Problem background: In June 2021, there were news articles about whether the Federal Aviation Authority would require airlines to weigh passengers before boarding. Indeed, a few years ago, it was reported that Finnair was weighing passengers in an effort to update the estimate of the average passenger weight that they had been using based on a 2009 survey by the European Aviation Safety Agency (Finnair was requesting passengers to have themselves weighed voluntarily). Though this sounds intrusive, there is good reason for airlines to have good estimates of passenger weights. A tragic illustration of this was the crash, in January 2003, of US Airways Express Flight 5481. This was a small Beechcraft 1900D plane with 19 passengers, that stalled seconds after take-off and crashed at the airport in Charlotte, N.C. The National Transportation Safety Board (NTSB) investigation found that one of the factors that contributed to this crash was the fact that the plane was overloaded and out of balance due to the use of incorrect passenger weight estimates, with the plane being apparently 580 lbs above its maximum allowable take-off weight, with a center of gravity below what was allowable. Apparently, though the pilots had computed the take-off weight of the plane, including the passengers, and found it to be within limits, they had done so using incorrect passenger weight estimates which had not been revised since 1936 by the Federal Aviation Administration. The NTSB found that these incorrect weight estimates underestimated the average passenger weight by more than 20 lbs., resulting in the pilots having underestimated the total weight of the 19 passengers. Problem: A 30 seater plane manufactured by a non-U.S. aircraft manufacturer has been certified to operate safely with a maximum passenger weight of 5850 lbs. The aircraft manufacturer's sales team has been pitching the aircraft model to a U.S. based airline, which is assessing its various operating parameters such as fuel efficiency as well as load capacity. The airline knows that based upon a recent survey by the National Center for Health Statistics, the weights of people in the U.S.A. have a mean of approximately 182 lbs and standard deviation of about 55 lbs. (i) From the values obtained from the comprehensive survey mentioned above and using a result called the Central Limit Theorem, it can be asserted that the total weight of 30 people chosen at random will be normally distributed with a mean of 5460 lbs and a standard deviation of 301 lbs. Using this fact, what is the probability that the total weight of a random collection of 30 U.S. passengers on board a full flight does not exceed the maximum allowable weight of 5800 lbs? (ii) Based upon your answer in (i), do you think that the airline should invest in this model for its U.S. operations? Justify your answer in a sentence or two