Coal is carried from a mine in West Virginia to a power plan

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 = 75 tons and standard deviation = 1.4 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 25 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.)

Solution

Normal Distribution
Mean ( u ) =75
Standard Deviation ( sd )=1.4
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 74.5) = (74.5-75)/1.4
= -0.5/1.4= -0.3571
= P ( Z <-0.3571) From Standard Normal Table
= 0.3605                  

b)
P(X < 74.5) = (74.5-75)/1.4/ Sqrt ( 25 )
= -0.5/0.28= -1.7857
= P ( Z <-1.7857) From Standard NOrmal Table
= 0.0371