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quiz3Sp2011_key - T | D(1-Pr D = 0 867 × 0074(1-1 ×(1...

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NAME: Stat 205 Quiz 3 Let D + denote that an individual is infected with Hepatitis C and D - denotes an individual is disease-free. Gonz´alez et al. (2008) discuss a test for Hepatitis C that has sensitivity Pr { T + | D + } = 0 . 867 and perfect specificity Pr { T - | D -} = 1 . 000 . The prevalence of Hepatitis C is Pr { D + } = 0 . 0074 in the general population. Here are some general rules for any events A and B : Pr { A } = Pr { A | B } Pr { B } + Pr { A | B C } Pr { B C } (law of total probability). Pr { B | A } = Pr { A | B } Pr { B } Pr { A } (Bayes’ rule). Pr { A C | B } = 1 - Pr { A | B } (compliment rule for conditional probability). 1. Find the probability that a test comes up positive Pr { T + } . Answer: Use the law of total probability and the complement rules Pr { T + } = Pr { T + | D + } Pr { D + } + Pr { T + | D -} Pr { D -} = Pr { T + | D + } Pr { D + } + (1 - Pr
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Unformatted text preview: T- | D-} )(1-Pr { D + } ) = 0 . 867 × . 0074 + (1-1) × (1-. 0074) = 0 . 00642 . 2. Find the probability of having the disease given the test comes up positive Pr { D + | T + } . Answer: Use Bayes’ rule Pr { D + | T + } = Pr { T + | D + } Pr { D + } Pr { T + } = . 867 × . 0074 . 00642 = 1 . 3. Is D + independent of T + ? Why or why not? Answer: No, they are dependent because Pr { D + | T + } = 1 which is different from Pr { D + } = 0 . 0074 . Knowing that the test came back positive T + changes the probabil-ity of having Hepatitis C from 0.0074 to 1....
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