Question

We operate a bottle filling factory. One of our machines fills a 32-ounce bottle with a target of 32.08 ounces of orange juice. Every 10 minutes or so we obtain a sample filled bottle and hold it for the quality control department who will weigh the approximately 50 such bottles to determine if the machine is performing correctly. If the average of the 50 bottles is less than 32 ounces, the machine is shut down and the entire production from that 8-hour shift is held until further tests are performed.

1. We know that over the years this machine delivers 32.082 ounce of orange juice with a standard deviation 0.01 ounces. What is the probability that a sample of 50 bottles has a mean volume of less than 32 ounces?

2. There are 1,008 8-hour shifts per year (allowing for holidays, maintenance and a two-week scheduled shutdown). In a typical year, hour many times do we expect the quality department to shut down the machine and hold that shift’s production?

Answer 1

Let X be the volume of orange juice in a bottle with

The distribution of X is not known

Let be the sample mean of 50 bottles

Using Central Limit Theorem , the sampling distribution of sample mean follow Normal with

mean = 32.080 ( population mean )

and standard error =

as sample size is large

that is ,

then

To find

= P(z < -58.57)

= 0.0000 (from z table)

Probability that a sample of 50 bottles have a mean volume less than 32 ounces is 0.0000

Number of times we expect quality department to shut down the machine

= 1008*

= 0

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