My Math Forum approximation problem

 January 24th, 2012, 06:44 PM #1 Senior Member   Joined: Nov 2009 Posts: 169 Thanks: 0 approximation problem The median age of residents of the United States is 31 years. If a survey of 100 randomly selected U.S. residents is to be taken, what is the approximate probability that at least 60 will be under 31 years of age? it seems like a normal approximate binomial question, but I don't know what median age mean, is it just the mean of the ages, or the median of the age, how do I use the approximation if I only kow the median of it?
 January 25th, 2012, 03:19 PM #2 Global Moderator   Joined: May 2007 Posts: 6,759 Thanks: 696 Re: approximation problem Median means half of the population is greater and half less. To calculate what you want exactly you need to use binomial distribution, summing from 60 to 100, where the total is 100. Alternatively you can use an approximate normal with mean = 50 and standard deviation = 5. The probability then is the probability of more than 2 standard deviations on one side.
 January 25th, 2012, 07:32 PM #3 Senior Member   Joined: Nov 2009 Posts: 169 Thanks: 0 Re: approximation problem so I used normal N~(50,5) p(y>=60)=p(y>=59.5)=p(z>=59.5-50/5)=p(z>=1.9)=0.0287 which does not make sense since as p=0.5, p(y>=60) should be something around 0.4, the probability I got is too small, anyone know why?
January 25th, 2012, 09:43 PM   #4
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Re: approximation problem

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 Originally Posted by 450081592 so I used normal N~(50,5) p(y>=60)=p(y>=59.5)=p(z>=59.5-50/5)=p(z>=1.9)=0.0287 which does not make sense since as p=0.5, p(y>=60) should be something around 0.4, the probability I got is too small, anyone know why?
It's not too small, that's why. I got .0228. (You've shifted by .5 for reasons I don't know - that might be a "book thing." But the 2 answers are close enough.) It's a fallacy that it should be .40. Just not sure I'd know how to explain why. From a normal distribution point of view, it's just cause that's bell shaped and 2 standard deviations to the right in that tail just isn't 40%. But explaining that as a binomial, I don't know that my brain can handle coming up with an explanation right now. Someone else can do that, but in the meantime you'll know your answer is OK assuming your book likes that .5 shift thingy. (Probably - I've seen that done, just didn't learn it that way.)

 January 26th, 2012, 12:20 PM #5 Global Moderator   Joined: May 2007 Posts: 6,759 Thanks: 696 Re: approximation problem Using the notation already used, the probability you want is P(z>2). I presume that is how Erimess got his answer.
 January 26th, 2012, 06:46 PM #6 Senior Member   Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: approximation problem Well, that's how Erimess got the answer she used.
January 27th, 2012, 01:29 PM   #7
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Re: approximation problem

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 Originally Posted by Erimess Well, that's how Erimess got the answer she used.
Sorry: Since the gender of a person is not normally displayed, I used he automatically - there is no word in English for he or she.

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