I wanted to post the question I wrote for the 2014 midterm in quantitative reasoning at MIT. The answers are italicized. It’s a good example of using quantitative reasoning for fisheries regulations planning.
It’s lobster season in Fiji. Villagers are allowed to take lobsters from a protected area for a two-week season every year in the summer. Lobster weights are distributed normally with a mean of 450 grams and a standard deviation of 120 grams. The national fisheries agency will not allow lobsters to be removed if they weigh less than 100 grams, as these are juveniles. If a lobster weighs 600 grams or more, the fisheries agency will not allow it to be removed since it is a “valuable spawner”.
a. What is the probability that a lobster chosen at random by a fisherman is too small to keep (i.e. it is 100 grams or less) show your work.
-Find the z score for the juvenile weight: (100-450)/120= ~2.92
-Look it up in the z table and see 0.4982
-Subtract this from 0.5 to get the area under the tail =.0018
-The probability of choosing a lobster this small is .0018
b. The chief has called you in to consult on behalf of his village. He says that the juvenile lobsters are too easy to catch, and that most fishermen have to throw most of their catch back into the water. He wants the size classification for juveniles changed to that they can keep the 100g lobsters. Do you think his perception is accurate?
No the likelihood of catching juveniles is so small that the fact they are catching so many is either untrue or it means that assuming the weights are normally distributed may not be correct.
c. What percentage of lobsters caught can the fishermen keep and bring to market? What can you say to the chief about the fairness of these regulations?
The allowable catch is between 100 and 600 grams. This is the area bounded by their two z scores, which we know is 2.92 for 100 g from part a. Now we need to get the z score for 600 g:
-The percentage from the mean to juveniles is .4982, and the percentage from the mean to valuable spawners is .3944. Add them to get the total area beneath the curve.
-A random lobster caught has an 89.26% chance of being kept since it is within the regulated allowable size. You can keep nearly 90% of everything you catch, the regulations do not need to be changed.