Question

A machine is used to fill Apple Juice bottles with juice. The machine has a known standard deviation of ? = 0.05 liters. The target mean fill volume is ? = 2.0 liters. A quality control manager obtains a random sample of 50 bottles. He will shut down the machine if the sample of these 50 bottles is less than 1.95 or greater than 2.1. What is the probability that the quality control manager will shut down the machine.?

Answer 1

Solution:

Given that ,

= 2.0

= 0.05

A sample of size n = 50 is taken from this population.

Let be the mean of sample.

The sampling distribution of the is approximately normal with

Mean = = 2.0

SD =      = 0.05/​50 = 0.00707106781

Now ,

P(Sample mean is less than 1.96 or greater than 2.1)

= 1 - { P( sample mean is in between 1.96 and 2.1) }

= 1 - { P(1.96 < < 2.1) }

= 1 - { P( < 2.1) - P( < 1.96) }

= 1 -  {P[( - )/ < (2.1 - 2.0)/0.00707106781] - P[( - )/ < (1.96 -2.0)/0.00707106781] }

= 1 - { P[Z < 14.14] - P[Z < -5.66] }

= 1 - { 1.0000 - 0.0000} (use z table)

= 0.0000