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Chapter 6, part C III. Normal Approximation of Binomial Probabilities When n is very large, computing a binomial gets difficult, especially with smaller pocket calculators. The Situation If a binomial problem has the following characteristics, you can use the normal probability distribution to approximate the binomial probability. • n>20 • np 5, and • n(1-p) 5 (recall that p is the probability of “success”) An Example A firm has found that 10% of their sales invoices contain errors. If the firm takes a sample of 100 invoices, what is the probability that 12 have errors? Steps to approximate with the normal 1. Calculate a mean and standard deviation: = np = 100(.10) = 10 np(1 p) 100(.10)(.90) 3 2. Create an interval around x=12 by adding and subtracting .5 from 12. =10 11.5 12 12.5 x Steps continued... 3. Find P(11.5 x 12.5) 4. Convert the range to z-scores. zL = (11.5-10)/3 = .5 zH = (12.5-10)/3 = .83 5. Use the standard normal probability table to find: P(.5 z .83) Steps continued... 6. Find P(0 z .83) - P(0 z .5) = .2967 - .1915 = .1052 The binomial solution to this same problem is .0988, so our normal approximation is fairly accurate. Check out this simulation (you browser needs to be Java compatible) and choose p and sample size n. IV. Exponential Probability Distribution The exponential is used to describe the time (and probability) that it takes to do something. For example, it can be used to calculate the probability that a delivery truck will be loaded in 15 to 30 minutes time. A. Exponential Probability Density function f ( x) 1 e x ( ) For x>0 and >0. As an example, let’s suppose that a delivery truck is loaded with a mean time of =10 minutes. B. Computing Probabilities with the Exponential The function f(x)=(1/10)e(-x/10) draws the curve below, but probabilities are still calculated as the area under the curve. For any x0, if you want the probability that the truck is loaded in less than that time, use the following formula: P( x x0 ) 1 e ( x0 ) A diagram of the exponential f(x) .10 f(x)=(1/10)e(-x/10) 0 5 10 20 30 x (time) Example probabilities Find the probability that the loading will take less than 5 minutes: P(x5) = 1-e(-5/10)=.3935 What about a loading time of less than 30 minutes? P(x30) = 1-e(-30/10)=.9502