C.16.MR:5.2 - Build the relation for an assessment of independent performance

The sought quantity p is the fraction of a stated population able to perform a specified action independently under specified conditions. An applicable assessment account supplies s, the probability of a positive response when that capability is present, and f, the probability when it is absent.

Partition the population by the capability. The probability of a positive response is

q = s*p + f*(1-p)
  = f + (s-f)*p

For s=0.9 and f=0.1, q=0.1+0.8*p. A sample positive fraction of 0.7 gives the estimate p=0.75 obtained by substituting the sample fraction. Its use also depends on sampling uncertainty, uncertainty in s and f and whether those rates apply to this population and procedure. The formula describes the expected response; the sample proportion estimates it.

If s=f, q is independent of p and the test does not distinguish the two conditions through this response. The failure is visible in the constructed relation. Further estimates of that same marginal positive rate cannot identify p through this relation.

Now change the procedure by offering hints. The earlier rates may cease to apply. Recover rates for the changed procedure or restore independent performance if that remains the target. The subject’s capability and assessment Methods establish those conditions; the general construction shows where they enter the inference.

This population estimate does not identify which particular people or agents have the capability. That different question requires an individual assessment account.