6 First, solve the problems without using Minitab, and then develop also Minitab solution (where appropriate). Submit both. 1. A design engineer would like to determine the optimum location for the mean value of an important quality characteristic X which is normally distributed with variance o = 1. The lower specification limit is LSL = 2, and the upper specification limit is USL = 6. The per unit production cost is $5. An item that meets specifications is sold for $15. If the X-value of an item is above USL, the item is re-worked at an additional cost of $3 to meet the specifications and such an item is sold for $13. If the X-value of an item is below LSL, the item must be scrapped. a) Derive a formula and draw a graph of the expected net profit function per unit produced. Find the optimum location of the mean of u*on the graph (submit Minitab plot). b) Using u in part a, calculate the optimum expected profit. 2. Samples of n items each are taken from a manufacturing process at regular intervals. Suppose the process mean is o =200 and the standard deviation is o =10. a) To control future production, what is the required sample size for the X chart with probability limits (a = 0.002) to detect the shift in the process mean from Mo-200 to -209 on the first or second sample following the shift with the probability 0.5? 3. Printed circuit boards are required to have a length of 0.25 in. Twenty subgroups of four boards each were used to obtain the data, which are given in the table below: d d d3 d4 Subgroup number 1 2 3 4 5 6 7 8 9 0.258 0.253 0.252 0.252 0.254 0.253 0.251 0.252 0.251 0.254 0.251 0.258 0.252 0.252 0.254 0.257 0.251 0.255 0.252 0.253 0.256 0.255 0.256 0.256 0.257 0.255 0.257 0.245 0.248 0.262 0.249 0.24 0.247 0.256 0.254 0.25 10 11 12 13 14 15 16 17 18 19 20 0.257 0.253 0.252 0.258 0.251 0.256 0.251 0.252 0.251 0.253 0.255 0.255 0.257 0.253 0.253 0.257 0.254 0.255 0.256 0.256 0.252 0.252 0.255 0.254 0.258 0.257 0.256 0.257 0.253 0.255 0.253 0.254 0.253 0.248 0.252 0.243 0.251 0.244 0.23 0.255 0.254 0.252 0.25 0.251 a) Set up X and R charts with 3 sigma limits. Is the process in control? If necessary, revise the control limits. Estimate the in-control process mean and standard deviation (o and ). (submit Minitab plot) b) If the specification limits are 0.255 and 0.0.245, find the proportion of defectives when the process is in control and also when the process mean is at the target = T. 4. Four ball bearings are sampled from a production process every hour for ten hours and the data are shown below: Hour 1 2 3 4 5 6 7 8 9 10 49.71 50.07 49.95 50.11 49.79 49.88 50.11 50.07 49.94 50.22 50.11 49.86 49.83 49.53 50.1 49.79 50.08 50.22 50.21 49.82 Data 49.63 49.91 49.8 50.39 50.54 49.97 50.01 50.21 50.03 50.24 49.95 49.92 50.08 49.94 49.99 50.2 50.1 50.15 49.97 49.77 a) Set up X and S charts with probability limits a = 0.002. Is the process in statistical control? If necessary, revise the control limits. Estimate the process mean and the standard deviation. b) To control future production, design an X chart with 3 sigma limits satisfying the following requirements: when the process mean shifts to = o + 1.47 o in., we want the ARL