Shortly after E1 was approved in 1985, a member of the U.S. Congress made a speech on the floor of the House of Representatives expressing the opinion that HIV was such a serious public health threat that everyone 18+ years old should be tested with E1. The goal in this final part of the problem is to fill out a new version of Table 1 with numbers quantifying what would have happened to the N = 175 million Americans under this Congresspersons

4

Table 2: Partially-filled-out table of expected numbers of people receiving HIV diagnoses under the Congresspersons plan.

Truth

HIV+ (=1) HIV (=0) Blood + (y1 =1) N[+(1)(1)]

495,000	N (1 ) (1 )
N (1 )	N (1 )

Test (y1 = 0) Total

Total 165,780,000

N 174,500,000

N

plan. If we knew for sure that = , we could just use that value of and the already- established values of (,), and multiply all of the resulting entries in Table 1 by N, but we dont know that for sure. Consider an unknown quantity (in AMS 131 we would have called it a random variable) with expected value E( | B) = .

(iv) By looking at the form of all 9 of the entries in Table 1 (including the margins) as func-

tions of (and remembering basic properties of expectation from AMS 131), briefly ex-

plain why we can obtain a table of expected cell and margin counts just by multiplying all

of the entries in Table 1 by N and then substituting in (, , ) = 500000 , 0.99, 0.95.175000000

Complete Table 2 by filling in the empty cells and margins; Ive given You a headstart on some of them. Briefly summarize the likely good and bad outcomes of the Con- gresspersons plan, when viewed as an instance of national health policy. In Your view, would the good outcomes outweigh the bad, or the other way around, or is it hard to come to a clear judgment? Explain briefly. (Note that were doing whats called a benefit-only analysis here, not a cost-benefit analysis, since weve not taken into ac- count how much administering 175,000,000 E1 tests would cost in time and money.)[25 points]