The discard limit in fish containing mercury is too high. How should it be chosen such that average levels are as small as possible? This is a question I'm trying to answer for my probability homework. I've been looking at it for the last 2 days and I'm not getting anywhere so I ask for help. Below is the question. After are my thoughts which i have moved from a comment into the main body.  

The mercury content in swordfish sold in some parts of the US is known
  to be normally distributed with mean 1.1 ppm and variance 0.25 pp$m^2$
  (see e.g. Lee and Krutchkoff and/or exercises 5.2 and 5.3 in Larsen),
  but according to the health authorities the average level of mercury
  in fish for consumption should not exceed 1 ppm. Fish sold through
  authorised retailers are always controlled by the FDA, and if the
  mercury content is too high, the lot is discarded. However, about
  $25\%$ of the fish caught is sold on the black market, that is, $25\%$
  of the fish does not go through the control. Therefore the rule
  "discard if the mercury content exceeds $1 ppm$" is not sufficient.
  How should the "discard limit" be chosen to ensure that the average
  level of mercury in the fish actually bought by the consumers is as
  small as possible?

Basically, I'm not sure about what is expected from me here. The 75% of fish tested have a mean mercury content above the allowed limit. So is it possible to somehow make the mean lower by by having a cutoff at a high point to lower the average? Is that even what they mean by 'choosing the discard limit'? If we discard the top end and somehow truncate the normal distribution will we then have a lower mean? Is that even a legal move? 
Furthermore I don't like the sentence 'How should the discard limit be chosen'. Surely its chosen by the medically qualified and isn't something you can just change? I feel like the wording is also causing some miscommunication. 
 A: If all of the fish were sold through controlled channels (i.e. not on the black market) then the discard limit given would ensure that no fish was sold containing more than $1$ppm of Mercury.  However, only $75\%$ of fish are sold that way; $25\%$ of the fish are sold through the black market.  The amount of Mercury in black market fish is normally distributed around a mean of $1.1$ppm with variance $0.25$ppm$^2$.  So you can calculate (which is what the question wants you to do): 


*

*what is the expected amount of Mercury you will get from a fish you buy, given that you don't know whether it's come through controlled sources or not, You know that any controlled-source fish has the same mean and distribution, with an additional cut-off at $1$ppm.


Once you have that, you need to determine a new cut-off point, which will be lower than $1$ppm, for which the expectation of Mercury content is exactly $1$ (or just under since you'll probably have to give a decimal).
The maths is up to you though :)
And, parenthetically, whether or not you like the wording of a question is irrelevant.  In real world terms though, the "medically qualified" have done their part: they've assessed the safe level as $1$ppm.  It's now down to the people who regulate the fishermen and fish-sellers to put rules in place that meet the medical requirements, and that's what the "discard limit" you dislike so much is.
