Answer all questions with explanation

Question 2 [17 Marks] (a) (Screening for Prostate Cancer). Measurements of prostate specific antigen (PSA) are often used as the initial screening test for prostate cancer. It is an appealing test, because it can be measured in urine, so it is non-invasive and inexpensive. It is controversial, though, and the National Institutes of Health and other organizations have issued conflicting guidelines for its use in the last 25 years. Recent studies by the American Cancer Society have estimated that the sensitivity of the test is approximately 0.51 or 51%, and the specificity is 0.91, or 91%. The estimated prevalence of undiagnosed prostate cancer in men age 60 - 70 is approximately 6,000 cases per 100,000 males in the population. Sensitivity (True Positive Rate) refers to the proportion of those who have the disease that received a positive result on this test. Specificity (True Negative Rate) refers to the proportion of those who do not have the disease that received a negative result on this test. i. 4 marks Suppose a 65 year old male has been told his PSA test was positive for prostate cancer. Calculate the probability that the man has prostate cancer.ii. 3 marks Construct a contingency table of expected counts for a hypothetical population of 100,000 males aged 60 - 70. (Hint: Row variable is whether or not have cancer, and column variable is test result. Also include row and column total.) ili. 3 marks Is the test result independent of whether a person has prostate cancer or not? Use the contingency table and the definition of independence to