Question

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 66 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean ? = 66 tons and standard deviation ? = 1.2 ton.

(a) What is the probability that one car chosen at random will have less than 65.5 tons of coal? (Round your answer to four decimal places.)


(b) What is the probability that 33 cars chosen at random will have a mean load weight x of less than 65.5 tons of coal? (Round your answer to four decimal places.)


(c) Suppose the weight of coal in one car was less than 65.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment?

Yes

No    

d) Suppose the weight of coal in 33 cars selected at random had an average x of less than 65.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Yes, the probability that this deviation is random is very small.

Yes, the probability that this deviation is random is very large.    

No, the probability that this deviation is random is very small.

No, the probability that this deviation is random is very large.

Answer 1

Let X denotes the weights of coal loaded in each car. X follows normal distribution with mean 66 tons and standard deviation 1.2 ton. The probability can be calculated using the standard normal transformation

(a)

Probability that one car chosen at random will have less than 65.5 tons of coal is given as P(X < 65.5). Using the standard normal transformation,

Probability that one car chosen at random will have less than 65.5 tons of coal is 0.3383

(b)

If X follows normal distribution with mean and standard deviation then the sample mean follows normal distribution with mean and standard deviation . So the mean load weight of 33 cars follows normal distribution with mean 66 tons and standard deviation .

Probability that 33 cars chosen at random will have a mean load weight less than 65.5 tons of coal is given as . Using the standard normal transformation,

Probability that 33 cars chosen at random will have a mean load weight less than 65.5 tons of coal is 0.0083.

(c)

If the weight of coal in one car was less than 65.5 tons, that fact does not make suspect that the loader had slipped out of adjustment as the probability that one car chosen at random will have less than 65.5 tons of coal is 0.3383 which indicates that 33.83% of the cars will have coal lesser than 65.5 tons. So it is an usual thing.

Correct option: NO

d)

If the weight of coal in 33 cars selected at random had an average less than 65.5 tons it would make fact to suspect that the loader had slipped out of adjustment, because the probability that 33 cars chosen at random will have a mean load weight less than 65.5 tons of coal is 0.0083 which indicates that only 0.83% of 33 cars selected at random will have average weight less than 65.5 tons. So if the weight of coal in 33 cars selected at random had an average less than 65.5 tons, it will be an unusual thing.

Correct option:

Yes, the probability that this deviation is random is very small.