A food processor manufactures a certain canned vegetable that has a printed label weight of 305 gm. The specifications for the net contents in each can are 305 gm + 10 gm. Individual cans are randomly selected from the end of the filling line, after sealing, and weighed. These are gross weights and reflect the weight of the empty cans, the lids, and the processed vegetable contents. The gross weights are in a state of statistical control with an average of 324.1 gm and a standard deviation of 4.5 gm. The cans and lids come from a supplier who provides data that shows that the processes producing these components is in statistical control with a process average and standard deviation for the cans of 15 gm and 2 gm, respectively; and for the lids the average and standard deviation are 3 gm and 0.3 gm, respectively.
a. Estimate the average and standard deviation of net weights of this canning operation.
b. Assuming a normal distribution, what percentage of the canning process is producing cans that are overweight (i.e., above the upper specification limit)? What percentage is underweight?
c. Assuming that the average of the canning operation can be brought to and controlled at the target weight of 305 gm, by what percentage would the variance of the gross fill weights have to be reduced so that the average fills are 3.5 standard deviations away from each specification limit? Assume that the variances of the cans and lids remain the same.