Question

Fifteen subjects are sampled from a population of one hundred, one quarter of which have COVID-19. What number of subjects in the sample does one expect to have COVID-19? Does the answer depend if the sampling is done with or without replacement? please show work...

Answer 1

Sampling with replacement -

In this case, the probability of random subjects having COVID-19 will be constant and equal to 0.25

Let X be the number of subjects having COVID-19 in a sample of 15 subjects.

Then X ~ Binomial(n = 15, p = 0.25)

Expected number of subjects having COVID-19 using Binomial distribution  = np = 15 * 0.25 = 3.75

Sampling without replacement -

In this case, the probability of random subjects having COVID-19 will not be constant and depends on number of subjects already sampled.

Let X be the number of subjects having COVID-19 in a sample of 15 subjects.

N be the population size. N = 100

k is number of COVID-19 cases in population. k = 25

n is samples size. n = 15

Then X ~ Hypoergeometric(N = 100, k = 25, n = 15)

Expected number of subjects having COVID-19 using Hypergeometric distribution  = nk/N = 15 * 25/100 = 3.75

Thus, the expected number of subjects having COVID-19 will be same if the sampling is done with or without replacement.