Question

a. 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 75 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean μ = 75 tons and standard deviation σ = 1.6 ton.

(a) What is the probability that one car chosen at random will have less than 74.5 tons of coal? (Round your answer to four decimal places.)


(b) What is the probability that 42 cars chosen at random will have a mean load weight x of less than 74.5 tons of coal? (Round your answer to four decimal places.)

Suppose the weight of coal in 42 cars selected at random had an average x of less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?(pick one)

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.

b. Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene. Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 7.6 minutes and a standard deviation of 2.3 minutes. For a randomly received emergency call, find the following probabilities. (Round your answers to four decimal places.)

(a) the response time is between 3 and 9 minutes


(b) the response time is less than 3 minutes


(c) the response time is more than 9 minutes

c. In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Do you take the free samples offered in supermarkets? About 64% of all customers will take free samples. Furthermore, of those who take the free samples, about 35% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 313 customers passed by your counter. (Round your answers to four decimal places.)

(a) What is the probability that more than 180 will take your free sample?


(b) What is the probability that fewer than 200 will take your free sample?


(c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.35, while P(sample) = 0.64.


(d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

Answer 1