Problem 3 (Total 16 points, 4 points for each subquestion) A manufacturing process has two 3D printing machines working independently to produce the same part with a cubic shape. Each part's quality is inspected by three dimensions L, W, and H. To understand how different machines affect the parts' shape (dimensional errors), an engineer suggests using PCA to analyze the correlation matrix of threedimensional measurement data (L,W, and H) of products made by each machine. The correlation matrix and the PCA analysis results are given below for each machine ( M1 and M2 ), respectively. M1=10.90.90.910.90.90.91;Eigenvector=0.57740.57740.577400.70710.70710.81650.40820.4082;Eigenvalue=2.80.10.1 M2=10.90.20.910.210.20.211;Eigenvector=0.70610.70800.00830.20550.19370.95930.67760.67910.2823;Eigenvalue=1.91.0850.0147 (1) Based on the PCA results, please help to interpret how many dominant variation patterns exist in each machine, which can cover 90% of the total variations for that machine (4 points, 2 points for each machine). (2) Based on the obtained first eigenvector of each machine, please discuss how the produced part is varied (shape or size) in each machine. Please provide your suggestion on how to adjust L,W, and H (adjust them in the same direction or not, and the relative magnate) to improve the part's dimensional quality for each machine. (4 points, 2 points for each machine) (3) For Machine 2, based on the PCA results, assuming the first PC feature (denoted as PC1 ) follows a normal distribution, what are the mean () and standard deviation ( ) of PC1 ? (2points) This distribution can be used to monitor PC1. Assume the control limits for monitoring PC1 with an upper control limit (UCL) and lower control limit (LCL) are calculated by UCL/LCL=z/2, please calculate the control limits based on =5% ( 2 points). (4) For Machine 2, assuming its covariance matrix is equal to its correlation matrix, and based on historical normal operation data, its produced parts' mean =101010. Now, if one produced part dimension (one sample) is x=9.910.69.9, what is its corresponding PC1 score (4 points)