Manufacturing and Product Development The 2013 Kawasaki Jet Ski Ultra 300LX is supercharged and intercooled, has a four-stoke inline-four engine, and is designed to have a load capacity of 496 pounds. If the passenger and gear weight exceed this capacity, the jet ski will not function properly and there will be a severe strain on the engine. To determine if customers are following the load capacity recommendation, a random sample of these jet skis is obtained from marinas around the country and each is carefully weighed as owners prepare for a ride. The summary statistics were n = 33 and x‾ = 498.42. Assume σ = 46.7.
a. Is there any evidence to suggest that the true mean weight of passengers and gear is greater than 496 pounds? Use a = 0.01.
b. Suppose the engineers have determined that the watercraft will overheat rapidly if the load is 525 pounds. Find the probability of a type II error if the true mean weight of passengers and gear is 525 pounds; that is, find β(525).