Prostate Cancer Screening (8 marks)

The initial screening test for prostate cancer often involves the measurements of prostate-specific antigen (PSA). The test can be measured in urine which makes it inexpensive and non-invasive. Higher levels of PSA in the blood can be caused by prostate cancer, among other things.

Test sensitivity is the ability of a test to correctly identify those with the disease (also called true positive rate), whereas test specificity is the ability of the test to correctly identify those without the disease (also called true negative rate). Recent studies have estimated that the sensitivity of the test is approximately 0.51 (or 51%) and the specificity of the test is 0.91 (or 91%). This problem addresses males in the age range of 60-70. Approximately 6,000 out of 100,000 males in this population are thought to have undiagnosed prostate cancer.

a) Draw a probability tree for this problem, label it properly. (2 marks).

b) Suppose that a male from this population (e.g., 65 years old) has been told that his PSA test was positive for prostate cancer. Utilize the probability tree you constructed in the previous part and Bayes' formula to calculate the probability that this individual has prostate cancer. (2 marks)

c) Populate (i.e., complete) the following contingency table for a hypothetical population of 100,000 males aged 60-70. In the table, rows represent the test status, columns represent the cancer status, and cells contain the number of people within each category. (2 marks)

	Cancer	No Cancer	Total
Positive
Negative
Total

d) Use your contingency table from the previous part to confirm your answer in part (b). (1 mark)

e) Use your contingency table to calculate the number of men in the 100,000 population who will have a positive PSA test. (1 mark)