Question

Suppose that a test for determining whether a person has been infected with COVID-19 has a 20% rate of false positives (i.e. in 20% of applications, the test falsely indicates that a person has been infected with COVID-19 when they actually have not). Now, consider the likelihood that when 100 healthy people are tested, the test falsely indicates that between 10 and 30 of them are infected.

a. Calculate the expected number of false positives (μ) in a sample of 100.

b. Calculate the standard deviation in the number of false positives (σ) in a sample of 100.

Answer 1

Solution:-

Given that:-

The test has a false positie rate of 20%. This is same as the probability that any given test is false positive is 0.20

Let X be the number of false positives out of 100 people tested. We can say that X has a Binomial distribution with parameters, number of trials (number of people tested) n=100 and success probability (the probability that any given test is false positive) p=0.20.

a) The expected value of X using the formula for the expectation of binomial distribution is

ans: The expected number of false positives in a sample of 100 is

b) The standard deviation of X using the formula for the standard deviation of binomial distribution is

ans: The standard deviation in the number of false positives in a sample of 100 is