# The diameter of the casting is an important quality characteristic. A coordinate measuring machine is used to measure the diameter of each casting at five different locations. Data for 20 castings are shown below

Problem 1:

The diameter of the casting is an important quality characteristic. A coordinate measuring machine is used to measure the diameter of each casting at five different locations. Data for 20 castings are shown below. Perform a process capability study using x-bar and R charts. The specification limits are: Lower specification limit: 11.7; Upper specification limit: 11.8

 Measurements Casting 1 2 3 4 5 1 11.7629 11.7403 11.7511 11.7474 11.7374 2 11.8122 11.7506 11.7787 11.7736 11.8412 3 11.7742 11.7114 11.753 11.7532 11.7773 4 11.7833 11.7311 11.7777 11.8108 11.7804 5 11.7134 11.687 11.7305 11.7419 11.6642 6 11.7925 11.7611 11.7588 11.7012 11.7611 7 11.6916 11.7205 11.6958 11.744 11.7062 8 11.7109 11.7832 11.7496 11.7496 11.7318 9 11.7984 11.8887 11.7729 11.8485 11.8416 10 11.7914 11.7613 11.7356 11.7628 11.707 11 11.726 11.7329 11.7424 11.7645 11.7571 12 11.7202 11.7537 11.7328 11.7582 11.7265 13 11.8356 11.7971 11.8023 11.7802 11.7903 14 11.7069 11.7112 11.7492 11.7329 11.7289 15 11.7116 11.7978 11.7982 11.7429 11.7154 16 11.7165 11.7284 11.7571 11.7597 11.7317 17 11.8022 11.8127 11.7864 11.7917 11.8167 18 11.7775 11.7372 11.7241 11.7773 11.7543 19 11.7753 11.787 11.7574 11.762 11.7673 20 11.7572 11.7626 11.7523 11.7395 11.7884

a) Set up the x-bar and r-charts for this process, assuming the measurements on each casting form a rational subgroup. Show the xbar and r-charts for the stable process. Ensure you have a stable process.

b) Calculate the Cp and Cpk indices based on the stable process.

c) Do your capability analysis using Minitab and show Minitab output.

d) Is the process capable?

e) Is the process centered?

Problem 2:

A paper mill uses a control chart to monitor the imperfections in finished rolls of paper. Production output is inspected for twenty days, and the resulting data are shown below. Set up a control chart for nonconformities per roll of paper. Remove any out of control points, and re-calculate the control limits and center line. Provide the final control chart center line and limits below. Show your equations

 Day Number of Rolls Produced Total Number of Imperfections Day Number of Rolls Produced Total Number of Imperfections 1 18 12 11 18 18 2 18 14 12 18 14 3 24 20 13 19 8 4 22 18 14 20 10 5 22 15 15 20 14 6 22 12 16 20 13 7 20 11 17 24 16 8 20 13 18 24 18 9 20 12 19 22 20 10 20 10 20 21 17

Problem 3:

The number of nonconforming switches in samples of size 150 is shown below. Create an np-chart with the following data and then remove the out of control points and provide the revised control charts. What are the final center lines, upper and lower control limits? Show your equations

 Sample 1 8 2 1 3 3 4 0 5 2 6 4 7 0 8 1 9 10 10 6 11 6 12 0 13 4 14 0 15 3 16 1 17 15 18 2 19 3 20 0

Problem 4

Bath concentrations are measured hourly in a chemical process. Data (in ppm) for the last 32 hours are shown in the table below. (Read down from left)

The process target uo=175 ppm

a. Construct a tabular Cusum for this process using h=5 and k=1/2

b. Provide your interpretation based on the chart

 Sample# Conc. Sample# Conc. Sample# Conc. Sample# Conc. 1 160 9 186 17 190 25 206 2 158 10 195 18 189 26 210 3 150 11 179 19 185 27 216 4 151 12 184 20 182 28 212 5 153 13 175 21 181 29 211 6 154 14 192 22 180 30 202 7 158 15 186 23 183 31 205 8 162 16 197 24 186 32 197

Problem 5:

The following data are temperature readings from a chemical process in degrees Celsius, taken every two minutes. Read the observations down from the left. The target value for the mean is 950.

a) Set up an EWMA control chart with lambda = 0.2 for this process. Show the EWMA chart.

b) Interpret the results

 953 985 949 937 959 948 958 952 945 972 941 946 939 937 955 931 972 955 960 954 948 955 947 928 945 950 966 935 958 927 941 937 975 948 934 941 963 940 938 950 970 957 937 933 973 962 945 970 957 940 946 960 949 963 963 933 973 930 952 968 942 943 967 960 940 965 935 959 965 950 969 934 936