A bottling company uses a filling machine to fill cans

A bottling company uses a filling machine to fill cans with an energy drink. The cans are supposed to contain 250 milliliters (ml). The machine, however, has some variability, so the standard deviation of the size is sigma = 3 ml. Assuming that the process mean is exactly equal to the target value (that is, 250 ml).

A sample of 6 cans is inspected each hour for process control purposes, and records are kept of the sample mean volume. What will be the mean and standard deviation of the numbers recorded? Averages are less variable than individual observations. Suppose that the can volumes vary according to a Normal distribution.

What is the probability that the volume of a single randomly chosen can differs from the target value by 1 ml or more? (4 decimal places) What is the probability that the mean volume of an simple random sample (SRS) of 6 cans differs from the target value by 1 ml or more?