Problem \#1: (3-D Charts) In this problem, wafer is processed one at a time by a machine. (They are not processed in a group or lot.) Within-piece variability is important and is to be tracked by a within-piece range RW chart. In addition, the process mean is to be monitored with an individual X chart and piece-to-piece variability is to be monitored with a moving range RP,M chart, with a moving window size of 2 . Part (a): Suppose that process mean is 100 . Suppose also that the piece-to-piece variability follows a Normal distribution with mean 0 and standard deviation P=2. Finally, suppose that within-piece variability follows a Normal distribution with mean 0 and standard deviation W=1. Develop the control limits for the three charts. Part (b): Suppose now that none of three process parameters are known and that five measurements have been collected at five different locations on each of 30 wafers. The collected data are organized in a 30 by 5 matrix, as shown in a companion Excel file entitled ISE235-Problem\#1-3D charts-with known parameters and with estimated parameters Develop the control limits for the three control charts. Also, plot on the three control charts the corresponding statistics for the 30 samples. Check if any of the plotted points fall outside of any of the control limits of the three control chart. Part (c): Now suppose that the five locational data are mistaken for data collected on five different wafers and the five phantom wafers constitute a subgroup (i.e., a sample of five wafers). Develop the control limits for the resulting X and R charts. Then, plot on the two control charts the corresponding statistics for the 30 samples. Then, check if any of the plotted points fall outside of any of the control limits of the two control chart. If so, explain what led to the control limit violations. \begin{tabular}{|r|r|r|r|r|r|} \hline Wafer & & & \\ No. & Location 1 Location & Location Location & Location 5 \\ \hline 1 & 96.5855 & 99.0047 & 99.3356 & 98.5455 & 101.046 \\ \hline 2 & 93.6057 & 96.4309 & 97.6695 & 94.3587 & 95.9539 \\ \hline 3 & 101.292 & 98.9796 & 99.8819 & 99.7914 & 102.443 \\ \hline 4 & 105.275 & 106.65 & 106.908 & 106.897 & 105.786 \\ \hline 5 & 104.769 & 103.411 & 102.769 & 103.624 & 104.482 \\ \hline 6 & 107.062 & 108.042 & 108.289 & 107.799 & 106.942 \\ \hline 7 & 90.851 & 90.9024 & 90.7924 & 90.0288 & 89.7801 \\ \hline 8 & 99.5218 & 98.9051 & 99.7057 & 99.2351 & 100.41 \\ \hline 9 & 105.18 & 105.117 & 104.372 & 103.759 & 104.903 \\ \hline 10 & 96.7768 & 95.3192 & 94.4455 & 96.6312 & 94.9506 \\ \hline 11 & 97.2189 & 96.6892 & 98.4487 & 97.6956 & 97.0172 \\ \hline 12 & 94.5062 & 94.2456 & 94.2731 & 93.6386 & 92.3837 \\ \hline 13 & 93.8553 & 93.3219 & 90.8222 & 93.4917 & 92.5314 \\ \hline 14 & 96.6683 & 95.6325 & 97.1423 & 96.0118 & 96.8718 \\ \hline 15 & 96.9994 & 95.9136 & 96.1003 & 95.4273 & 97.3308 \\ \hline 16 & 90.8979 & 90.2487 & 92.4448 & 90.0731 & 90.9655 \\ \hline 17 & 97.9941 & 98.1072 & 97.4737 & 97.451 & 97.757 \\ \hline 18 & 99.4631 & 97.5518 & 100.059 & 96.5825 & 97.7768 \\ \hline 19 & 99.5149 & 99.3406 & 101.362 & 100.311 & 98.8575 \\ \hline 20 & 99.6528 & 97.6207 & 98.5683 & 98.3439 & 98.0731 \\ \hline 21 & 97.7124 & 97.9809 & 97.2035 & 98.2449 & 97.2147 \\ \hline 22 & 98.49 & 97.6076 & 97.2321 & 98.0882 & 97.8046 \\ \hline 23 & 105.343 & 107.642 & 106.178 & 106.205 & 104.046 \\ \hline 24 & 98.2599 & 99.7155 & 99.4302 & 99.1887 & 100.363 \\ \hline 25 & 100.063 & 99.4202 & 98.914 & 100.51 & 97.877 \\ \hline 26 & 100.053 & 95.9032 & 98.8694 & 98.8882 & 98.2486 \\ \hline 27 & 107.136 & 108.562 & 108.587 & 108.265 & 108.698 \\ \hline 28 & 104.215 & 103.536 & 104.265 & 101.041 & 102.374 \\ \hline 29 & 109.626 & 110.901 & 109.436 & 110.792 & 109.74 \\ \hline 30 & 96.9631 & 98.5765 & 97.8298 & 96.6577 & 97.7155 \\ \hline \end{tabular}