Question

In a sample of 5000 people, 2% of the population have COVID. In testing, there is a false positive rate of 1.6% and a false negative rate of 4%.

a. Compile the results in a chart or a tree diagram.

b. In this sample, how many people have COVID?

c. What percent of the population have a positive test for COVID?

d. What is the probability that a patient has COVID given a positive test?

e. What is the probability that a patient has COVID given a negative test?

f. You have a relative convinced they must have COVID because their test was positive. What would you tell them, based on your answer to questions #c & d. Be specific.

g. What percent of all people have “accurate tests”?

please show all work and calculations

Answer 1

We would be looking at the first 4 parts here as:

a) We are given here that 5000 of the people are tested, 2% have COVID.

Also, as false positive rate is 1.6%, therefore,
P(+ | no COVID) = 0.016,
P(- | no COVID) = 1 - 0.016 = 0.984
As the false negative rate is 4%, therefore,
P(- | COVID) = 0.04,
P(+|COVID) = 1 - 0.04 = 0.96

P(+ and COVID) = P(+|COVID)P(COVID) = 0.96*0.02 = 0.0192
P(- and COVID) = P(- | COVID)P(COVID) = 0.04*0.02 = 0.0008
P(+ and no COVID) = P(+ | no COVID) P(no COVID) = 0.016*0.98 = 0.01568
P( - and no COVID) = P(- | no COVID)P(no COVID) = 0.984*0.98 = 0.96432

Therefore this can be shown in a table format as:

	COVID	No COVID
Positive	0.0192	0.01568
Negative	0.0008	0.96432

b) In the sample, total number of COVID patients
= 2% of 5000

= 0.02*5000

= 100

c) % of population with positive test for COVID, from the above table, we can add the first row values to get:
= 0.0192 + 0.01568

= 0.03488

Therefore 3.488% is the required percentage here.

d) Given a positive test, probability that the person does have COVID is computed using Bayes theorem here as:  
= P(COVID and +) / P( + )
= 0.0192 / 0.03488
= 0.5505

Therefore 0.5505 is the required probability here.