Question 1:
Scenario: Leakage from underground gasoline tanks at service stations can damage the environment. It is estimated that 25% of these tanks leak. You examine 15 tanks chosen at random, independently of each other.
a) What is the mean number of leaking tanks in such samples of 15?
b) What is the probability that 10 or more of the 15 tanks leak?
c) Now you do a larger study, examining a random sample of 1,000 tanks nationally. What is the probability that at least 275 of these tanks are leaking?
Question 2:
Scenario: Testing for HIV. Enzyme immunoassay (EIA) tests are used to screen blood specimens for the presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always correct. Here are approximate probabilities of positive and negative EIA outcomes when the blood tested does and does not actually contain antibodies to HIV:
Test Results
+ -
Antibodies present 0.9985 0.0015
Antibodies absent 0.006 0.994
Suppose that 1% of a large population carries antibodies to HIV in their blood.
a) Draw a tree diagram for selecting a person from this population (outcomes: antibodies present or absent) and for testing his or her blood (outcomes: EIA positive or negative)
b) What is the probability that the EIA is positive for a randomly chosen person from this population?
c) What is the probability that a person has the antibody, given that the EIA test is positive? (Comment: this exercise illustrates a fact that is important when considering proposals for a widespread testing for HIV, illegal drugs, or agents of biological warfare: if the condition being tested is uncommon in the population, many positives will be false-positives.
Question 3:
Scenario: Testing for HIV, continued. The previous exercise gives data on the results of EIA tests for the presence of antibodies to HIV. Repeat part (c) of that exercise for two different populations:
a) Blood donors are prescreened for HIV risk factors, so perhaps only 0.1% (0.001) of this population carries HIV antibodies
b) Clients of a drug rehab clinic are a high-risk group, so perhaps 10% of this population carries HIV antibodies
c) What general lesson do your calculations illustrate?
Question 4:
a) Find the number z such that the proportion of observations that are less than z in a standard Normal distribution is 0.8.
b) Find the number z such that 35% of all observations from a standard Normal distribution are greater than z.
Question 5:
Scenario: The NCAA requires Division 1 athletes to score at least 820 on the combined mathematics and verbal parts of the SAT to complete in their first college year (higher scores required for students with poor high school grades). In 2000, the scores of 1,260,000 students taking the SATs were approximately normal with mean 1019 and standard deviation 209. The NCAA considers a student a "partial qualifier" eligible to practice and receive an athletic scholarship, but not to compete, if the combined SAT score is at least 720.
a) What percent of all SAT scores are less than 720?
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