M 362K, Spring 03, Smith

## Assignment for Friday, September 26:

#### To Hand In:

Remember: Reasoning and clarity of explanation are important!

1. p. 58 #46

2. p. 104 #12

3. Medical tests for diseases (for example, the ELISA test for HIV) are
not perfect. When such a test is given, the result is described as either
positive or negative. Positive is supposed to mean the person has the disease;
negative is supposed to mean the person does not have the disease. However,
since the test is not perfect, there are always some people who are false
positives (that is, the test result is positive, but they don't really have
the disease), some who are true positives (that is, their test result is positive
and they really do have the disease), some false negatives, and some true
negative (defined similarly). Let H be the event, "Has the disease". Let
P be the event, "Test result is positive."

a. Describe each of the events first in words (but not
using the words union, intersection, or complement) and then in terms of
H and P, using the symbols for intersection, union, and complement of an
event.

i. False negative
ii. True negative.

b. In a study reported in 2001 to the Centers for Disease Prevention
and Control (CDPC) on the effectiveness of a certain test for HIV,
99% of tests were positive when HIV was present, and 98.7% of tests were
negative when HIV was not present. The National Institute of Allergies and
Infectious Diseases estimated that about 0.3% (three tenths of a percent)
of US residents were infected with HIV at that time. Using this estimate,
find the probability that a (randomly chosen) US resident had HIV if they
tested positive on this test at the end of 2002. (Be sure to work out your
answer numerically, for use in part (c).)

c. A 1996 study estimated that 14% of injection drug
users were infected with HIV. Using this estimate, find the probability that
a (randomly selected) injected drug user who tests positive on this test has
HIV. (Be sure to work out your answer numerically, for use in part (c).)

d. Discuss your findings in parts (b) and (c) and their
implications for someone being tested for HIV.