A Warehuse contains a large population of many thousands of

A Warehuse contains a large population of many thousands of machine parts x=the thickness at a particular location on the part. From previous studies x is known to be approximately normally distributed, the population mean of x is 10.8 mm, and the population standard deviation of x is 0.9.

You take a random samle of 20 machine parts and compute the sample mean.

A) What is the probability that your sample mean is less than 10.4mm?

B) What is the probability that your sample mean is between 10.4 & 10.6mm?

Solution

Mean ( u ) =10.8
Standard Deviation ( sd )=0.9
Number ( n ) = 20
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
a)
P(X < 10.4) = (10.4-10.8)/0.9/ Sqrt ( 20 )
= -0.4/0.2012= -1.9876
= P ( Z <-1.9876) From Standard NOrmal Table
= 0.0234                  

b)
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 10.4) = (10.4-10.8)/0.9/ Sqrt ( 20 )
= -0.4/0.2012
= -1.9876
= P ( Z <-1.9876) From Standard Normal Table
= 0.02343
P(X < 10.6) = (10.6-10.8)/0.9/ Sqrt ( 20 )
= -0.2/0.2012 = -0.9938
= P ( Z <-0.9938) From Standard Normal Table
= 0.16016
P(10.4 < X < 10.6) = 0.16016-0.02343 = 0.1367