not cause the alarm to sound. The following table shows the results of their test. Part 1: Metal Detectors The metal detector at the entrance of a courthouse is used to detect the presence of a weapon when someone enters the building. The metal detector should only sound, indicating a positive test result, when a weapon is present. A metal detector should remain silent, indicating a negative test result, when a weapon is not present. 1. Classify each of the following scenarios as a true positive, false positive, true negative, or false negative. Totals (a) Positive (alarm sounds) Weapon True Positive Harmless Item False Positive 133 271 False Negative True Negative Negative (no alarm) 8 104 141 375 Totals 404 112 516 (4 x 2 pt = 8 pt) Answer as a percentage rounded to two decimal places. Show your calculations. a. A person walks through the metal detector and the alarm did not sound. The person did not have a weapon. 2. What is the probability of a positive test result from the metal detector? (4 pts) b. A person walks through the metal detector and the alarm did not sound. A weapon was found on the person. 3. What is the probability of a positive test result from the metal detector given that a weapon is present? This is the sensitivity of the metal detector. (4 pts) C. A person walks through the metal detector and an alarm sounds. This person did not have a weapon. 4. What is the probability of a negative test result from the metal detector given that a weapon was present? This is the false negative rate of the metal detector. (4 pts) d. A person walks through the metal detector and an alarm sounds. A weapon was found on the person. 5. What is the probability of a negative test result from the metal detector given that a harmless item was present? This is the specificity of the metal detector. (4 pts) With the current settings the metal detector was able to find most metal objects regardless of type. With such a high sensitivity this would cause many follow up searches which increase the amount of time someone would need to spend to go through the security check. Using all the same items as before, more tests were conducted with lower sensitivity. Note that the metal detector can only look for the presence of metal and cannot discern if the metal item is a weapon or not. 7. The following table shows the results of their testing, but it is not complete yet. Copy your answers from p. 2 questions 3, 4, 5, and 6, into the correspondingly numbered cells in the bottom row of the table. 6. What is the probability of a positive test result from the metal detector given that a harmless item was present? This is the false positive rate of the metal detector. (4 pts) 3. (1 pt) Sensitivity False Negative Rate Specificity False Positive Rate 62.00% 38.00% 62.50% 37.50% 76.00% 24.00% 47.50% 52.50% 86.00% 14.00% 33.75% 66.25% 4. 5. 6. Answer the following questions in complete sentences. 8. As the Sensitivity of the metal detector increases, what happens to the False Negative Rate? In context of the metal detector, explain what this means in practice? (3 pts) Part 2: Disease Detection Suppose a new rapid screening test is being developed to detect a disease. Patients who take the new test also take an older test to confirm the results. The older test is 100% accurate, but since it is slower and more invasive, it is not a good choice for testing the general population. It is estimated that the new test has a sensitivity of 97% and a specificity of 95%. 11. Consider a population of 50,000 patients where a (b) 9% disease prevalence is assumed. Fill in the missing numbers in the breakdown below based on the use of the new test. Show your work. (6 x 2 pt = 12 pt) a. Number of Infected People 9% of Population Population 50,000 b. Number of Healthy People (d). 91 % of Population 9. As the Sensitivity of the metal detector increases, what happens to the Specificity? In context of the metal detector, explain what this means in practice? (3 pts) c. Number of Positive Test Results (97% of Infected Get a Positive Result) d. Number of Negative Test Results (3% of Infected Get a Negative Result) e. Number of Positive Test Results (5% of Healthy Get a Positive Result) f. Number of Negative Test Results (95% of Healthy Get a Negative Result) True Positives (TP) False Negatives (FN) False Positives (FP) True Negatives (TN) 10. As the Sensitivity of the metal detector increases, what happens to the False Positive Rate? In context of the metal detector, explain what this means in practice? 12. Fill in the table based on the results in problem 11 above. Has Disease (Infected) (8 pts) Does Not Have Disease (Healthy) Totals (3 pts) Test Positive Test Negative Totals 50,000