The PSA blood test is designed to detect prostate cancer. Suppose that of men who have this disease, the test fails to detect prostate cancer in 1 in 4, and of men who do not have it, 1 in 10 have...

The PSA blood test is designed to detect prostate cancer. Suppose that of men who have this disease, the test fails to detect prostate cancer in 1 in 4, and of men who do not have it, 1 in 10 have positive test results (so-called false-positive results).

Let C (C¯) denote the event of having (not having) prostate cancer and let + (−) denote a positive (negative) test result.

a. Which is true: P(− | C)=1/4 or P(C | −)=1/4? P(C¯ | +) = 1/10 or P(+ | C¯)=1/10?

b. Find the sensitivity and specificity of this test.

c. Of men who take the PSA test, suppose P(C)=0.04. Find the cell probabilities in the 2 × 2 table for the joint distribution that cross-classifies Y = diagnosis with X = true disease status.

d. Using (c), find the marginal distribution for the diagnosis and show that P(C | +) = 0.238. (In fact, the National Cancer Institute estimates that only about 25% of men who have a slightly elevated PSA level, 4–10 ng/mL, actually have prostate cancer.18)