Question

1. ) Many companies must monitor the effluent that is discharged from their plants into rivers and waterways. It is the law that the concentration of arsenic, which is used in the production of integrated circuits, be below a certain limit.

Suppose that the concentration of arsenic in an effluent from a particular company is below the limit 95% of the time. There is a test that measures whether the concentration is below limit. The testing method, of course, is not perfect: it has a sensitivity of 90%; that is, 90% of above-limit concentrations are correctly identified as being above limit. The specificity of the test is 92%; that is, 92% below-limit concentrations are correctly identified as being below limit.

1. Suppose a sample of effluent is identified having an above-limit concentration of arsenic. What is the probability that it actually does?
2. Suppose a sample of effluent is identified as having a below-limit concentration of arsenic. What is the probability that the test is correct?

(Hint: Define two events as follows:
A: sample has an above-limit concentration of arsenic
B: test says sample has an above-limit concentration of arsenic)

Answer 1

Let us assume we have 1000 samples then out of them, 950 has below the limit arsenic concentration

Out of the remaining 50, only 45 (90% of 50) are classified as above limit by the test, and out of the 950 test correctly identifies 874 samples (0.92*950)

So the confusion matrix will be like as follows ( Here 1 is above limit identification)

P / A	1	0
1	45	76
0	5	874

Now,

for first case,

If the affluent is identified as having above limit, then we are in the first row of the confusion matrix,

Thus, probability of it actually having will be 45 / (45 + 76) = 0.3719

Fir second case,

If the affluent is identified as below limit, then we are in the second row of the confusion matrix

Thus, probability of it actually true will be 874/(874+5) = 0.9943