1. The diameter of bearing shafts that are manufactured in your factory follow a normal distribution with a mean of 1 inch and a standard deviation of 0.002 inch.
a. If a purchase is looking for shafts that have a diameter of 1 ± 0.003 inch, what fraction of your manufactured bearing shafts will be rejected by the purchaser?
b. You decide to improve the quality of you product and set up a target that no more that 1% of your manufactured shafts should be rejected. In order to ensure this, what should the standard deviation of the new sample be?
2. Imagine that a dam is being designed for flood control in a community. The historical stream flow record illustrates that a hydrologic extreme event (flood) occurred once in 100 years.
a. What is the chance that such an event will occur at least once during the life time of the dam? I.e. in the next 30 years?
b. The elevation (i.e. water level in the dam) during floods follows an exponential distribution with an average elevation of 5 feet. The dam is designed to hold up to 12 feet of water. The community downstream of the dam will be inundated if the dam overtops or spills. What is the probability of overtopping the dam if a flood occurs?