Six Sigma Is One Approach for Setting Quality Expectations for a Given

Six Sigma is one approach for setting quality expectations for a given process or output. As pointed out in Chapter 17 of your text, the term "Six Sigma" comes from statistics: in a normal distribution, the area (probability) outside of +/− six (6) standard deviations from the mean value is exceedingly small. You are provided the following values from a standardized normal distribution (i.e., a distribution with a mean of zero and a standard deviation of 1.0): $\begin{array}{|c|c|}\hline \mathbf{Z} & \mathbf{p} \\\hline 0.00 & 0.50000000 \\\hline 1.00 & 0.15870000 \\\hline 1.50 & 0.06681000 \\\hline 1.75 & 0.04006000 \\\hline 2.00 & 0.02275000 \\\hline 2.50 & 0.00621000 \\\hline 3.00 & 0.00135000 \\\hline 3.50 & 0.00023270 \\\hline 4.00 & 0.00003169 \\\hline\end{array}$ Note: The Z values in the above table refer to the number of sigmas to the right of center (i.e., to the right of Z = 0). The listed probabilities, p, refer to the area to the right of the chosen Z point, as illustrated by the graph below: Required:
1. Given the above, what is the total probability (area under the curve) corresponding to Z = 0 +/− 1.0 sigma, rounded to two decimal places (e.g., 0.34817 = 34.82%)? How many units (out of 1,000 outputs) would have one or more defects for a process operating at one-sigma performance level? Round your answer to nearest whole number.
2. What is the total probability (area under the curve) corresponding to Z = 0 +/− 3.0 sigma, rounded to two decimal places? How many units (out of 1,000 outputs) would have one or more defects for a process operating at three-sigma performance level? Round your answer to nearest whole number.
3. Under a four-sigma control level, what is the total area in the two tails of the distribution, rounded to six (6) decimal places (e.g., 0.00004169 = 0.00417%). What is the total probability (area under the curve) corresponding to Z = 0 +/− 4.0 sigma, rounded to two decimal places? How many units (out of 100,000 outputs) would have one or more defects for a process operating at four-sigma performance level? Round your answer to nearest whole number.
4. What general conclusion can you draw based on the preceding results?