Question

A machine that fills cardboard boxes with cereal has a fill weight whose mean is 12.02oz, with a standard deviation of 0.03 oz. A case consists of 12 boxes randomly sampled from the output of the machine. 2(a) If I buy a case of cereal, what is the expected total weight of the cereal I have bought? 2(b) By how much, in standard deviation terms, will my total weight purchased deviate from the expectation? In other words, what is the standard deviation of the total weight of a case? 2(c) What is the long-run theoretical average weight per box, in a case? 2(d)By how much will the average weight per box of my randomly-selected case deviate from the long run theoretical average? In other words, what is the standard deviation of the mean, for a sample of size n=12 ? Show any work you perform. 2(e) How many boxes must be included in a case for the standard deviation of the average weight per box to be 0.005 oz?

Answer 1

The expected total weight of a case of cereal is 144.24 oz, with a standard deviation of 0.07 oz. The long-run theoretical average weight per box is 12.02 oz, with a standard deviation of 0.005 oz.

Step-by-step explanation:

The expected total weight of a case of cereal can be calculated as the mean weight of a box (12.02 oz) times the number of boxes in a case (12). This gives an expected total weight of 144.24 oz. The standard deviation of the total weight of a case can be calculated by multiplying the standard deviation of the fill weight (0.03 oz) by the square root of the number of boxes in a case (12). This gives a standard deviation of 0.07 oz.

The long-run theoretical average weight per box can be calculated using the same mean weight of 12.02 oz. The standard deviation of the mean weight per box for a sample of size n=12 can be calculated by dividing the standard deviation of the fill weight (0.03 oz) by the square root of the sample size (12). This gives a standard deviation of 0.005 oz.

The average weight per box of a randomly-selected case may deviate from the long-run theoretical average, although the amount of deviation will depend on the size of the sample. The larger the sample size, the less deviation there will be. In order to achieve a standard deviation of 0.005 oz for the average weight per box of a case, the number of boxes must be increased to a size of 144 (12^2). This would result in a case containing 12 boxes of 12.02 oz each, yielding an expected total weight of 144.24 oz.

In conclusion, the expected total weight of a case of cereal is 144.24 oz, with a standard deviation of 0.07 oz. The long-run theoretical average weight per box is 12.02 oz, with a standard deviation of 0.005 oz. To achieve a standard deviation of 0.005 oz for the average weight per box of a case, the number of boxes must be increased to a size of 144.