Final answer:
Yes, the Central Limit Theorem does apply to the binomial distribution. As the sample size increases, the distribution of sample means or sums approaches a normal distribution, even if the original population is not normally distributed. The sample size is a critical factor in applying the CLT.
Step-by-step explanation:
Central Limit Theorem and Binomial Distribution
The Central Limit Theorem (CLT) does apply to the binomial distribution. When you have a binomial distribution with parameters n (number of trials) and p (probability of success), and if n is large enough, the distribution of the sample means will approach a normal distribution. This application of the CLT is especially useful when n is large, as calculating binomial probabilities directly can be quite complex. The mean of the sample means will be equal to the original mean (np), and the variance will be the original variance (np(1-p)) divided by the sample size.
Applying the CLT for sums tells us that for a large sample size n, the distribution of the sum of sample means will also tend toward a normal distribution. If the original population has a mean μx and a standard deviation σx, then the mean of the sums is nμx and the standard deviation is (√n) (σx).
For example, if a student takes samples of size 50 from a population with a mean of 80 and a standard deviation of four, and calculates the sum of each sample, according to the CLT, the expected distribution of these sums will be normal with a mean of 50 * 80 = 4000 and a standard deviation of √50 * 4 = √200, approximated to a normal distribution.
Sample size and the original population's distribution are central to determining when to apply the central limit theorem.
The larger the sample size, the more the sample means tend toward the population mean, μ, in accordance with the law of large numbers.