GEORGETOWN UNIVERSITY SCHOOL OF BUSINESS ADMINISTRATION STATISTICAL QUALITY CONTROL. Professor José-Luis Guerrero-Cusumano

GEORGETOWN UNIVERSITY SCHOOL OF BUSINESS ADMINISTRATION STATISTICAL QUALITY CONTROL Professor José-Luis Guerrero-Cusumano guerrerj@gunet.georgetown.edu Office: Telephone; 202-687.4338 Meeting Times: At CBN 203 Thursday 4:15pm-6:45pm Office Hours: Tuesday and Thursday from 3:00 p.m. -4:15 p.m. 1

STATISTICAL QUALITY CONTROL The goal of most companies is to conduct business in such a manner that an acceptable rate of return is obtained by the shareholders. A quality control system approach is concerned with the improvement of quality for products and services, this approach is continually evaluated and modified to meet the changing needs of the customer. Therefore it creates mechanism to rapidly modified product, or process design, manufacture, and service to meet customer requirements so that company remains competitive. In Industry, quality improvement efforts have quantitative methods as their foundation. Quantitative methods are used to understand the behavior of processes. Being able to utilize statistics to make decisions, change processes and improve company performance requires an understanding of advanced useful statistical tools. LIST OF TOPICS I.PHILOSOPHY AND FUNDAMENTALS. 1.Introduction to Quality Control and the Total Quality System. 2.Some Philosophies and Their Impact on Quality. 3.Quality Management Practices, Tools, and Standards. II. QUALITY FUNCTION DEPLOYMENT 4. The Measurement of Correlations 5. Houses of Quality III. STATISTICAL FOUNDATIONS AND METHODS OF QUALITY IMPROVEMENT. 6.Fundamentals of Statistical Concepts and Techniques in Quality Control and improvement. 7.Graphical Methods of Data Presentation and Quality Improvement. IV. STATISTICAL PROCESS CONTROL. 8.Statistical Process Control Using Control Charts. 9.Control Charts for Variables. 10.Control Charts for Attributes 11.Process Capability Analysis. The 6 Sigma Challenge. 12. Advanced Univariate Control Charts. V: MULTIVARIATE QUALITY CONTROL 2

13. The Multivariate Normal Distribution 14. Chi-square and Hotelling T-square distributions VI. ELEMENTS OF DESIGN OF EXPERIMENT AND ANALYSIS OF VARIANCE Reference Materials for This Class I) Materials Required: Course Packet for MGMT-576-10 PowerPoint Lecture Notes II) Computer notes located at the following directory s:\guerrero-statistical quality control Namely: 1) The Syllabus 2) PowerPoint Presentations Textbooks Recommended: Layth C. Alwan Statistical Process Control. Mc. Graw Hill,2000, ISBN 0-256-11939-2. Farnum Nicholas, Modern Statistical Quality Control and Improvement, Duxbury, 1994, ISBN 0-534-20304-3. McClave, J., Benson, P.G. and Sincich, T. Statistics for Business and Economics, 7th edition, Prentice-Hall. Mitra Amitawa, Fundamental of Quality Control and Improvement, 2 nd ed.,1998, Prentice Hall, ISBN 0-13-645086-5. Summers Donna, Quality, 2 nd Ed., Prentice Hall, 2000, ISBN 0-13-099924-5. Name of the Case: Classification Your Due Date Work Exercises for Group February 23 rd CHAPTER CONTROL CHART CONCEPTS VARIABLES CONTROL CHARTS ATTRIBUTES CONTROL CHARTS CAPABILITY INDEXES Work Quality Function Deployment in the EXERCISE-CASE Group February 23 rd 3

Automobile Industry Work Multivariate Quality Control Final Research Project Group Work February 29 th The final research paper should have: 1) An extensive bibliographical search about the topic 2) The most relevant qualitative aspects of the problem 3) Real life examples. 4) Conclusions and Recommendations 5) Power point presentations, Excel and other computer output should be used and developed. GRADES DIVISIONS: Class Participation 15% Group Projects (Cases and Exercises) 60% Exam (Individual) 25%. Date Topic Comments January 13 th PHILOSOPHY AND FUNDAMENTALS January 20 th QUALITY FUNCTION DEPLOYMENT-. STATISTICAL FOUNDATIONS AND METHODS OF QUALITY IMPROVEMENT January 27 th February 3 rd CONTROL CHART FOR VARIABLES- CAPABILITY ANALYSIS- THE 6 SIGMA APPROACH CONTROL CHARTS FOR ATTRIBUTES ADDITIONAL UNIVARIATE CONTROL CHARTS. February MULTIVARIATE CONTROL OF PROCESSES 10 th Pearson and Spearman Correlation Coefficients Binomial-Poisson-Normal Distribution THIS CLASS WILL BE RESCHEDULE February ELEMENTS OF DESIGN OF EXPERIMENT AND 17 th ANALYSIS OF VARIANCE. Multivariate Normal Distribution. 4

EXERCISES FOR CHAPTER CONTROL CHART CONCEPTS CCC.1 CCC.2 CCC.3 CCC.4 CCC.5 CCC.6 Are rational subgroups also random samples? Explain why or why not. For a stable process, one wants all the points in a control chart to fall within the control limits. However, one does not want all the points on the chart to fall close to, or exactly on, the centerline. Why? Suppose that for subgroups of size 50 the proportion nonconforming in each subgroup is charted. Furthermore, suppose the process is in control with an overall average nonconformance rate of = 0.05. (a) If shifts to 0.10, what is the probability that this shift is detected in the next subgroup after the shift occurs? That is, what is the probability that the next falls outside the control limits? (b) Generalize the result in part (a) to shift in of any size, δ. In Exercise CCC.3, what is the average number of subgroups inspected before an `out of control' signal is given? From an chart, when a process shifts off target by 2 sigmas, the probability of detecting this shift at the next subgroup can be shown to be p d = PP(z > 3-2/n). (a) Generalize this result to a shift of k sigmas. (b) Using the result in part (a), plot the average run length (ARL) versus k for samples of size n = 1, 5, 10, and 20. How frequently should subgroup data be collected for a control chart? List several factors involved in this decision and give supporting examples. CCC.7 The theoretical control limits for an chart are µ " 3σ / /n. For a given process (i.e., for a fixed µ and σ), these control limits become narrower as the subgroup size increases. Does this mean that the `out of control' signals will become more likely for control charts based on larger subgroup sizes? CCC.8 CCC.9 Three-sigma limits are used with most control charts. In terms of the probability of detecting special causes, (a) describe the effect of widening the limits, say, to 3.1-sigma limits; (b) describe the effect of narrowing the control limits using, say, 2-sigma limits. Show that ARL = 1/p d. (Hint: Let Y = number of subgroups until the first `out of control' signal is given; find the distribution of Y; then find the expected value of Y.) CCC.10 Suppose that a measuring instrument used to obtain process data for an and R chart is out of calibration, so that each of its reported measurements is off by +δ units from the true value. What effect will this have on the signals given by the and R charts? CCC.11 A canning machine has 15 different heads, each of which affixes lids to cans passing by on a conveyor system. Two methods have been proposed for collecting subgroup data from this process: (1) taking periodic samples of five cans from the finished cans exiting the machine, and (2) taking periodic samples of 15 cans exiting the machine. Compare these two methods. If one or more of the heads is out of adjustment and causing inadequate sealing, which of the two methods is more likely to detect such a problem? CCC.12 When two `out of control' rules (with false alarm rates α 1 and α 2 ) are applied simultaneously, the overall false alarm rate is given by 1 - (1 - α 1 )(1 - α 2 ). Generalize this statement to the case where any number of rules, k, are used together (assume independence). 5

CCC.13 An chart appears as follows: (a) (b) Explain what is wrong with such a pattern. Give at least two reasons or scenarios for how such a pattern might arise in practice. CCC.14 Solder joints that connect electronic components to printed circuit boards are examined and classified as either acceptable or not acceptable by an inspector. The inspector then records the number of unacceptable joints per board. (a) If all the circuit boards are of the same type, what kind of control chart do you recommend for this process? (b) When different types of circuit boards are inspected, both the number of unacceptable joints and the total number of joints are recorded for every circuit board. For this type of data, what kind of control chart would you recommend? CCC.15 Insurance companies publish standard tables of average heights and weights of Americans. Weight ranges show weights without clothes for people aged 25-29, while heights are measured without shoes. For example, one such table suggests that healthy men who are 5 ft, 11 in. tall with a medium frame should weight between 152 and 165 lb. For a man 5 ft., 11 in. tall, are these limits best thought of as specification limits, process limits, or control limits? CCC.16 Refer to the list of rules for special causes (a) For independent measurements from a normal distribution, calculate the probability of the event in rule 2 (eight successive points on the same side of the centerline). (b) Under the same assumptions as in part (a), find the probability associated with rule 5 (four out of five points on the same side of the centerline, all four at a distance exceeding 1 sigma from the centerline). CCC.17 The list of `out of control' rules cannot be used on standardized control charts. 1 To show this, develop a list of seven (or more) subgroup means and their associated subgroup sizes that has the following property: the unstandardized means show no particular pattern, but the standardized means form an increasing sequence. 1 See Nelson (1989). 6

EXERCISES FOR CHAPTER VARIABLES CONTROL CHARTS (VCC) VCC.1 For a given process, the control limits of the chart become closer together as the subgroup size n is increased (i.e., the A 2 factor shrinks as n increases; see table of Constants). Does this mean that charts based on large subgroup sizes are more likely to have some subgroups that fall outside their control limits than charts based on smaller subgroup sizes? VCC.2 Instead of constructing and R charts for 30 subgroups of size 4, a friend suggests the alternative of simply finding the averages of all 30 subgroups and then creating an individuals chart for these averages. Explain what is wrong with this procedure. (Note: Do not solve problem VCC.3) VCC.3 An R chart is used to monitor the combined output of six identical machines. For the last 25 samples of size 5, the R chart appears as follows: (a) (b) Does the chart indicate that the process is in statistical control? Explain what could be happening to cause the R chart to have this form. VCC.4 Prior to shipment, thrust washers supplied to the automotive industry go through a five-step process of sanding, stripping, punching, and baking. The important quality characteristics of each washer are its thickness, inside diameter, and outside diameter. 2 The following data represent 25 subgroups of five washers, with each subgroup sampled from a different production lot: Lot # x 1 x 2 x 3 x 4 x 5 1 0.0767 0.0771 0.0774 0.0768 0.0776 2 0.0771 0.0771 0.0776 0.0774 0.0776 3 0.0773 0.0773 0.0772 0.0776 0.0772 4 0.0772 0.0776 0.0779 0.0770 0.0778 5 0.0769 0.0777 0.0775 0.0772 0.0775 6 0.0767 0.0772 0.0773 0.0774 0.0772 7 0.0772 0.0776 0.0773 0.0775 0.0766 8 0.0775 0.0773 0.0770 0.0769 0.0771 9 0.0774 0.0772 0.0773 0.0775 0.0770 10 0.0774 0.0773 0.0777 0.0772 0.0776 11 0.0770 0.0774 0.0774 0.0773 0.0772 12 0.0780 0.0775 0.0767 0.0773 0.0775 13 0.0764 0.0775 0.0776 0.0774 0.0777 14 0.0781 0.0772 0.0772 0.0773 0.0775 15 0.0775 0.0772 0.0776 0.0774 0.0772 16 0.0773 0.0769 0.0776 0.0773 0.0769 17 0.0770 0.0772 0.0775 0.0773 0.0775 2 See Chaudhry and Higbie (1990). 7

18 0.0773 0.0774 0.0778 0.0771 0.0774 19 0.0773 0.0774 0.0774 0.0774 0.0773 20 0.0776 0.0770 0.0771 0.0777 0.0779 21 0.0779 0.0768 0.0769 0.0771 0.0770 22 0.0780 0.0771 0.0776 0.0774 0.0779 23 0.0769 0.0771 0.0773 0.0771 0.0769 24 0.0773 0.0771 0.0780 0.0773 0.0773 25 0.0772 0.0777 0.0773 0.0767 0.0773 (a) (b) (c) (d) Construct and R charts for these data. Do the charts constructed in part (a) indicate that the washer stamping process is in statistical control? Using the centerline from the R chart, obtain an estimate of the standard deviation of washer thickness. Suppose the specification limits for washer thickness are 0.0755 and 0.0795 in. Using the result in part (a), approximately what proportion of the washers has thicknesses that exceed one of the specification limits? VCC.5 Galvanized coatings on pipes protect them from rust. A certain coating process for large pipes calls for an average coating weight of 200 lb. per pipe. 3 The lower specification limit is 180 lb. per pipe, but there is no upper specification, since extra coating material only provides more protection for the pipe. The following data show the coating weights of 30 pipes sampled at a rate of one per shift (read across). 216 202 208 208 212 202 193 208 206 206 206 213 204 204 204 218 204 198 207 218 204 212 212 205 203 196 216 200 215 202 (a) (b) (c) Construct an individual chart for these data Does the chart in part (a) indicate that the coating process is in statistical control? From the chart in part (a), how is the process performing from the point of view of the customer? How does the producer view these results? VCC.6 For the data of Exercise VCC.5, construct a CUSUM chart using h = 5 sigma-hat, k = 0.5 sigma-hat and w = 2 sigma-hat (estimate sigma-hat from the data). Interpret the resulting chart. VCC.7 For the data of Exercise VCC.5, construct a EWMA chart using a parameter of lambda = 0.20. Interpret the resulting chart. 3 See Weaver (1990). 8

VCC.8 Certain manufactured parts are required to have a length of 0.254. Twenty subgroups of three parts each were used to form and R charts for the part lengths. To simplify the data-gathering process, the measurements were reported as deviations from the nominal length in units of 0.001 (e.g., a recorded value of -3 refers to a measured length of 0.251). In this format, the data on the 20 subgroups are given here. Subgroup number d 1 d 2 d 3 1 4 0-2 2-1 -3-1 3-2 4 2 4-2 -2 1 5 0-2 2 6-1 0 2 7-3 3 3 8-2 -3 1 9-3 1 3 10 3 1 1 11-1 3 0 12-2 -1 4 13 4-1 3 14-3 3 2 15 2 0 3 16-3 1-1 17-2 2 1 18-3 2-1 19-1 -2 0 20 1-2 -1 Construct and R charts for these data. From the extended list of `out of control rules' in Section 6.4, are there any indications that this process is not in control? VCC.9 Construct a CUSUM chart for the data of Exercise VCC.8. Use h = 5 sigma-hat, k = 0.5, sigma-hat and w = 2 sigma-hat (estimate sigma-hat from the range chart). Interpret the resulting chart and compare these results to those of Exercise VCC.8. VCC.10 Construct an EWMA chart for the data of Exercise VCC.8. Use a parameter of lambda = 0.30. Compare the resulting chart to those in Exercises VCC.8 and VCC.9. 9

VCC.11 VCC.12 VCC.13 VCC.14 VCC.15 VCC.16 In an EWMA chart, explain the effect on the chart of using a value of lambda that is very close to 1. What is the effect of using a ë that is close to 0? Explain why all `out of control' rules do not apply to DUSUM or EWMA chart. What is the minimum value of the process coefficient of variation that results in a positive lower control limit, LCL of an x-bar chart? (Use R-bar /d 2 to estimate the process standard deviation and use to estimate the process average.) A drilling tool that machines metal parts eventually wears out and periodically must be replaced. If the hole diameters drilled by this machine are monitored on a control chart, describe the type of pattern you would expect to see in the points plotted on the chart. Process data that do not closely follow a normal distribution must sometimes be transformed so that they appear normal. One popular transformation used for positive data is to take logarithms of the original data. For a given set of positive measurements, suppose that two charts are constructed, one from the raw data and one from the logarithms (any base) of the raw data. If a point falls beyond the 3-sigma control limits on the chart for the logarithms, must the corresponding points fall outside the control limits on the chart of the raw data? In a process that produces molded plastic containers, hourly samples of size 3 were used to create control charts for a critical dimension. For the most recent 30 samples, the measurements were: Hour Measurements 1.36.39.36 2.33.35.30 3.51.41.42 4.42.37.34 5.39.39.38 6.33.41.45 7.43.39.41 8.41.32.32 9.37.42.36 10.26.42.32 11.36.32.36 12.38.47.35 10

13.29.45.39 14.44.38.43 15.38.37.37 16.31.43.38 17.39.49.35 18.43.36.38 19.40.45.32 20.40.40.32 (a) (b) Construct a range chart for this data. Is there any `out of control' condition indicated on the chart? Construct an x-bar chart for this data and check for any signs of special causes. VCC.17 Each hour a 3-ft length is cut from a continuous extruded sheets of plastic. The weights of these cross sections are used to monitor the uniformity of the extrusion process. The weights (in pounds) of the last 20 cross sections are: Hour Weight (lb) 1 169 2 164 3 169 4 178 5 178 6 183 7 181 8 195 9 184 10 179 11 216 12 170 13 168 14 182 15 177 16 164 17 182 18 148 19 176 20 162 11

Construct an individuals chart for this data. Are there any signs of the presence of special causes? VCC.18 Sand is an important component in a process that produces molds for cylinder blocks (Krishnamuoorthi 1990). Foundry workers have determined that the compactibility of the sand is of key importance in making goods molds. Compactibility is measured as the percent reduction in volume in a fixed amount of sand after being compacted with a standard force. Because testing sand samples is time consuming, an individuals chart is used to monitor the compactibility. The following data represent compactibility measurements (in percent) from 30 successive samples taken from the molding process (read across and down): 44 39 49 41 38 44 40 43 40 41 33 31 30 46 45 48 45 42 40 45 44 41 49 39 41 40 48 42 36 39 (a) (b) Construct an individuals chart for this data. Does the chart indicate that there are any problems with the sand compactibility? VCC.19 CUSUM charts have many applications in chemical industries, in which numerous chemical characteristics must be maintained close to specified target levels. To ensure the chemical purity of a commercial organic chemical, measurements of the level of a certain intermediate chemical material are taken every 4 hr. Data from 22 samples appear below: Sample number Level 1 15.3 2 15.7 3 14.4 4 14.0 5 15.2 6 15.8 7 16.7 8 16.6 9 15.9 10 17.4 11 15.7 12 15.9 13 14.7 14 15.2 15 14.6 12

16 13.7 17 12.9 18 13.2 19 14.1 20 14.2 21 13.8 22 14.6 (a) (b) (c) Given that the target chemical level is 15 and that the process standard deviation is known to be about 1, construct a CUSUM chart for this data. Use k = 0.5 and h = 5.0ó. Construct an individuals chart of the data and compare its performance to the CUSUM chart in part (a). Using a parameter value of lambda =.20, construct an EWMA chart of this data and compare its performance to the CUSUM chart in part (a). 13

EXERCISES FOR ATTRIBUTES CONTROL CHARTS (ACC) ACC.1 The following table contains the number of accidents on the work site across 40 operating divisions of a certain company: 4 Number of Accidents Number of Accidents Unit March- June July- October November- February Unit March- June July- October November- February 1 2 1 2 21 3 4 1 2 1 3 3 22 0 3 1 3 2 4 0 23 1 4 1 4 1 2 4 24 3 1 2 5 1 3 1 25 1 4 4 6 1 1 1 26 1 0 0 7 4 8 8 27 1 0 0 8 0 0 0 28 1 0 0 9 2 1 2 29 0 0 0 10 1 0 2 30 0 0 0 11 3 2 0 31 0 0 12 12 2 6 3 32 1 0 1 13 0 3 1 33 2 3 2 14 0 0 0 34 0 0 0 15 1 0 1 35 0 2 3 16 2 2 4 36 0 0 0 17 0 3 2 37 0 0 0 18 0 0 3 38 0 0 0 19 2 0 4 39 0 1 0 20 2 6 7 40 1 1 1 Construct an np chart for these data. 4 See Bicking (1991). 14

ACC.2 Off-color flaws in aspirins are caused by extremely small amounts of iron that change color when wet aspirin material comes into contact with the sides of drying containers. 5 The flaws are not harmful but are nonetheless unattractive to consumers. At one Dow Chemical plant, out of every batch of aspirin, a 250-lb sample is taken and the number of off-color flaws is counted. The following table shows the number of flaws per 250-lb sample obtained over a 25-day period: Sample number Number of flaws Sample number Number of flaws 1 46 14 49 2 51 15 48 3 56 16 59 4 57 17 53 5 37 18 61 6 51 19 63 7 47 20 42 8 34 21 45 9 30 22 43 10 44 23 42 11 47 24 39 12 51 25 38 13 46 Construct an appropriate control chart for these data and examine it for any evidence of lack of statistical control. ACC.3 Under what conditions on is the lower control limit of a c chart positive? ACC.4 Explain the difference in the actions taken on a process when a point on a p chart exceeds the upper control limit versus the actions taken when a point falls below the lower control limit. ACC.5 Under what circumstances does the adage "variables data are stronger than attributes data" apply? ACC.6 For a fixed subgroup size, n, what values of lead to a positive lower control limit on a p chart? 5 See Bemowski (1988). 15

ACC.7 For a fixed value of, how large does the subgroup need to be to yield a positive lower control limit on a p chart? ACC.8 To establish the subgroup size n for a p chart, a probability of 0.90 is specified for finding at least one nonconforming item in any sample of n items. If the process has an average nonconformance rate of about 4%, what subgroup size should be used? ACC.9 After passing through a painting operation, individual items are inspected for surface flaws, which are counted and recorded on a c chart. The painting process is currently in statistical control, with an overall average of 0.5 flaw per item inspected. Approximately what proportion of the inspected items will have two or more surface flaws? ACC.10 The following data show the number of fabric flaws, c i, and the number of square feet of material inspected, n i, for 30 samples of material from a continuous roll of fabric: i c ni i ci ni i ci n 1 12 3.9 11 0 8.4 21 21 5.2 2 18 9.0 12 14 6.8 22 6 5.6 3 27 6.7 13 9 4.4 23 16 8.0 4 64 9.2 14 16 5.2 24 27 8.9 5 11 3.6 15 0 7.8 25 21 5.3 6 13 6.7 16 29 9.8 26 12 3.1 7 25 8.3 17 18 8.8 27 19 6.2 8 22 5.6 18 28 7.1 28 14 4.8 9 43 6.1 19 10 3.3 29 42 8.3 10 17 4.2 20 47 5.9 30 19 4.7 (a) From these data, construct a control chart for the number of flaws per square foot of fabric. 16

(b) Interpret the chart in part (a). ACC.11 The following data show the number of nonconforming items per lot for 30 lots, each of size 50 (read across): 4 3 0 2 2 2 0 1 1 0 3 2 1 1 0 0 2 4 2 5 0 0 1 1 0 3 2 1 2 4 (a) Construct a control chart for the proportion of nonconforming items per lot. (b) Interpret the chart in part (a). ACC.12 Forty consecutive automobile dashboards are examined for signs of pinholes in the plastic molding. The numbers of pinholes found are listed here (read across): 6 2 3 2 5 2 2 3 2 4 9 4 0 5 0 6 5 4 2 3 3 1 4 1 7 3 3 5 7 3 6 7 6 4 5 3 8 5 4 3 (a) Construct a control chart for the number of pinholes per dashboard. (b) Interpret the chart in part (a). 17

GUIDELINES FOR USING CAPABILITY INDEXES Much of the popularity of capability indexes is undoubtedly due to their ease of calculation and apparent ease of interpretation. As we have see, though, their proper interpretation is subject to many conditions. The following list summarizes these conditions and may serve as a checklist of a capability analysis: 1. Statistical control Before any capability analysis can be performed, the process must be brought into a state of statistical control. Control is established with a variables control chart. 2. Measurement error The magnitude of the errors produced by the measurement system should be ascertained, since the error variation imposes limits on the achievable values of the capability indexes. 3. Distribution types The exact form of the distribution of process measurements can never be exactly known. If is necessary to assume that some distribution (such as the normal) governs the process variability. Some effort should be dedicated to making a sound choice for the underlying distribution. a. Skewed distributions often arise when one-sided specification limits are used (Gunter 1989). The normal distribution does not apply in such situations. b. Some processes involve automated sorting, which may produce truncated distributions that underestimate the true process variation. The normal distribution again is not appropriate in this situation (Gunter 1989). 4. Sampling variation Even for controlled processes, the C p index is a statistic and is therefore expected to exhibit natural sampling variability from sample to sample. The amount of this variation can be estimated (Kane 1986). 5. Software If capability analyses are done with SPC software, then it is important to know which method is used to estimate the process variation, ó. Methods based on control chart centerlines ( and ) may give very different results than those based on the standard deviation of the combined data set. 18

EXERCISES FOR USING CAPABILITY INDEXES (CI) CI.1 CI.2 CI.3 CI.4 Why must a process be in statistical control before its capability is measured? A process has a C p of 1.2 and is centered on its nominal value. What proportion of the specification limits are used by the process's measurements? Explain how it is possible for all the measurements in a given sample to be within the specification limits, while the same data yield a nonzero estimate of the proportion of the process that exceeds specifications. Suppose a measuring measurement is very precise but has an unknown offset, delta, that is, each reading from the instrument is exactly ä units higher (or lower if delta is negative) than the true value. What effect does this have on the C p and C pk indexes generated from such measurements. CI.5 A computer printout shows that a certain process has a C p of 1.6 and a C pk of 0.9. Assuming that the process is in control, what do these indexes say about the capability of the process? CI.6 A process with specification limits of 5 " 0.01 has a C p of 1.2 and a C pk of 1.0. What is the estimated process average from which these indexes are derived? CI.7 Show algebraically that C pk can never exceed C p. CI.8 CI.9 For a certain process, and R control charts based on subgroups of size 5 have centerlines of 14.5 and 1.163, respectively. Given that the process has specification limits of 12 and 16, calculate C p, C pu, C pl, and C pk. Show that the C p index of a process that is known to follow an exponential distribution with mean ì is given by C p = (U - L)/6ì. CI.10 (a) (Difficult) Assuming that process measurements, X, follow a normal distribution, show that the following equation always holds for processes with a two-sided tolerance: proportion out of specification = P(z larger or equal than 3C pk ) + P(z larger or equal than 6C p - 3C pk ) (b) (c) What properties of the normal distribution are needed to prove this statement? Does the equation hold for any other distributions? Suppose, for a given process, that C p and C pk are estimated to be 1.4 and 0.90, respectively. Assuming that the process follows a normal distribution, estimate the proportion of the measurements that will be out of specification (Report your answer in parts per million). 19

(d) For a centered process with C p = 2.0, what proportion of the measurements will be out of specification? CI.11 Suppose that a measuring instrument has a rated accuracy of "1%; that is, readings are within 1% of the true dimension being measured. When this instrument is used to obtain data on a given process, what is the largest (i.e., best) value that the C p index could attain? State any assumptions made in obtaining this estimate. 20