1- Suppose two doctors, A and B, test all patients coming into a clinic for syphilis. Let events A ={doctor A makes a positive diagnosis) and B' = {doctor B makes a positive diagnosis). Suppose doctor A diagnoses 10% of all patients as positive, doctor B diagnoses 17% of all patients as positive, and both doctors diagnose 8% of all patients as positive. (10 pts) a. Are the events A, B independent? Suppose a patient is referred for further lab tests if either doctor A or B makes a positive diagnosis. (10 pts) b. What is the probability that a patient will be referred for further lab test? 2- A certain virus infects one in every 200 people. A test used to detect the virus in a person is positive 80% of the time when the person has the virus and 5% of the time when the person does not have the virus. Let A be the event "the person is infected" and B be the event "the person tests positive." (20 pts) a. Using Bayes' Theorem, when a person tests positive, determine the probability that the person is infected b. Using Bayes' Theorem, when a person tests negative, determine the probability that a person is not infected 3- Skinfold thickness over the triceps muscle in the arm is an anthropometric measure that varies with states of health. Table 1 below lists skinfold measurement at the midpoint of the triceps in eight men with chronic lung disease and nine comparably aged controls. (20 pts) a. Compare the groups with side-by-side stem-plots b. Calculate group means, medians, variances, and standard deviations. C. Draw a boxplot for each data, and determine any outlier? Table 1. Chronic lung disease: 9.1 10.9 11.4 15.3 18.4 12.5 20.7 25.2 Controls: 10.4 19.6 20.6 23.8 24.7 32.8 31.7 36.4 39.2