Many homes in the United States are heated with oil. Most heating oil that is used in central Massachusetts is shipped by super tankers into Providence, R.I., and then transported in tanker trucks to their distribution points in Worcester County.

In Providence, the tanker trucks are filled using a pumping mechanism. The pumping mechanism is set to shut off when the weight of the load reaches 72,000 pounds. The weight of trucks is regulated by federal and state laws, resulting in heavy fines for violators. When the pumping mechanism is functioning properly it will pump, on average 72,000 pounds into each truck, with a standard deviation of 700 pounds. The weights are distributed normally. If the mean weight of the loads is greater than 72,000 pounds the pumping mechanism is overfilling.

Just across the Massachusetts state line on route 146 is weighing station. State troopers randomly pull over trucks and weigh them, to make sure they are not overweight.

(a) If the pumping mechanism is functioning properly, what is the probability that the weight of the oil in a randomly selected oil truck will be 72,560 pounds or more? Show your work

(b) Suppose that the weight of oil in a randomly selected oil truck is 72,560 pounds. Would that fact make you suspect that pumping mechanism is overfilling the tanker trucks? Justify

(c) If the pumping mechanism is functioning properly, what is the probability that the mean weight of the oil in a random sample of 10 oils trucks will be 72,560 pounds or more? Show your work.

(d) Based on your answer in part (c), if a random sample of 10 oil trucks had a mean oil weight of 72,560 pounds or more, would you suspect that the loading mechanism was overfilling the tanker trucks? Justify your answer.

(e) In part d, the sample was only 10 trucks. Is it still ok to use normal distribution to estimate the probability in part d? Why or why not?