Question

An article suggests that substance concentration (mg/cm^3) of influence to a reactor is normally distributed with mew=0.5 and delta=0.1.

a) What is the probability that the concentration is less than 0.3?

b) How would you characterize the largest 5% of all concentration values?

Answer 1

Part a)

X ~ N ( µ = 0.5 , σ = 0.1 )
P ( X < 0.3 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 0.3 - 0.5 ) / 0.1
Z = -2
P ( ( X - µ ) / σ ) < ( 0.3 - 0.5 ) / 0.1 )
P ( X < 0.3 ) = P ( Z < -2 )
P ( X < 0.3 ) = 0.0228

Part b)

X ~ N ( µ = 0.5 , σ = 0.1 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.05 = 0.95
To find the value of x
Looking for the probability 0.95 in standard normal table to calculate Z score = 1.6449
Z = ( X - µ ) / σ
1.6449 = ( X - 0.5 ) / 0.1
X = 0.6645 ≈ 0.7
P ( X > 0.6645 ) = 0.05