The number of Btus of petroleum and petroleum products used

The number of Btu’s of petroleum and petroleum products used per person in the united states in 1975 was normally distributed with mean 153 million Btu’s and standard deviation 25 million Btu’s. Approximately what percentage of the population used between 128 and 178 million Btu’s during that year? Approximately what percentage of the population used in excess of 228 million Btu’s?

Solution

For the Normal (153,25² ) variable, we use standardisation by subtracting 153 from X and dividing the result by 25. This is the Z score.

P(128 < X < 178) = P( (128-153)/25 < Z < (178-153)/25 ) = P( -1 < Z < 1) = P(Z<1) - P(Z<-1) = 0.6827 = 68.27%

P(X>228) = P(Z > (228-153)/25 ) = P(Z>3) = 0.00135 = 0.135%