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 μ = 66 tons and standard deviation σ = 0.5 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 27 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 or No

Suppose the weight of coal in 27 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

µ = 66, σ = 0.5

a) probability that one car chosen at random will have less than 65.5, P(X < 65.5) =

= P( (X-µ)/σ < (65.5-66)/0.5 )

= P(z < -1)

Using excel function:

= NORM.S.DIST(-1, 1)

= 0.1587

b) µ = 66, σ = 0.5, n = 27

P(X̅ < 65.5) =

= P( (X̅-μ)/(σ/√n) < (65.5-66)/(0.5/√27) )

= P(z < -5.1962)

Using excel function:

= NORM.S.DIST(-5.1962, 1)

= 0.0000

c) No, that fact would not make us suspect that the loader had slipped out of adjustment.

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