The defect length of a corrosion defect in a pressurized ste

The defect length of a corrosion defect in a pressurized steel pipe is normally distributed with mean value 30 mm and standard deviation 7.8 mm. a. What is the probability that defect length is at most 20 mm? Less than 20 mm? b. What is the 75th percentile of the defect length distribution-that is, the value that separates the smallest 75% of all lengths from the largest 25%? c. What is the 15th percentile of the defect length distribution? d. What values separate the middle 80% of the defect length distribution from the smallest 10% and the largest 10%?

Solution

Normal Distribution
Mean ( u ) =30
Standard Deviation ( sd )=7.8
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 20) = (20-30)/7.8
= -10/7.8= -1.2821
= P ( Z <-1.2821) From Standard Normal Table
= 0.0999                  

b)
P ( Z < x ) = 0.75
Value of z to the cumulative probability of 0.75 from normal table is 0.674
P( x-u/s.d < x - 30/7.8 ) = 0.75
That is, ( x - 30/7.8 ) = 0.67
--> x = 0.67 * 7.8 + 30 = 35.2572                  

c)
P ( Z < x ) = 0.15
Value of z to the cumulative probability of 0.15 from normal table is -1.036
P( x-u/s.d < x - 30/7.8 ) = 0.15
That is, ( x - 30/7.8 ) = -1.04
--> x = -1.04 * 7.8 + 30 = 21.9192