A production process is checked periodically by a quality co

A production process is checked periodically by a quality control inspector. The inspector selects simple random samples of 30 finished products and computes the sample mean product weight . If test results over a long period of time show that 5% of the  values are over 2.1 pounds and 5% are under 1.9 pounds, what are the mean and the standard deviation for the population of products produced with this process?

What is the population mean? (to 1 decimal)


What is the population standard deviation (to 2 decimals)?

Solution

Normal Distribution
Mean ( u ) =0
Standard Deviation ( sd )=1
Normal Distribution = Z= X- u / sd ~ N(0,1)                  

P(X > 2.1) = 0.05
P(X < 1.9) = 0.05

P(X > 2.1) = 0.05
P((X - )*sqrt(30)/x > (2.1 - )*sqrt(30)/x ) = 0.05
P(Z > z) = 0.05
(2.1 - )*sqrt(30)/x ) = 1.645 ----(1)

P(X < 1.9) = 0.05
P((X - )*sqrt(30)/x < (1.9 - )*sqrt(30)/x ) = 1.645
P(Z < - z) = 0.05
(1.9 - )*sqrt(30)/x ) = - 1.645 ----(2)

5.4772*Eqn(1) + 5.4772Eqn(2)

11.502 = 1.645*x + 5.4772*
10.4067 = - 1.645*x + 5.4772*
--------------------------------------...
21.9089= 10.9544*

= 2

Substitue = 2 => 11.502 = 1.645*x + 5.4772*
x = 0.3323