 7. In the second START of this two-part series, discussing SPRTs for continuous variables, we will obtain the ASN for a reduced number of values of the parameter of interest.