The breaking strength in kgmm for a certain type of syntheti

The breaking strength (in kg/mm) for a certain type of synthetic fabric commonly used for sutures follows a normal distribution with mean 1.86 and standard deviation 0.27. A simple random sample of 80 pieces of fabric is drawn.
(a) What is the probability that the sample mean breaking strength is less than 1.8 kg/mm? (b) Find the 80th percentile of the sample mean breaking strength.

Solution

Normal Distribution
Mean ( u ) =1.86
Standard Deviation ( sd )=0.27
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 1.8) = (1.8-1.86)/0.27
= -0.06/0.27= -0.2222
= P ( Z <-0.2222) From Standard Normal Table
= 0.4121                  
b)
P ( Z < x ) = 0.8
Value of z to the cumulative probability of 0.8 from normal table is 0.842
P( x-u/s.d < x - 1.86/0.27 ) = 0.8
That is, ( x - 1.86/0.27 ) = 0.84
--> x = 0.84 * 0.27 + 1.86 = 2.0873