We are about to select subjects to run a diagnostic test from a small sample population. The sample consists of three groups of people divided based on age. Group A (age ≤ 25 years) has 15 people, group B

Instructions: You have 1 hour to complete the exam.  Calculators of phones are allowed. Feel free to use scrap paper. Please show all work. The exam is worth 25% of your grade.

You are free to use notes or books.

1. We are about to select subjects to run a diagnostic test from a small sample population. The sample consists of three groups of people divided based on age. Group A (age ≤ 25 years) has 15 people, group B (age > 25 and age ≤ 50 years) has 13 people, and group C (age > 50 years) has 12 people. (5 points)

Each subject is selected randomly from the sample and separated from it (sampling without replacement). Based on this information, answer the following questions.

• What is the probability of choosing a person belonging to group A as the first selection?
• What is the probability of choosing a person belonging to group A or group C as the first selection?
• What is the probability of choosing a person from group A as the first selection, and a person from group C as the second selection?
• Given that the first person chosen is from group B, what is the probability that the next person chosen will be from group C?
• What is the probability of choosing one person from group A and one person from group C, in any order, for the first two selections?

2. (2 Points) Two independent events have the following probabilities: P(A) = 4, P(B) = 0.8. Compute:

• P(A∩B)

• P(A|B)

3. In the Brownsville area of Brooklyn, 31% of the people are (hypothetically) known to be infected with a strain of influenza virus. In the same population, 65% of the people already infected have complaints of severe headache. Among the people who don’t have the infection, only 10% report severe headache. Given that a selected person from this population suffers from severe headache, what is the probability that the person is infected by the virus, P(Infection | Headache)? (5 points)

4. A clinician assumes that the prior probability of a patient having a disease is 22. The clinician will choose intervention A if the probability of the disease is at least 0.75 after a diagnostic test. (5 points).

• What is the required likelihood ratio for intervention A to be selected?

• Given that the sensitivity of a diagnostic test is 0.9 and specificity is 0.8. Is the clinician likely to choose this test based on this information?

5. You go to see the doctor about a sore throat that has been bothering you off and on for a few weeks. The doctor tests you for a new variant of strep throat, which has infected 1 in 200 people in Brooklyn. The test sensitivity is 99% and the specificity is 97%. You test positive. What is the probability that you have a new variant of strep throat?  (5 points)

6. (5 Points)

From the table above, which provides information about a specific diagnostic test, compute:

• Sensitivity
• Specificity
• Negative likelihood Ratio

Equation Sheet Midterm 2020