Date: 03/26/99 From: Anonymous Subject: Probability that a general pop will live past 85 My question is: It is estimated that the probability that the general population will live past their 85th birthday is 5.4%. Out of a sample of 600 people, what is the probability that fewer than 30 will live beyond their 85th birthday? I have tried to use the standard normal distribution to approximate but cannot seem to work through it. Can you help?
Date: 03/26/99 From: Doctor Mitteldorf Subject: Re: Probability that a general pop will live past 85 You can do this problem exactly, without a normal approximation, if you use a computer. The probability that exactly n people will live past 85 is a product of two factors: one is 0.054^n * (1-0.054)^(600-n). This reflects the probability that n people will be in the 0.054 group and (600 - n) will be in the group with much higher probability (1-0.054). The second factor is from Pascal's triangle, and it is the number of ways that those n people can be selected from 600: C(600, n) = 600!/(n!(600 - n)!) With a short computer program, you can add up the terms for 0 through 29, and find an exact answer = 0.3071. If you want to use the normal approximation, the formula you need is that the mean, quite transparently, is 600*0.054 = 32.4. The standard deviation, not so obviously, is sqrt(600*0.054*(1 - 0.054)) = 5.536. Now you need either a computer to calculate the area under the normal curve, or a table of these values. The table is indexed by z: the number of standard deviations from the mean. The question arises, do you take (32.4 - 30), (32.4 - 29) or something in between? (The normal approximation really only applies to the extent that 30 is a "large number" so 1/30 should not matter. But 30 is not such a large number, so it does matter.) For the in-between value (32.4 - 29.5), I get 0.3003 for the probability, which is moderately close to the actual value of 0.3071. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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