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 on the problem and how to tackle it.
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 ppm^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%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 1ppm" 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. Any thoughts on how to interpret the problem are gratefully appreciated.
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 ppm^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%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 1ppm" 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. Any thoughts on how to interpret the problem are gratefully appreciated.