3 (35 Points): (3-D Charts, sample size=1) Sputter deposition is a physical vapor deposition (PVD) method of depositing thin films by sputtering. It is important that the thickness of the thin film is uniform across all locations of a piece. Therefore, the issue is not just the average thickness of a piece (theoretically obtainable as the average thickness across the infinite many locations on the piece), but also the within-piece variability of the thickness. Suppose that the actual locations of thickness measurement are of no significance, and what matters is the variability among the measurements taken at the different locations. Suppose also that pieces are processed one by one individually, not as a lot or batch. Part (a): (20 Points) The management has decided to use the (X, RP,M, Rw) charts, i.e., the piece-to-piece (or time) individual X chart, piece-to-piece moving range and within- piece range charts, with a moving window of size 2. Usually, 25 or 30 samples are required for determining the control chart limits. For the purpose of this exam, we use only 4 pieces, as four samples. The thickness is measured at 9 different locations on a piece. The data collected are summarized in the following table. Note that, for each individual piece, the average thickness of the 9 measurements and the within-piece range of thickness have been calculated for you and are given in column Aver and column Rw, respectively. Piece L1 L2 L3 L4 L5 L6 L7 L8 L9 Aver Rw 1 113 113 113 112 113 113 113 113 114 113 2 2 101 100 99 102 99 100 100 101 98 100 4 3 108 107 106 105 105 107 107 108 110 107 5 4 99 102 100 101 100 99 99 100 100 100 3 Determine the center line and the upper control chart limits for the RP,M and Rwcharts. (There is no need to calculate the lower control limits for them. There is no need to calculate the limits for the X chart.) Part (b): (10 Points) The management decided to use an S chart to monitor the within- piece variability. The within-piece S has been calculated for each of the four pieces (using the same data, of course), and the result is summarized in the following table. Piece L1 L2 L3 L4 L5 L6 1 113 113 113 112 113 2 101 100 99 3 108 107 106 4 99 102 99 105 105 102 100 101 100 113 100 107 99 L7 L8 L9 Aver S 113 113 114 113 0.50 100 101 98 100 1.22 107 108 110 107 1.58 99 100 100 100 1.00 What is the Upper Control Limit and the Centerline of the within-piece S chart, i.e., the Sw chart. (There is no need to calculate the lower limits for them.) Part (c): (5 Point) In this problem, what would be a good reason to use this Sw chart instead of the Rw chart? Explain in up to two sentences