More on "six-sigma quality." The originators of the "six-sigma quality" idea reasoned as follows. Short-term process variation is described by σ. In the long term, the process mean μ will also vary. Studies show that in most manufacturing processes, ±1.5σ is adequate to allow for changes in μ. The six-sigma standard is intended to allow the mean μ to be as much as 1.5σ away from the center of the specifications and still meet high standards for percent of output lying outside the specifications.

(a) Sketch the specification limits and a Normal distribution for process output when C_p = 2 and the mean is 1.5σ away from the center of the specifications.

(b) What is C_pk in this case? Is six-sigma quality as strong a requirement as C_pk ≥ 2?

(c) Because most people don't understand standard deviations, six-sigma quality is usually described as guaranteeing a certain level of parts per million of output that fails to meet specifications. Based on your sketch in part (a), what is the probability of an outcome outside the specification limits when the mean is 1.5σ away from the center? How many parts per million is this?

(You will need software or a calculator for Normal probability calculations, because the value you want is beyond the limits of the standard Normal table.)