Here is the problem:
We have a random sample of 48 discrete random variables X1...X48, independent and identically distributed, picked from a population at random with replacement (meaning that I could possibly repick the same one various time, not sure if it's the right word. I'm french xD). We know that the Xi follow a geometric law with parameter of 1/4. Give the estimate with the central limit theorem, of the probability that at least 20 of the 48 random variables be less than 3.
Here is my attempt of a solution :
X -> G(1/4)
p = 1/4
We know that for a geometric law E(X) = 1/p = 4 and that Var(X) = (1-p)/p^2 = 12
The central limit theorem is as follow : [X - E(X)]/sqr_root(Var(x)) -> N(0;1)
so in my case I have (X - 4)/sqr_rt(12)
I seek P(X<3)
P(X<3) = P( X-4/sqr_rt(12) < (3-4)/sqr_rt(12) ) = P(Z < - 0,28) = *** HERE IS MY PROBLEM *** I don't know how to interpret what "at least 20 of the 48 random variables" implies...
Should it be P(Z < -0,28) = 20/48 - 0,1141 (from the normal distribution table value of 0,28) ?
Or should it be something else???
Thanks for your help.