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 88 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean μ = 88 tons and standard deviation σ = 1.3 ton.

(a) What is the probability that one car chosen at random will have less than 87.5 tons of coal? (Round your answer to four decimal places.)


(b) What is the probability that 36 cars chosen at random will have a mean load weight x of less than 87.5 tons of coal? (Round your answer to four decimal places.)

Answer 1

a)

Given,

= 88, = 1.3

We convert this to standard normal as

P( X < x) = P( Z < x - / )

So,

P( X < 87.5) = P( Z < 87.5 - 88 / 1.3)

= P( Z < -0.3846)

= 1 - P( Z < 0.3846)

= 1 - 0.6497

= 0.3503

b)

Using central limit theorem ,

P( < x) = P (Z < x - / / sqrt(n)  )

So,

P( < 87.5) = P( Z < 87.5 - 88 / 1.3 / sqrt(36) )

= P( Z < -2.3077)

= 1 - P( Z < 2.3077)

= 1 - 0.9895

= 0.0105