Find the approximate probability that a carton of 64 boxes contain fewer than 6300 clips in all?

Question

A company packs paper clips in boxes labeled “100 pieces.” We have counted the clips in some of those boxes and found that the number in a box varies form as low as 93 to over 100.  Suppose that the number of clips in boxes produced by this company has a mean of 100 and a standard deviation of 8.  A carton is made up of 64 boxes of clips.  Suppose a carton of 64 boxes can be considered as a random sample of 64 boxes.  Let X-bar be the mean number of clips per box for a carton of 64 boxes.

Find the approximate probability that a carton of 64 boxes contain fewer than 6300 clips in all, more than 100 shy of 6400.

I know I'm supposed to use the Standard units z=(x-mu)/standard deviation
I guess I just don't understand the way my teacher worded the question.

slowjerry · Accepted Answer

The random variable which represents the sum of 64 random variables x_bar  (each representing one box )
will have an expected mean of 64*100 
and an expected variance of 64* 64 ( thats the standard deviation of 8 squared to get the variance of one box then multiplied by the 64 boxes)

so that's a mean of 6400 and a standard deviation of 64.

and the prob of the random variable having a value of less than 6300 is that equivalent to the standardized  variable  
   ............     z having a value of less than    (6300-6400)/64

which i shall leave you to look up.

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Find the approximate probability that a carton of 64 boxes contain fewer than 6300 clips in all?

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