Question

In: Statistics and Probability

4) Consider two medical tests, A and B for a virus. Test A is 90% effective...

4) Consider two medical tests, A and B for a virus. Test A is 90% effective at recognizing the virus when it is present, but has a 5% false positive rate. Test B is 80% effective at recognizing the virus when it is present, but only has a 1% false positive rate. The two tests are independent (i.e., they use different means for identifying the virus). The virus is carried by 2% of all people. If you could use only one of the two tests to identify the virus, which would you choose? Justify your answer mathematically. How much more certain can you be if you can use both tests? << how do you answer this bold part?

P(Test A = + | Virus = Present) = 0.90

P(Test A = + | Virus = Absent) = 0.05

P(Test B = + | Virus = Present) = 0.80

P(Test B = + | Virus = Absent) = 0.1

P(Virus = present) = 0.02

P(Virus =Absent) = 0.98

Baye’s Theorem:
P(H|E) = P(E|H) * P(H) / P(E|H) * P(H) + P(E| not H) * p(not H)

P(Virus = present | Test A = +) = P(Test A = +| Virus = present) * P(Virus = present) / P(Test A = +| Virus = present) * P(Virus = present) + P(Test A = + | Virus = Absent) * P(Virus = Absent)

P(Virus = present | Test B = +) = P(Test B = +| Virus = present) * P(Virus = present) / P(Test B = +| Virus = present) * P(Virus = present) + P(Test B= + | Virus = Absent) * P(Virus = Absent)

Values placed in formula:
Test A = (0.90)(0.02)/(0.90)(0.02)+(0.05)(0.98) = 0.2686

Test B = (0.80) (0.02)/ (0.80) (0.02) + (0.01) (0.98) = 0.6201
the value obtained for B is higher, so it can be said the result of B is more reliable. Therefore, I would use test B.

Expert Solution

the probability that virus is present in the people is

The probability that the virus is not present in the people is

we need to find which test has the probability higher to give right decisions

The probability that the test A gives the right decision = Probability that the person has virus and test positive + the probability that the person has no virus and test negative

=

Similarly, The probability that the test B gives the right decision = Probability that the person has a virus and test positive + the probability that the person has no virus and test negative

=

Surely test B has the probability higher of giving the right decision

The result you found and the explainantion above can both be combined to support that test B is more effective than test A

orchestra answered 2 years ago

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