The filling machine at Natrona Bottling Company fills bottle

The filling machine at Natrona Bottling Company fills bottles of Red Ribbon Cherry soda with mean of 20.04 ounces, with a standard deviation of 0.02 ounces. Suppose I buy a 6 pack of bottles, which can be considered randomly (independently) sampled from the output of the machine.

(a) If I buy a whole 6 pack, what is the expected total volume of soda I have bought? (I.e., what is the mean of the total volume of a 6 pack?)

(b) By how much, in standard deviation terms, will my total volume purchased deviate from the expectation? In other words, what is the standard deviation of the total volume of a 6 pack? [Hint, this is similar to the ‘apple/orange’ example in lecture 9, and the example on page 3 of lecture 10.]

(c) What is the long-run theoretical average volume per bottle, in a 6 pack? (This should be somewhat obvious. It could be proved formally using the rules for expected value, but you don’t need to show the proof.) (d) By how much will the average volume per bottle of my randomly-selected 6 pack deviate from the long run theoretical average? In other words, what is the standard deviation of the mean, for a sample of size n=6? Show any work you perform

Solution