1. The tensile strength and diameter of a textile fiber are two important quality characteristics that are to be jointly controlled. The quality engineer has decided to use n=5 fiber specimens in each sample. 20 preliminary samples have been taken. Construct a control chart to monitor this process. Assume there is only mean shift in strength and diameter and there is no variance change. Please set up a T2 control chart to monitor the strength and diameter. (a= 0.01) What are the Phase I control limits and control limits for future monitoring? Data Format • X1: tensile strength • X2: diameter • 1st column: sample mean of X1 • 2nd column: sample mean of X2 • 3rd column: sample variance of X1 • 4th column: sample variance of X2 • 5th column: sample variance between X1 and X2 2. Consider aT2 control chart for monitoringp= 6 quality characteristics. Suppose that the subgroup size isn= 3 and there are 30 preliminary samples available to estimate the sample covariance matrix. (a) Find the phase II control limits assuming thata=0.005 . (b) Compare the control limits from part (a) to the chi-square control limit. What is the magnitude of the difference in the two control limits? (c) How many preliminary samples would have to be taken to ensure that the exact phase II control limit is within 1% of the chi-square control limit? (Hint: calculate the phase II control limit for differentmand find “m” such that the exact phase II control limit is smaller than 101% of the chi-square control limit) 3. a)The data shown here come from a production process with two observable quality characteristics,x1andx2. The data are sample means of each quality characteristic, based on samples of sizen= 25. Assume that mean values of the quality characteristics and the covariance matrix were computed from 50 preliminary samples: Construct aT2 control chart using these data. Use the phase II limits. Sample number X1 X2 1 58 32 2 60 33 3 50 27 4 54 31 5 63 38 6 53 30 7 42 20 8 55 31 9 46 25 10 50 29 11 49 27 12 57 30 13 58 33 14 75 45 15 55 27 b) Suppose that the sample mean vector and sample covariance matrix provided here the actual population parameters. What control limit would be appropriate for the control chart? Apply this limit to the data and discuss any differences in results that you find in comparison to the original choice of control limit.
