The density of a chemical solution is normally distributed w

The density of a chemical solution is normally distributed with mean 0.0046 and variance 9.6 * 10^(-3).

(a) What is the probability that the density is between 0.004 and 0.005?

(b) What is the 95th percentile of the density level?

Solution

Normal Distribution
Mean ( u ) =0.0046
Variance = 9.6 * 10^(-3)
Standard Deviation ( sd )=0.09797
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 0.004) = (0.004-0.0046)/0.09797
= -0.0006/0.09797 = -0.0061
= P ( Z <-0.0061) From Standard Normal Table
= 0.49756
P(X < 0.005) = (0.005-0.0046)/0.09797
= 0.0004/0.09797 = 0.0041
= P ( Z <0.0041) From Standard Normal Table
= 0.50163
P(0.004 < X < 0.005) = 0.50163-0.49756 = 0.0041                  
b)
P ( Z < x ) = 0.95
Value of z to the cumulative probability of 0.95 from normal table is 1.645
P( x-u/s.d < x - 0.0046/0.09797 ) = 0.95
That is, ( x - 0.0046/0.09797 ) = 1.64
--> x = 1.64 * 0.09797 + 0.0046 = 0.1658