A type of assembly is produced by gluing 5 identical wafers together in a sandwich-like arrangement. The critical dimension of this assembly is its width, the sum of the widths of the 5 wafers. The specifications for the assembly width are from 4.95 inches to 5.05 inches, with a target of 5.00 inches. The manufacturer wants at least 99.5% of the assemblies to meet specifications. Based on a lot of evidence, the individual wafer widths are normally distributed with mean 1 inch and some standard deviation σ.
a. One engineer, Bob Smith, argues that the company should try to achieve a value of σ (through appropriate changes in the process) such that 99.5% of all individual wafer widths are between 0.99 inch and 1.01 inches. What value of σ is necessary to achieve this?
b. A second engineer, Ed Jones, argues that Smith is solving the wrong problem. Instead, he says, the company should try to achieve a value of σ such that the sum of 5 normally distributed random values has only a 0.5% chance of falling outside the interval from 4.95 inches to 5.05 inches. What value of σ is necessary to achieve this?
c. Which of the engineers is solving the "correct" problem? If the company follows the advice of the wrong engineer, will it err on the high side (too many assemblies out of specifications) or the low side? Is there any disadvantage to erring on the low side?