Dr Genius has developed an experimental vaccine for an allergy to cat fur. He ne

Question

Dr Genius has developed an experimental vaccine for an allergy to cat fur. He needs to test it out on some people to make sure it works. He has a list of 4000 people from his clinic, and 1250 of those have an allergy to cat fur. His receptionist randomly calls in 50 different people from his list, and he checks how many of them have cat fur allergies. (a) Explain in words what X is in terms of this story. What values can it take? (b) Why is this a Hypergeometric r.v. situation? What are the parameters? (c) What are the expected number of people in his sample who have cats fur allergies?

Explanation / Answer

a) X is the number of success, i.e. how many people out of the 50 calls had cat fur allergies.
X can take values ranging from 0 to 50, meaning, none of the call recipients had cat fur allergy to all 50 of them having cat fur allergies.

b) This is a case of sampling without replacement from a finite population and we are interested in the number of success, therefore we need to use hypergeometric distribution

X ~ Hyp(N = 4000, n = 50, K = 1250), where N is the total population size, n is the number of draws or calls in this case, K is the number of success present in the population.

c) mean of hypergeometric distribution is given by
n*(K/N) = 50*(1250/4000) = 15.625

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