STAT 495, Fall 2003 Homework Assignment #5 1. In a study performed by Ronald Beals of General Motors, the amount of chrome plating on intake valves was measured. One measure was the length (in millimeters) of chrome plating on the valve stem. Subgroup means for 50 subgroups, each of size n=4 are plotted on the two versions of the X chart and appear on the last pages of the homework. (a) From the control chart with only the center line, upper and lower control limits, identify those subgroups (by number) that fall outside the control limits. (b) From the control chart with only the center line, and and 1, 2 and 3 “sigma” limits use all 8 of Nelson’s rules and indicate whether or not there are alarms. If there are alarms, indicate at which subgroups the alarms sound. (c) The raw data for this problem are given on the website and WebCT. Use JMP to verify your answers in (a) and (b). 2. For the following assume we have a normal distribution and set up control limits based on the theoretical values of µ and σ. (a) Use the rule that sounds an alarm when a single point falls outside zone A (3-‘sigma’ limits). If the process shifts an amount of 0.5σ how long, on average, will it take an X chart to sound an alarm using subgroups of size n=4? n=5? n=10? n=20? (b) We wish to set up a control chart that will sound an alarm, on average, within 10 subgroups if the process shifts 0.65σ. What subgroup size should be used? (c) If we use a rule that sounds an alarm when a single point falls outside zone B (2-‘sigma’ limits) what is the ARL of an X chart for a process that is in control and centered on the process mean? (d) For the rule in (c), a single point outside zone B (2-‘sigma’ limits) how long, on average, will it take an X chart to sound an alarm when the process shifts 0.5σ. Use subgroups of size n=4, n=5, n=10 and n=20. (e) Compare your results in (d) to those in (a). What general statement can you make about the affect narrowing your control limits has on the ARL? (f) Suppose the process experiences a mule kick that does not affect the process mean (it remains at µ) but doubles the process standard deviation, (after the kick the process has standard deviation 2σ). What is the Average Run Length for the X̄ chart to sound an alarm for this special cause? (g) Will a larger subgroup size sound the alarm sooner for this special cause? Explain briefly. 3. A sheet metal company is a major supplier of mild steel sheet metal to a leading automobile manufacturer. In the process cast metal slabs at elevated temperatures are fed between cylindrical rollers. This flattens the slabs into sheets. The sheets are then cold rolled to the desired thickness. The quality characteristic for this study is thickness of the sheet. The product tolerances set by the purchaser are 2.2500 ± 0.008 mm. The thickness (in millimeters) of 32 sheets produced during a 4-hour period are given below. Hour 1 Sheet Thickness 1 2.233 2 2.240 3 2.247 4 2.252 5 2.250 6 2.251 7 2.252 8 2.251 Sheet 9 10 11 12 13 14 15 16 Hour 2 Thickness 2.234 2.240 2.251 2.246 2.252 2.255 2.255 2.250 Sheet 17 18 19 20 21 22 23 24 Hour 3 Thickness 2.234 2.242 2.250 2.249 2.249 2.252 2.255 2.248 Sheet 25 26 27 28 29 30 31 32 Hour 4 Thickness 2.232 2.239 2.251 2.244 2.253 2.251 2.252 2.251 (a) Construct a histogram for these data. Comment on the general shape of the distribution of sheet thickness. (b) Given that X = 2.247 and M R = 0.005742, calculate an estimate of the process standard deviation from the average moving range. Also, compute control limits for a moving range and an individuals chart. (c) Construct individuals and moving range charts. You can use JMP or Minitab or some other program to do this. (d) Comment on the appearance of each chart. Are there any points outside 3-‘sigma’ limits? (e) Note that the first two sheets of every hour are low in thickness. This can be attributed to the fact that scrap collected during the previous hour is fed into the furnace at the beginning of the next hour. The steel obtained during scrap reprocessing is more ductile and weaker in grain strength. Once identified this special cause could be removed. Recalculate control limits and rechart the data by excluding the first two observations for each hour (i.e. sheets 1, 2, 9, 10, 17, 18, 25 and 26 from the calculation of the moving ranges and X). (f) Comment on the appearance of each chart. Are there any points outside 3-‘sigma’ limits? (g) What can you say about the ability of the process to produce sheets within the stated tolerances?