Statistics Question

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 89 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean μ = 89 tons and standard deviation σ = 0.8 ton.

(a) What is the probability that one car chosen at random will have less than 88.5 tons of coal? (Round your answer to four decimal places.)

(b) What is the probability that 35 cars chosen at random will have a mean load weight x of less than 88.5 tons of coal? (Round your answer to four decimal places.)

(c) Suppose the weight of coal in one car was less than 88.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment?

YesNo    

Suppose the weight of coal in 35 cars selected at random had an average x of less than 88.5 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

Maya

(a) We want p (X<88.5) = P (Z < (88.5 - 89)/0.8) = P (Z < -0.625).

Using an area table of areas under the standard normal gives this to be 0.2660

So a 27% chance approximately.

(b) The distribution of the mean here is normal, with μ = 89, σ = 0.8 / √35

So p (mean < 88.5) = p (Z < (88.5-89) / (0.8/√35) = P (Z < -3.70) = 0.0001 ie 0.01 %

(c) No (high probability that this can happen at random)

Yes (very low probability this can happen at random).

Mike