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According to Boeing data, the 757 airliner carries 200 passengers and has doors with a mean height of 72 inches. Assume for a certain population of men we have a mean of 69.0 inches and a standard deviation of 2.8 inches.
: Normal Approximation to the Binomial
Historically, being able to compute binomial probabilities was one of the most important applications of the central limit theorem. Binomial probabilities with a small value for n (say, 20) were displayed in a table in a book. To calculate the probabilities with large values of n , you had to use the binomial formula, which could be very complicated. Using the normal approximation to the binomial distribution simplified the process. To compute the normal approximation to the binomial distribution, take a simple random sample from a population. You must meet the conditions for a binomial distribution :
Recall that if X is the binomial random variable, then X ~ B ( n, p ). The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np and 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). Then the binomial can be approximated by the normal distribution with mean μ = np and standard deviation σ = . Remember that q = 1 – p . In order to get the best approximation, add 0.5 to x or subtract 0.5 from x (use x + 0.5 or x – 0.5). The number 0.5 is called the continuity correction factor and is used in the following example.
Suppose in a local Kindergarten through 12 th grade (K - 12) school district, 53 percent of the population favor a charter school for grades K through 5. A simple random sample of 300 is surveyed.
Let X = the number that favor a charter school for grades K trough 5. X ~ B ( n, p ) where n = 300 and p = 0.53. Since np >5 and nq >5, use the normal approximation to the binomial. The formulas for the mean and standard deviation are μ = np and σ = . The mean is 159 and the standard deviation is 8.6447. The random variable for the normal distribution is Y . Y ~ N (159, 8.6447). See The Normal Distribution for help with calculator instructions.
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