Please show me specific process of the correct answer. 2.1 An article in the New

Question

Please show me specific process of the correct answer.

2.1 An article in the New York 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 and let + (-) denote a positive (negative) test result. a. Which is true: P(-IC) = 1/4 or PCI-)= 1/4? PCI+)= 2/3 or the event of having (not having) prostate cancer -2/3? b. W hat is the sensitivity of this test? c. Of men who take the PSA test, suppose P(C) - 0.01. Find the cell prob- abilities in the 2 x 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.

Explanation / Answer

a) true options are P(-|C)=1/4

P(+|C--) =2/3

b)

sensitivity =P(+|C) =1-P(-|C) =1-1/4 =3/4

c)

from above: P(C and +)=P(C)*P(+|C)=0.01*1/4=0.0025

P(C and -)=P(C)-P(C and +)=0.01-0.0025 =0.0075

similarly P(C-- and +)=0.99*(2/3)=0.66

P(C-- and -)=0.99-0.66=0.33

d)

from above marginal distribution of diagnosis:

P(+)=0.6675

P(-)=0.3325

e)

P(C|+)=0.0075/0.6675=0.011236

hence given test result comes posiitve probability that person actually have disease is 0.011236.

C C-- total + 0.0075 0.66 0.6675 - 0.0025 0.33 0.3325 total 0.01 0.99 1

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