About 1.5 in a 1000 of those in a particular population are diseased. A diagnostic test is touted as being very reliable, specifically, it is said to be “99% accurate”; that is, the manufacturer of the diagnostic states that 99% of diseased persons would correctly test positive (say Pr+|D = 0.99) and 99% of non-diseased persons would correctly test negative (say Pr(− | not D = 0.99). (A) If a randomly selected person from this population tests postive for the disease, what are the chances that this person actually has the disease? Put another way (from a public health perspective), if all persons in this population were tested, about what fraction of those testing positive would actually have the disease? That is, what is Pr(D|+)? Hint: It is NOT equal to Pr(+ | D) = 0.99!!! B) If symptoms of the disease caused the person to come in to be tested and this person tests positive, then what are the chance that this person actually has the disease? You may assume that of those people with the same symptoms as this person, about 40% have the disease. You may also assume that the diagnostic accuracy results (the “99% accurate”) are the same for this symptomatic population as they were for the larger population. [We might say that the diagnostic results are independent of symptom status given disease status.] Hint: Consider the modified statement of the problem, whereby only the first sentence, “About 1.5 in 1000 of ...” is changed. In particular, replace the first sentence by, “About 40% of those in a particular population are diseased.” C) If a randomly selected person from this population is tested three times and tests postive for the disease each time, then what are the chances that the person actually has the disease? You may assume that Pr(+ + +| D) = Pr(+ | D) Pr(+ | D) Pr(+ | D) and Pr( + + + | not D) = Pr( + | not D) Pr ( + | not D) Pr(+ | not D). [We might say that, given disease status, the repeated test trials are independent and identically distributed (more later)]
About 1.5 in a 1000 of those in a particular population are diseased. A diagnostic test is touted as being very reliable, specifically, it is said to be “99% accurate”; that is, the manufacturer of the diagnostic states that 99% of diseased persons would correctly test positive (say Pr+|D = 0.99) and 99% of non-diseased persons would correctly test negative (say Pr(− | not D = 0.99).
(A) If a randomly selected person from this population tests postive for the disease, what are the chances that this person actually has the disease? Put another way (from a public health perspective), if all persons in this population were tested, about what fraction of those testing positive would actually have the disease? That is, what is Pr(D|+)? Hint: It is NOT equal to Pr(+ | D) = 0.99!!!
B) If symptoms of the disease caused the person to come in to be tested and this person tests positive, then what are the chance that this person actually has the disease? You may assume that of those people with the same symptoms as this person, about 40% have the disease. You may also assume that the diagnostic accuracy results (the “99% accurate”) are the same for this symptomatic population as they were for the larger population. [We might say that the diagnostic results are independent of symptom status given disease status.] Hint: Consider the modified statement of the problem, whereby only the first sentence, “About 1.5 in 1000 of ...” is changed. In particular, replace the first sentence by, “About 40% of those in a particular population are diseased.”
C) If a randomly selected person from this population is tested three times and tests postive for the disease each time, then what are the chances that the person actually has the disease? You may assume that Pr(+ + +| D) = Pr(+ | D) Pr(+ | D) Pr(+ | D) and Pr( + + + | not D) = Pr( + | not D) Pr ( + | not D) Pr(+ | not D). [We might say that, given disease status, the repeated test trials are independent and identically distributed (more later)].
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