Dr. I.M.A. Genius has developed an experimental vaccine for an allergy to cat fu

Question

Dr. I.M.A. 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 situation? What are the parameters?
c. What are the expected number of people in his sample who have cat fur allergies?
d. What is the probability that exactly 20 of the people in his sample have cat fur allergies? Just set up the equation, but don’t try to solve it.
e. What would be a good approximation to use in this situation? What are the parameters?
f. What is the approximate probability that 20 of the people in his sample have cat fur allergies? Set up the equation and solve.

Explanation / Answer

a)X Hyp(N = 4000, n = 50, M = 1250)

b) In this case , a sample size n=50 is randoly selected without replacement from a population of N=4000 items
In the population k=1250
N-k =4000- 1250 =2750 items

c)E[C] = n*M/N = 50*1250/4000 = 15.625

d) P(C = 20) = (1250 20 )(2750 30 )/(4000 50 )

e) Since N = 4000 > 20 × n = 1000. We can use C Binomial(n = 50, p =M/N =1250/4000 = 0.3125) to approximate C.

f) P(C = 20) P(C = 20) =(50 20) (0.3125)^20(1 0.3125)^30 = 0.0488

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