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Question about normal distribution in probabilty

  1. Jan 12, 2014 #1
    I have a final exam in probability and I faced a question that made me think of the logic and the concept of the normal distribution.
    Here is the question:

    A food industry company imports oil in big tanks and refills bottles of different sizes with it. One of the main filling sizes is the 3000 ml, where the nominal value is 3000 ml but the actual size is a normally distributed random variable N(3000,4). The filling process acceptable specifications are 3000 ± 10 ml.
    a) Determine the proportion of bottles that will be out of specifications.

    My question is:
    Based on my knowledge, I know that the normal distribution function would represent the probability that a single bottle would be out of specifications and not the proportion of bottles that will be out of specifications.

    Am I right? If so, how can I solve such a problem?
  2. jcsd
  3. Jan 12, 2014 #2


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    Staff: Mentor

    You are right that with a finite number of bottles, you cannot determine this proportion - the best thing you can do is calculate the expected proportion, which is the same as the probability that a single bottle does not meet the specifications.
    In the limit of an infinite amount of bottles, the fraction is the same as the probability for a single bottle.
  4. Jan 12, 2014 #3
    I'm not an expert on statistics, but I think that the 4 in the N(3000,4) should tell you something about the shape of the distribution curve. I would use that to determine the equation of the distribution curve and then integrate between 10 and infinity at each end to determine the proportion that was outside of the specification.
  5. Jan 15, 2014 #4


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    The normal distribution here gives the distribution of fill amount per bottle. When you calculate the chance that the fill amount is not to specification, whatever that probability is, it can be interpreted in two (equivalent) ways.

    1) It is the probability that when you randomly select one bottle its fill amount is not within specification

    2) It is the percentage of all bottles that are not filled to specifications
  6. Jan 15, 2014 #5
    You are right. They are being a little bit sloppy. I think that they mean determine the EXPECTED proportion of bottles that will be out of specifications.

    The only other meaning it could have would be that they wanted you to calculate the distribution of the random variable that is the number of bottles out of spec in a given sample. But that is much too hard for an undergraduate exam. Besides, if that is what they wanted they would have written so.
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