Photoresists in semiconductor manufacturing. Photoresists are light-sensitive polymeric resins used in the manufacturing of a variety of products, including printing plates, printed circuit boards, flat-panel liquid crystal displays, magnetic recording heads, and integrated circuit devices. An important quality variable is the flow resistance of the photoresist. A process for monitoring the flow resistance of the photoresist during the hard-bake step in the manufacture of semiconductor wafers was explored in Quality and Reliability Engineering International (April 2020). In phase 1 of the analysis (where the manufacturer believes the process is in control), five wafers were selected each day for 25 consecutive days during the hard-bake step and the flow resistance (in microns) measured for each. These data are displayed in the first table below. Data for Exercise 13.22 (Phase I: Process In Control) Day Flow Resistance Measurements 1 1.3235 1.4128 1.6744 1.4573 1.6914 2 1.4314 1.3592 1.6075 1.4666 1.6109 3 1.4284 1.4871 1.4932 1.4324 1.5674 4 1.5028 1.6352 1.3841 1.2831 1.5507 5 1.5604 1.2735 1.5265 1.4363 1.6441 6 1.5955 1.5451 1.3574 1.3281 1.4198 7 1.6274 1.5064 1.8366 1.4177 1.5144 8 1.4190 1.4303 1.6637 1.6067 1.5519 9 1.3884 1.7277 1.5355 1.5176 1.3688 10 1.4039 1.6697 1.5089 1.4627 1.5220 11 1.4158 1.7667 1.4278 1.5928 1.4181 12 1.5821 1.3355 1.5777 1.3908 1.7559 13 1.2856 1.4106 1.4447 1.6398 1.1928 14 1.4951 1.4036 1.5893 1.6458 1.4969 15 1.3589 1.2863 1.5996 1.2497 1.5471 16 1.5747 1.5301 1.5171 1.1839 1.8662 17 1.3680 1.7269 1.3957 1.5014 1.4449 18 1.4163 1.3864 1.3057 1.6210 1.5573 19 1.5796 1.4185 1.6541 1.5116 1.7247 20 1.7106 1.4412 1.2361 1.3820 1.7601 21 1.4371 1.5051 1.3485 1.5670 1.4880 22 1.4738 1.5936 1.6583 1.4973 1.4720 23 1.5917 1.4333 1.5551 1.5295 1.6866 24 1.6399 1.5243 1.5705 1.5563 1.5530 25 1.5797 1.3663 1.6240 1.3732 1.6887 Source: Hu, X., et al. “Conditional median run length performance of the synthetic X chart with unknown process parameters,” Quality and Reliability Engineering International, Vol. 36, No. 3, April 2020 (Table 11).

a. Find the center line, UCL and LCL of an x-chart for monitoring the flow resistance.

b. Construct a control chart for the mean flow resistance (i.e., an x-chart).

c. Verify that the process is in control.

d. For phase 2 of the analysis, data were collected for 30 additional days. (These data are shown in the second table.) Plot the means for these new data on the control chart of part

b. Is the process still in control? Data for Exercise 13.22 (Phase II: New Data) Day Flow Resistance Measurements 1 1.5355 1.3560 1.4248 1.4066 1.4845 2 1.3943 1.3660 1.3652 1.3120 1.3454 3 1.4853 1.6622 1.5949 1.5672 1.4172 4 1.6494 1.3485 1.3658 1.4224 1.4438 5 1.4880 1.2845 1.4007 1.7353 1.3979 6 1.3145 1.5022 1.6025 1.6966 1.5955 7 1.3414 1.7716 1.5354 1.4946 1.8508 8 1.4445 1.4738 1.6045 1.4276 1.2715 9 1.4412 1.5162 1.5457 1.6428 1.4893 10 1.3401 1.3293 1.4099 1.2980 1.2536 11 1.4971 1.7211 1.3929 1.5711 1.6654 12 1.6600 1.4887 1.5895 1.3696 1.6815 13 1.4323 1.5495 1.3693 1.4100 1.2206 14 1.4393 1.3272 1.6480 1.3692 1.7616 15 1.1398 1.5981 1.5886 1.5563 1.3073 16 1.3538 1.5284 1.5132 1.3533 1.5040 17 1.2610 1.2966 1.4705 1.3105 1.6348 18 1.4121 1.4760 1.5248 1.5529 1.4921 19 1.5488 1.4024 1.9990 1.4065 1.8143 20 1.6985 1.4864 1.4531 1.5242 1.8829 21 1.7398 1.4324 1.2365 1.4172 1.4907 22 1.4525 1.4415 1.7573 1.5104 1.1839 23 1.4220 1.3151 1.5615 1.0853 1.4153 24 1.5926 1.4215 1.3728 1.5399 1.3428 25 1.5435 1.5850 1.8626 1.7480 1.3553 26 1.5848 1.7317 1.5526 1.6259 1.3008 27 1.5867 1.2959 1.5141 1.6066 1.4817 28 1.2480 1.8266 1.3533 1.4340 1.7860 29 1.5511 1.6989 1.3103 1.4971 1.3626 30 1.4563 1.4953 1.2047 1.5066 1.4279 13.23 What characteristic of a process is an R-chart designed to monitor? 13.24 In practice, x@ and R-charts are used together to monitor a process. However, the R-chart should be interpreted before the x@chart. Why?