Problem: Five samples of n = 5 boxes of snacks were collected from a machine that fills boxes of snacks. We would like to construct an chart to monitor the performance of the machine. To develop the chart, the weight of five consecutive boxes of snacks, five times each day, are sampled. Based on the following five samples,

Sample Weight of snack boxes Total

1 23.95, 23.98, 24.11, 23.90, 24.10 120.04

2 24.01, 24.07, 23.93, 24.09, 23.98 120.08

3 24.14, 24.07, 24.08, 23.98, 24.02 120.29

4 23.91, 24.04, 23.89, 24.01, 23.95 119.80

5 24.03, 24.04, 24.01, 23.98, 24.10 120.16

1. What is the number of samples and what is the sample size?
   Which number is used to find a control chart constant such as A ₂ .
2. Calculate the centerline for constructing an x-bar chart.
3. Calculate the upper/lower control limits of the x-bar chart.

4. Graph the x-bar chart.

5. Calculate the centerline for constructing an R chart.

6. Calculate the upper and lower control limits of R chart.

7. Graph the R-chart.

8. Using Excel re-answer parts 2, 3, 4, 5 and 6 above.

9. Based on both x-bar and R-charts is the process in control? Why or why not?

II. A Tire co. periodically tests its tires for tread wear under simulated road conditions. To study and control the manufacturing process, 20 samples of 3 radial tires, were chosen from different shifts over several days of operation, with the following results. Develop R and Xbar charts. Is the process in control?

III . The following are quality control data for a manufacturing process at a chemical co. The data show the temperature in degrees of centigrade at five points in time during a manufacturing cycle. The co. is interested in using control charts to monitor the temperature of its manufacturing process. Construct the Xbar chart and R chart. What conclusions can be made about the quality of the process?

IV. An automotive industry supplier produces pistons for several models of automobiles. Twenty samples each consisting of 200 pistons, were selected when the process was known to be operating correctly. The number of defective pistons found in the samples, follow.

1. What is the estimate of the proportion defective for the piston manufacturing process in control?
2. Compute a p-chart for the manufacturing
   process, assuming each sample has 200 pistons.
3. With the results of part (2), what conclusion should be made if a sample of 200 has 20 defective pistons?
4. Compute the upper and lower control limits for an np-chart.
5. Answer part (3) using the results of part (4).

V. A local newspaper has 10 delivery boys who each deliver the morning paper to 50 customers every day. The owner decides to record the percentage of papers delivered on time for a 10-day period and to construct a p chart to see whether the percentage of papers delivered on time is too erratic. Based on the following table, answer the following questions:

Number of Papers Proportion of Papers

Day Delivered on Time Delivered on Time

1 458 91.6

2 447 89.4

3 464 92.8

4 450 90.0

5 432 86.4

6 484 96.8

7 457 91.4

8 494 98.8

9 476 95.2

10 468 93.6

4630 925.7

1. What is the numerical value of the center line for the p

chart?

1. 0.926 b) 0.911 c) 0.885 d) 0.500

1. What is the numerical value of the lower control limit for the p chart?
   1. 0.920 b) 0.81 c) 0.891 d) 0.798
2. What is the numerical value of the upper control limit for the p chart?
   1. 0.926 b) 0.961 c) 0.979 d) 1
3. Which expression best characterizes the p chart?
   1. Cycles b) Trend c) In-control d) Out-of-control

Multiple Choice Questions:

1. Variation due to the inherent variability in a system of operation is called
   1. special or assignable causes.

1. common or chance causes.
2. explained variation.
3. the standard deviation.

1. Which of the following is not one of Deming's 14 points?
   1. Belief in mass inspection.
   2. Create constancy of purpose for improvement of product or service.
   3. Adopt and institute leadership.
   4. Drive out fear.
2. The principal focus of the control chart is the attempt to separate special or assignable causes of variation from common causes of variation. What cause of variation can be reduced only by changing the system?
   1. Special or assignable causes
   2. Common causes
   3. Total causes
   4. None of the above
3. Once the control limits are set for a control chart, one attempts to
   1. discern patterns that might exist in values over time.
   2. determine whether any points fall outside the control limits.
   3. Both of the above.
   4. None of the above.
4. A process is said to be out of control if
   1. a point falls below the upper or above the lower control lines.
   2. A point falls within three standard deviations around the mean.
   3. Either of the above.
   4. None of the above.
5. Which famous statistician developed the 14 Points of Quality?
   1. Shewhart
   2. Deming
   3. Chebyshev
   4. Taguchi
6. If the lower control limit of a P chart is negative,
   1. A mistake has been made in the computations
   2. Use the absolute value of the lower limit
   3. It is set to zero
   4. It is set to one

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