Find an approximated probability:
Sixteen numbers are selected independently and at random from the interval [ 0, 1 ]. Find an approximated probability that the mean of these sixteen, numbers does not exceed 0.6.
Let Xi, i = 1, 2, ..., 16 be the selected numbers. Then Xi are i.i.d. each having a uniform distribution on the interval [O, 1 ]. Hence
μ = E (Xi) =1/2 and σ2 = Var Xi =1/12
Hence required probability is
=Φ (1.39) = .9177
The utilization of the central limit theorem in approximating a few probability distributions is given in the subsequent section.