Question

Two percent of the population has a certain condition for which there are two diagnostic tests. Test A, which costs $1 per person, gives positive results for 80% of persons with the condition and for 5% of the persons without the condition. Test B, which costs $100 per person, gives positive results for all persons with the condition and negative results for all persons without it.
(a) Suppose that Test B is given to 150 persons, at a cost of $15; 000. How many cases of the condition would one expect to detect?
(b) Suppose that 2000 persons are given Test A, and then only those who test positive are given Test B. Find the expected cost.
(c) What is the expected number of cases you will detect from (b)?

Answer 1

(a)2% of 150 = 3 expected to have the condition

As test B gives positive results for all persons with the condition

expected number of detection = 3

(b) Cost of test A = 2000*$1= $2000

2% of 2000= 40 expected to have the condition and 2000-400= 1960 expected to not have the condition

Test A has 80% true positive rate and 5% false positive rate

Out of which test A is expected to detect positively

80% of 40 + 5% of 1960= 320+ 98 = 130

(true positive =32 , false positive 98)

130 are given test B

cost of test B = 130*$100= $13,000

Total expected cost = $13,000+$2000= $15,000

Expected cost per person = 15000/2000= $7.50

(c) Test A detects positive for 130 persons

true positive =32 , false positive 98

As test B gives positive results for all persons with the condition

expected number of detection = 32