![]() ![]() Then the binomial can be approximated by the normal distribution with mean μ = np and standard deviation σ = n p q n p q. We subtract 0.5 if we are looking for the probability that is greater than or equal to that number. We add 0.5 if we are looking for the probability that is less than or equal to that number. So, for whatever value of x we are looking at (the number of successes). The product >5 is more or less accepted as the norm here.). To ensure this, the quantities npĪnd nq must both be greater than five ( np > 5 and nq > 5 the approximation is better if they are both greater than or equal to 10. Similar to the shape of the normal distribution. The shape of the binomial distribution needs to be Recall that if X is the binomial random variable, then X ~ B( n, p). Each trial has the same probability of a success, p.The outcomes of any trial are success or failure.There are a certain number, n, of independent trials.You must meet the following conditions for a binomial distribution: To compute the normal approximation to the binomial distribution, take a simple random sample from a population. Using the normal approximation to the binomial distribution simplified the process. To calculate the probabilities with large values of n, you had to use the binomial formula, which could be very complicated. Binomial probabilities with a small value for n (say, 20) were displayed in a table in a book. Historically, being able to compute binomial probabilities was one of the most important applications of the central limit theorem. ![]()
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