An article in the NewYork Times (February 17, 1999) about the PSA blood test for detecting prostate cancer stated that, of men who had this disease, the test fails to detect prostate cancer in 1 in 4 (so called false-negative results), and of men who did not have it, as many as two-thirds receive 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 |+) = 2/3 or P(+| ¯ C) = 2/3?

b. What is the sensitivity of this test?

c. Of men who take the PSA test, suppose P(C) = 0.01. Find the cell probabilities in the 2 × 2 table for the joint distribution that cross classifies Y = diagnosis (+,−) with X = true disease status (C, ¯C ).

d. Using (c), find the marginal distribution for the diagnosis.

e. Using

(c) and (d), find P(C|+), and interpret.