The Central Limit Theorem Preview CT-STEM

Applications

From our formula sheet:

\(\mu_\bar X = \mu\) \(\sigma_\bar X = \frac {\sigma}{\sqrt n}\)

Trains carry iron ore from a mine in Brazil to an aluminum processing plant in Peru in hopper cars. Filling equipment is used to lode ore into the hopper cars. When functioning properly, the actual weights of ore loaded into each car by the filling equipment at the mine are approximately normally distributed with a mean of 70 tons and a standard deviation of 0.9 tons. If the mean is greater than 70 tons, the loading mechanism is overfilling.

Questions

Please answer the questions below.

Question 4.1

a) If the filling equipment is functioning properly, what is the probability that the weight of the ore in a randomly selected car will be 70.7 tons or more? Show your work.

Question 4.2

b) Suppose that the weight of ore in a randomly selected car is 70.7 tons. Would that fact make you suspect that the loading mechanism is overfilling cars? Justify your answer.

Question 4.3

c) If the filling equipment is functioning properly, what is the probability that a random sample of 10 cars will have a mean weight of 70.7 tons or more? Show your work.

Question 4.4

d) Based on your answer in part (c), if a random sample of 10 cars had a mean ore weight of 70.7 tons, would you suspect that the loading mechanism was overfilling the cars? Justify your answer.

Notes

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