A factory produces metal links for building long chains used inthe shipping and construction industry. The factory’s R&D unitmodels the length of an individual metal link (in ?m) by a randomvariable ?? with expectation ?0 m and variance ?. ?4 ??^?. Thelength of a chain is equal to the sum of the lengths of individuallinks that comprise the chain. The factory sells ?0 ? long chainsas its final product, which it makes by combining ?002 individualmetal links together. The factory guarantees that the chain lengthis not shorter than ?0 ? − if by chance a chain is too short, thecustomer is reimbursed and a new chain is issued for free.
a. For what percentage of chains would the factory need toreimburse clients and deliver new chains free of cost?
b. The sales team notices that the factory has been handing outa much larger fraction of free chains than expected as per youranswer above. After further probing, the research lab reports thatthe exact expectation is ?. ?? ?m and not ? ?m as was previouslyreported based on rounding-off approximation. Do you think theresearch team committed a serious mistake by rounding off to thenearest whole number instead of using the exact value forexpectation? Why?
