Question: Normal Approximation to the Binomial Distribution Consider a binomial experiment with n trials and probability of success p. If we assign a 0 to each failure and a 1 to each success, then the binomial random variable can be defined as a sum. By the central limit theorem, as n increases the distribution of X (the sum) approaches a normal distribution with mean np and variance np(1 - p). Suppose X is a binomial random variable with n = 30 and probability of success p = 0.5.
a. Construct a probability histogram for the binomial random variable X. Find P(12 ≤ X ≤ 16) using the binomial distribution.
b. Find the approximate normal distribution for X. Find P(12 ≤ X ≤ 16) using the normal distribution for X.
c. Compare the probabilities found in parts (a) and (b).
d. Find P(11.5 ≤ X ≤ 16.5) using the normal distribution for X. Compare this answer with the probability in part (a). Why do you think this is a much better approximation?