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Consider the following example in which 20 samples of size 4 were taken from the production line at regular time intervals. Each item in the sample was measured and the results recorded in the table below: Sample Measures within the Sample 1 2.81 2.79 2.78 2.80 2 2.81 2.79 2.80 2.83 3 2.79 2.77 2.80 2.81 4 2.81 2.78 2.79 2.81 5 2.82 2.79 2.80 2.78 6 2.78 2.79 2.82 2.80 7 2.80 2.80 2.81 2.79 8 2.79 2.80 2.80 2.82 9 2.78 2.78 2.81 2.80 10 2.81 2.79 2.80 2.81 11 2.78 2.82 2.79 2.80 12 2.80 2.82 2.81 2.79 13 2.78 2.79 2.78 2.81 14 2.81 2.82 2.80 2.79 15 2.78 2.82 2.81 2.79 16 2.79 2.80 2.79 2.82 17 2.80 2.81 2.80 2.78 18 2.80 2.80 2.79 2.82 19 2.78 2.82 2.79 2.79 20 2.77 2.81 2.82 2.80

The above data is available on Canvas in the Excel format. Answer the following questions. 1. In Excel, calculate the average of each sample. What is the grand mean?

2. Calculate the range of each sample. What is the average range? 3. What is the sample size? 4. what are the upper and lower control limits of the X-bar chart? 5. What are the upper and lower control limits of the R chart? 6. Set up statistical process control charts for X-bar and R. The X-bar chart will have three horizontal lines as follows: the center line is set at the grand mean, the upper line corresponds to the upper control limit and the lower line to the lower control limit. The R chart also has three horizontal lines defined in the same manner.

After the control chart was established, daily sampling continued as follows: Time Measurements in each Sample 8:00 2.80 2.81 2.81 2.79 8:30 2.81 2.79 2.80 2.79 9:00 2.78 2.82 2.79 2.81 9:30 2.78 2.79 2.81 2.80 10:00 2.80 2.81 2.80 2.80 10:30 2.80 2.81 2.80 2.79

7. Plot each sample on the X-bar and R charts

8. Is the operation in control?