Question

Current estimates are that somewhere between 1% and 5% of the U.S. population hasbeen infected with Covid-19. A serological test for Covid-19 is supposed to determine if a personhas antibodies for that virus, which would indicate a previous infection. Some claim that if a persontests positive, then the person should be able to return to normal activity since they would haveimmunity (though whether an infected person develops immunity seems to be currently unknown).One particular test has a “specificity” of 95.6%, which means that the probability of a false positiveis 4.4%. The test has a “sensitivity” of 93.8%, which means that the probability of a false negativeis 6.2%. These numbers look impressive, and the numbers for this test are much better than most.However, notice that a false positive rate of around 4.4% is substantial compared to the proportionof people infected (1% to 5%).(1) Suppose that 5% of the people have been infected. Given that the test comes back positive fora randomly selected person, what is the probability that that person has been infected withCovid-19?(2) Suppose that 1% of the people have been infected. Given that the test comes back positive fora randomly selected person, what is the probability that that person has been infected withCovid-19?(3) Even if we assume that a prior infection bestows immunity, is it reasonable for a person receiv-ing a positive serological test to assume that they have immunity?

Answer 1

Answer:-

Given That:-

Current estimates are that somewhere between 1% and 5% of the U.S. population has been infected with Covid - 19.

P: the tests comes back positive

I: the person is infected

N: the tests comes back negative

H: the person is not infected

Data:

P(P|I) = 0.938

P(P|H) = 0.044

P(N|I) = 0.062

P(N|H) = 0.956

(a) Suppose that 5% of the people have been infected. GIven taht the test comes back positive for a randomly selected person, What is the probability that person has been infected with Covid-19?

P(I) = 0.05

P(H) = 1 - P(I) = 0.95

P(I|P) = \frac{P(P|I)P(I)}{P(P|I)P(I) +P(P|H)P(H) }

P(I|P) = \frac{0.938*0.05}{0.938*0.05+0.044*0.95 }

= 0.528748591

(b) Suppose taht 1% of the people have been infected. Given that the test comes back positive for a randomly selected person, What is the probability that person has been infected with Covid-19?

P(I) = 0.01

P(H) = 1 - P(I) = 0.99

P(I|P) = \frac{P(P|I)P(I)}{P(P|I)P(I) +P(P|H)P(H) }

P(I|P) = \frac{0.938*0.01}{0.938*0.01+0.044*0.99 }

= 0.177181715

(c) Even if we assume that a prior infection bestows immunity, is it reasonable for a person receiving a positive serological test to assume that they have immunity?

It is not reasonable to reach that conclusion since the probability of the person actually being infected given that the serological test was positive is very low. At best it is 0.5287, which does not provide sufficient confidence to assume immunity.

Plz like it......,