What must be true about the variation of a process before

Problem 1E What is a control chart? Describe its use.

Explain why rational subgrouping should be used in constructing control charts.

When a control chart is first constructed, why are the centerline and control limits treated as trial values?

Even if all the points on an \(bar{x}\) - chart fall between the control limits, the process may be out of control. Explain. Text Transcription: bar{x}

Which process parameter is an \(\bar{x}\) used to monitor?

Problem 7E Use the six pattern-analysis rules described in Figure 13.22 to determine whether the process being monitored with the shown below is out of statistical control.

What must be true about the variation of a process before an \(\bar{x} \text {-chart }\) is used to monitor the mean of the process? Why?

Problem 89SE Bayfield Mud Company case. In their text Quantitative Analysis of Management (2005), B. Render (Rollins College) and R. M. Stair (Florida State University) present the case of the Bayfield Mud Company. Bayfield supplies boxcars of 50-pound bags of mud treating agents to the Wet-Land Drilling Company. Mud treating agents are used to control the pH and other chemical properties of the cone during oil drilling operations. Wet-Land has complained to Bayfield that its most recent shipment of bags were underweight by about 5%. (The use of underweight bags may result in poor chemical control during drilling, which may hurt drilling efficiency, resulting in serious economic consequences.) Afraid of losing a longtime customer, Bayfield immediately began investigating their production process. Management suspected that the causes of the problem were the recently added third shift and the fact that all three shifts were under pressure to increase output to meet increasing demand for the product. Their quality-control staff began randomly sampling and weighing six bags of output each hour. The average weight of each sample over the last 3 days is recorded in the table below along with the weight of the heaviest and lightest bags in each sample. Does it appear that management’s suspicion about the third shift is correct? Explain?

Consider the next \(\bar{x}\)-chart shown. a. Is the process affected by special causes of variation only, common causes of variation only, or both? Explain. b. The means for the next five samples in the process are 27, 29, 32, 36, and 34. Plot these points on an extended \(\bar{x}\)-chart. What does the pattern suggest about the process?

Use Table IX in Appendix D to find the value of \(A_2\) for each of the following sample sizes. a. n = 3 b. n = 10 c. n = 22

Twenty-five consecutive samples of size n = 5 were collected to construct an \(\bar{x} \text {-chart }\). The sample means and ranges for the data are shown in the next table. a. Calculate the mean of the sample means, \(\overline{\bar{x}}\), and the mean of the sample ranges, \(\bar{R}\). b. Calculate and plot the centerline and the upper and lower control limits for the \(\bar{x} \text {-chart }\). c. Calculate and plot the A, B, and C zone boundaries of the \(\bar{x} \text {-chart }\). d. Plot the 25 sample means on the \(\bar{x} \text {-chart }\) and use the six pattern-analysis rules to determine whether the process is under statistical control.

The data in the following table were collected for the purpose of constructing an \(\bar{x}\) chart. a. Calculate \(\bar{x}\) and R for each sample. b. Calculate \(\bar{x}\) and \(\bar{R}\). c. Calculate and plot the centerline and the upper and lower control limits for the \(\bar{x}\) chart. d. Calculate and plot the A, B, and C zone boundaries of the \(\bar{x}\) chart. e. Plot the 20 sample means on the \(\bar{x}\) chart. Is the process in control? Justify your answer.

Detecting gender-related employment disparities. In the Federal Reserve Bank of Atlanta, Working Paper Series (Oct. 2008), researchers presented a novel approach to identifying firms that discriminate against women. The researchers used the Equal Employment Opportunity (EEOC) Systematic Gender Disparity Scorecard as a measure of the degree to which a firm complies with EEOC guidelines for eliminating gender bias. (EEOC Scorecard values range from 0 to 1, with larger values indicating a greater gender disparity.) In a hypothetical example, a sample of firms was selected for each of 30 different types of firms in the service industry (e.g., food service, financial services, oil and gas service, etc.) The mean scorecard values for the 30 firms types are plotted in the \(\bar{x}\)-chart shown above. The centerline and upper and lower control limits are shown on the chart. a. Identify the rational subgroups used to construct the chart. b. Identify the key variable plotted on the chart. c. What are the approximate values of \(\bar{x}\), UCL, and LCL? d. What conclusions can you draw from the chart? Are there any firm types that should concern the EEOC? Why?

Quality control for irrigation data. Most farmers budget water by using an irrigation schedule. The success of the schedule hinges on collecting accurate data on evapotranspiration (ETo), a term that describes the sum of evaporation and plant transpiration. The California Irrigation Management Information System (CIMIS) collects daily weather data (e.g., air temperature, wind speed, and vapor pressure) used to estimate ETo and supplies this information to farmers. Researchers at CIMIS demonstrated the use of quality-control charts to monitor daily ETo measurements (IV International Symposium on Irrigation of Horticultural Crops, Dec. 31, 2004). Daily minimum air temperatures (\(^{\circ}C\)) collected hourly during the month of May at the Davis CIMIS station yielded the following summary statistics (where five measurements are collected each hour): \(\overline{\bar{x}}=10.16^{\circ} \text { and } R=14.87^{\circ}\). a. Use the information provided to find the lower and upper control limits for an \(\bar{x}\)-chart. b. Suppose that one day in May the mean air temperature at the Davis CIMIS station was recorded as \(\bar{x}=20.3^{\circ}\). How should the manager of the station respond to this observation?

Pain levels of ICU patients. Various interventions are available for nurses to help relieve patients’ pain (e.g., heat/cold applications, breathing exercises, massage). The journal Research in Nursing & Health (Vol. 35, 2012) demonstrated the utility of statistical process control in determining the effectiveness of a pain intervention. The researchers presented the following illustration. Pain levels (measured on a 100-point scale) were recorded for a sample of 10 intensive care unit (ICU) patients 24 hours post surgery each week for 20 consecutive weeks. The next table provides the means and ranges for each of the 20 weeks. To establish that the pain management process is “in control,” an \(\bar{x}\)-chart is constructed. a. Compute the value of the centerline for the \(\bar{x}\)-chart. b. Compute the value of \(\bar{R}\). c. Compute the UCL and LCL for the \(\bar{x}\)-chart. d. Plot the means for the 20 weeks on the \(\bar{x}\)-chart. Is the pain management process “in control”? e. After the 20th week, a pain intervention occurred in the ICU. The goal of the intervention was to reduce the average pain level of ICU patients. To determine if the intervention was effective, the sampling of ICU patients was continued for 8 more consecutive weeks. The mean pain levels of these patients were (in order): 71, 72, 69, 67, 66, 65, 64, and 62. Plot these means on the \(\bar{x}\)-chart. f. Apply pattern-analysis rules to the extended \(\bar{x}\)-chart, part e. Do you detect a shift in the mean pain level of the patients following the intervention? Explain.

Problem 15E CPU of a computer chip. The central processing unit (CPU) of a microcomputer is a computer chip containing millions of transistors. Connecting the transistors are slender circuit paths only .5 to .85 micrometer wide. A manufacturer of CPU chips knows that if the circuit paths are not .5–.85 micrometer wide, a variety of problems will arise in the chips’ performance. The manufacturer sampled four CPU chips six times a day (every 90 minutes from 8:00 a.m. until 4:30 p.m.) for 5 consecutive days and measured the circuit path widths. These data and Minitab were used to construct the shown below. a. Assuming that = .335, calculate the chart’s upper and lower control limits, the upper and lower A–B boundaries, and the upper and lower B–C boundaries. b. What does the chart suggest about the stability of the process used to put circuit paths on the CPU chip? Justify your answer. c. Should the control limits be used to monitor future process output? Explain.

Problem 16E Cereal box manufacturing. A machine at K-Company fills boxes with bran flakes cereal. The target weight for the filled boxes is 24 ounces. The company would like to use an to monitor the performance of the machine. To develop the control chart, the company decides to sample and weigh five boxes of cereal each day (at 8:00 and 11:00 a.m. and 2:00, 5:00, and 8:00 p.m.) for 20 consecutive days. The data are summarized in the SPSS printout below. a. Construct an from the summary statistics. b. What does the chart suggest about the stability of the filling process (whether the process is in or out of statistical control)? Justify your answer. c. Should the control limits be used to monitor future process output? Explain. d. Two shifts of workers run the filling operation. Each day the second shift takes over at 3:00 p.m. Will the rational subgrouping strategy used by K-Company facilitate or hinder the identification of process variation caused by differences in the two shifts? Explain.

Improving public health waiting times. Statistical process IIl control was utilized in an effort to improve the service wic1, delivery at a Women, Infants, and Children (WIC) health WIc2 clinic in Minnesota (American Journal of Public Health, Sep. 2009). The health department sampled patients for 10 consecutive days and recorded each patient's lobby waiting time (in minutes). The mean lobby waiting times were (in order): 25.5, 16.5,22.5, 18,13, 10.5, 14,4.5, 12.5, and 14 minutes. These means were plotted in an \(\bar{x}\)-chart to monitor the waiting time process. a. Compute the value of the centerline for the \(\bar{x}\)-chart. b. Critical boundaries for the \(\bar{x}\)-chart are listed as follows: UCL = 31.5, Upper A-B = 26, Upper B-C = 20.5, Lower B-C = 9.5, Lower A-B = 4, LCL = 0. Plot the 10 means in an \(\bar{x}\)-chart; include the critical boundaries on the chart. c. Refer to part b. Apply pattern-analysis rules to the chart. Do you detect the presence of any special causes of variation? d. After a 3-month time period, during which the clinic implemented new procedures, the health department conducted a second sampling of patients over 14 consecutive days. The mean lobby waiting times for these 14 days were (in order): 11.5, 12.5, 14.5, 11, 12.5,11.5, 13.5, 11.5, 7.5, 13.5,7, 10.5, 13.5, and 2 minutes. Critical \(\bar{x}\)-chart boundaries for the new data are listed as follows: UCL = 21.5, Upper A-B = 18, Upper B-C = 14.5, Lower B-C = 7.5, Lower A-B=4, LCL = .5. Construct an \(\bar{x}\)-chart for the new data. Do you detect the presence of any special causes of variation? e. Place the two \(\bar{x}\)-charts side by side. The health department concluded that a process shift occurred after the clinic implemented its new procedures. Do you agree?

Selecting the best wafer-slicing machine. Silicon wafer slicing is a critical step in the production of semiconductor devices (e.g., diodes, solar cells, transistors). Yuanpei (China) University researchers used control charts to aid in selecting the best silicon wafer-slicing machine (Computers & Industrial Engineering, Vol. 52, 2007). Samples of n = 2 wafers were sliced each hour for 67 consecutive hours and bow measurements (a measure of precision) were recorded. The resulting \(\bar{x}\)-chart for one of the machines tested revealed that the cutting process was out of control on the 19th, 40th, and 59th hours. For each of these 3 hours, the mean bow measurement fell above the upper control limit. Assume the mean bow measurements are normally distributed. a. If the process is in control, what is the probability that a mean bow measurement for a randomly selected hour will fall above the upper control limit? b. If the process is in control, what is the probability that 3 of 67 mean bow measurements fall above the upper control limit?

Military aircraft bolts. A precision parts manufacturer produces bolts for use in military aircraft. Ideally, the bolts should be 37 centimeters in length. The company sampled four consecutively produced bolts each hour on the hour for 25 consecutive hours and measured them using a computerized precision instrument. The data are presented in the table below. a. What process is the manufacturer interested in monitoring? b. Construct an \(\bar{x}\) - chart from the data. c. Does the chart suggest that special causes of variation are present? Justify your answer. d. Provide an example of a special cause of variation that could potentially affect this process. Do the same for a common cause of variation. e. Should the control limits be used to monitor future process output? Explain. Text Transcription: bar{x}

Detecting under-reported emissions. The Environmental Protection Agency (EPA) regulates the level of carbon dioxide (\(CO_2\)) emissions. Periodically these emissions measurements are under-reported due to leakage or faulty equipment. Such problems are often detected only by an expensive test (RATA) that is typically conducted only once per year. Just recently, the EPA began applying an automated control chart methodology to detect under-measurement of emissions data (EPRI CEM Users Group Conference, Nashville, TN, May 13, 2008). Each day, the EPA collects emissions data by measuring \(CO_2\) concentration for each of 6 randomly selected hours. The daily average \(CO_2\) levels for each of 30 days are shown in the table above. The EPA considers these values to truly represent emissions levels because the RATA test was recently performed and showed no problems with under-reporting. The lower and upper control limits for the averages were established as LCL = 12.26 and UCL = 13.76. a. Construct a control chart for the daily average \(CO_2\) levels. b. Based on the control chart, describe the behavior of the measurement process. c. The following average \(CO_2\) levels were determined for a later 10-day period: 12.7, 12.1, 12.0, 12.0, 11.8, 11.7, 11.6, 11.7, 11.8, 11.7. Make an inference about the potential under-reporting of the emissions data for this 10-day period. Daily Average CO2 Measurements for 30 Consecutive Days 1 2 3 4 5 6 7 8 9 10 11 12 13 14 12.9 13.2 13.4 13.3 13.1 13.2 13.1 13 12.5 12.5 12.7 12.8 12.7 12.9 16 17 18 19 20 21 22 23 24 25 26 27 28 29 12.9 12.8 12.7 13.2 13.2 13.3 13 13 13.2 13.2 13.4 13.1 13.3 13.4

Chunky data. BFI Consulting, a leading provider of statistical process control software and training in the United States, recently alerted its clients to problems with “chunky” data. In an April 2007 report, BPI Consulting identifies “chunky” data as data that results when the range between possible values of the variable of interest becomes too large. This typically occurs when the data are rounded. For example, a company monitor lug the time it takes shipments to arrive from a given supplier rounded off the data to the nearest day. To show the effect of chunky data on a control chart, BP1 Consulting considered a process with a quality characteristic that averages about 100. Data on the quality characteristic for a random sample of 3 observations collected each hour for 40 consecutive hours are given in the table above. (The data are saved in the CHUNKY file.) (Note: BPl Consulting cautions its clients that out-of- control data points in this example were actually due to the measurement process and not to an “out-of-control” process.) a. Show that the process is “in control,” according to Rule 1, by constructing an -chart for the data. b. Round each measurement in the data set to a whole number and then form an -chart for the rounded data. What do you observe?

What characteristic of a process is an R-chart designed to monitor?

In practice, \(\bar{x}\) - and R-charts are used together to monitor a process. However, the R-chart should be interpreted before the \(\bar{x}\)-chart. Why? Text Transcription: bar{x}

Use Table IX in Appendix D to find the values of \(D_3\) and \(D_4\) for each of the following sample sizes. a. n = 4 b. n = 12 c. n = 24

Construct and interpret an R-chart for the data in Exercise 13.10 (p. 13-29). a. Calculate and plot the upper control limit and, if appropriate, the lower control limit. b. Calculate and plot the A, B, and C zone boundaries on the R-chart. c. Plot the sample ranges on the R-chart and use pattern analysis Rules 1–4 of Figure 13.22 to determine whether the process is under statistical control.

Construct and interpret an R-chart for the data in Exercise 13.11 (p. 13-29). a. Calculate and plot the upper control limit and, if appropriate, the lower control limit. b. Calculate and plot the A, B, and C zone boundaries on the R-chart. c. Plot the sample ranges on the R-chart and determine whether the process is in control.

Construct and interpret an R-chart and an \(\bar{x}\)-chart from the sample data shown below. Remember to interpret the R-chart before the \(\bar{x}\)-chart.

Detecting gender-related employment disparities. Refer to the Federal Reserve Bank of Atlanta, Working Paper Series (October 2008) study of gender-related employment disparities. Recall that the researchers used the Equal Employment Opportunity (EEOC) Systematic Gender Disparity Scorecard as a measure of the degree to which a firm complies with EEOC guidelines for eliminating gender bias. An R-chart for the data obtained by sampling firms for each of 30 different service industry firm types is shown above. The centerline and upper and lower control limits are shown on the chart. a. What are the approximate values of \(\bar{R}\), UCL, and LCL? b. What conclusions can you draw from the chart? Are there any firm types that should concern the EEOC? Why?

Problem 29E Pain levels of ICU patients. Refer to the Research in Nursing & Health (Vol. 35, 2012) study of the effectiveness of a pain intervention, Exercise 13.13 (p. 13-30). Recall that pain levels (measured on a 100-point scale) were recorded for a sample of 10 ICU patients 24 hours post surgery each week for 20 consecutive weeks. The data are repeated in the accompanying table. Now, you want to check process variation using an R-chart a. Compute the centerline for the chart. b. Compute the UCL and LCL for the R-chart. c. Plot the ranges for the 20 weeks on the R-chart. Does the variation of the pain management process appear to be “in control”? d. Recall that after the 20th week, a pain intervention occurred in the ICU. The ranges of the pain levels for the samples of patients over the next 8 consecutive weeks were (in order): 22, 29, 16, 15, 23, 19, 30, and 32. Plot these ranges on the R-chart. e. Apply pattern-analysis rules to the extended R-chart, part d. What do you observe? Source: Based on “Statistical Process Control in Nursing Research” by Denise F. Polit and Wendy Chaboyer from RESEARCH IN NURSING AND HEALTH, February 2012, Volume 35(1).

Quality control for irrigation data. Refer to Exercise 13.14 (p. 13-30) and the monitoring of irrigation data by the CIMIS. Recall that daily minimum air temperatures \((^{\circ}C)\) collected hourly during the month of May at the Davis CIMIS station yielded the following summary statistics (where five measurements are collected each hour): \(\overline{\bar{x}}=10.16^{\circ} \text { and } \bar{R}=14.87^{\circ}\) a. Use the information provided to find the lower and upper control limits for an R-chart. b. Suppose that one day in May the air temperature at the Davis CIMIS station had a high of \(24.7^{\circ}\) and a low of \(2.2^{\circ}\). How should the manager of the station respond to this observation?

Problem 32E Cola bottle filling process. A soft-drink bottling company is interested in monitoring the amount of cola injected into 16-ounce bottles by a particular filling head. The process is entirely automated and operates 24 hours a day. At 6:00 a.m. and 6:00 p.m. each day, a new dispenser of carbon dioxide capable of producing 20,000 gallons of cola is hooked up to the filling machine. To monitor the process using control charts, the company decided to sample five consecutive bottles of cola each hour beginning at 6:15 a.m. (i.e., 6:15 a.m., 7:15 a.m., 8:15 a.m., etc.). The data for the first day are saved in the file. An SPSS descriptive statistics printout for the data is shown on the next page. a. Will the rational subgrouping strategy that was used enable the company to detect variation in fill caused by differences in the carbon dioxide dispensers? Explain. b. Construct an R-chart from the data. c. What does the R-chart indicate about the stability of the filling process during the time when the data were collected? Justify your answer. d. Should the control limit(s) be used to monitor future process output? Explain. e. Given your answer to part c, should an be constructed from the given data? Explain.

Problem 33E Lowering the thickness of an expensive blow-molded container. Quality (Mar. 2009) presented a problem that actually occurred at a plant that produces a high-volume, blow-molded container with multiple layers. One of the layers is very expensive to manufacture. The quality manager at the plant desires to lower the average thickness for the expensive layer of material and still meet specifications. To estimate the actual thickness for this layer, the manager measured the thickness for one container from each of two cavities every 2 hours for 2 consecutive days. The data (in millimeters) are shown in the following tables. SPSS output for Exercise 13.32 a. Construct an R-chart for the data. b. Construct an x-chart for the data. c. Based on the control charts in parts a and b, c/omment on the current behavior of the manufacturing process. As part of your answer, give an estimate of the true average thickness of the expensive layer.

CPU of a computer chip. Refer to Exercise 13.15 (p. 13-30), where the desired circuit path widths were .5 to .85 micro-meter. The manufacturer sampled four CPU chips six times a day (every 90 minutes from 8:00 a.m. until 4:30 p.m.) for 5 consecutive days. The path widths were measured and used to construct the accompanying Minitab R-chart. a. Calculate the chart’s upper and lower control limits. b. What does the R-chart suggest about the presence of special causes of variation during the time when the data were collected? c. Should the control limit(s) be used to monitor future process output? Explain. d. How many different R values are plotted on the control chart? Notice how most of the R values fall along three horizontal lines. What could cause such a pattern?

What characteristic of a process is a p-chart designed to monitor?

Precision of scale weight measurements. The Journal of Quality Technology (July 1998) published an article examining the effects of the precision of measurement on the R-chart. The authors presented data from a British nutrition company that fills containers labeled “500 grams” with a powdered dietary supplement. Once every 15 minutes, five containers are sampled from the filling process, and the fill weight is measured. The table to the right lists the measurements for 25 consecutive samples made with a scale that is accurate to .5 gram, followed by the table below, which gives measurements for the same samples made with a scale that is accurate to only 2.5 grams. Throughout the time period over which the samples were drawn, it is known that the filling process was in statistical control with mean 500 grams and standard deviation 1 gram. a. Construct an R-chart for the data that is accurate to .5 gram. Is the process under statistical control? Explain. b. Given your answer to part a, is it appropriate to construct an \(\bar{x}\)-chart for the data? Explain. c. Construct an R-chart for the data that is accurate to only 2.5 grams. What does it suggest about the stability of the filling process? d. Based on your answers to parts a and c, discuss the importance of the accuracy of measurement instruments in evaluating the stability of production processes.

Problem 38E In each of the following cases, use the sample size formula to determine a sample size large enough to avoid constructing a p-chart with a negative lower control limit. a. p 0 = .01 b. p0 = .05 c. p0 = .10 d. p0 = .20

Replacement times for lost ATM cards. In an effort to reduce customer dissatisfaction with delays in replacing lost automated teller machine (ATM) cards, some retail banks monitor the time required to replace a lost ATM card. Called replacement cycle time, it is the elapsed time from when the customer contacts the bank about the loss until the customer receives a new card (Management Science, Sept. 1999). A particular retail bank monitors replacement cycle time for the first five requests each week for replacement cards. Variation in cycle times is monitored using an R-chart. Data for 20 weeks are presented below. a. Construct an R-chart for these data. b. What does the R-chart suggest about the presence of special causes of variation in the process? c. Should the control limits of your R-chart be used to monitor future replacement cycle times? Explain. d. Given your conclusion in part b and the pattern displayed on the R-chart, discuss the possible future impact on the performance of the bank.

Chunky data. Refer to Exercise 13.21 (p. 13-32) and the hourly data collected by BPI consulting. Recall that the data are saved in the file. a . Construct an R-chart for the data. Is the process variation in control? b. Round each measurement in the data set to a whole number, like in Exercise 13.21b. Form an R-chart for the rounded data. What do you observe?

Rental car call center study. A worldwide rental car company receives about 10,000 calls per month at its European call center. These calls typically involve customer issues with the level of service or the billing/invoice process. In an effort to reduce the proportion of issues that are not resolved on the customer’s first call, management conducted a thorough study of the call center’s procedures. The results were published in the International Journal of Productivity and Performance Management (Vol. 59, 2010). After making major changes at the call center, management constructed a p-chart to monitor the process improvements. Assume that 18 calls to the center were sampled each day for 60 consecutive days. The article reported that the proportion of all calls in the sample that had unresolved issues at the end of the call was .107. (This was a major improvement over the previous unresolved-first-call rate of .845.) a. What is the centerline for the p-chart? b. Compute the lower and upper control limits for the p-chart. c. When the proportions of daily calls that resulted in unresolved issues are plotted on the p-chart, all fall within the LCL and UCL boundaries. What does this imply about the process?

The proportion of defective items generated by a manufacturing process is believed to be 8%. In constructing a p-chart for the process, determine how large the sample size should be to avoid ending up with a negative lower control limit.

To construct a p-chart, 20 samples of size 150 were drawn from a process. The proportion of defective items found in each of the samples is listed in the next table. a. Calculate and plot the centerline and the upper and lower control limits for the p-chart. b. Calculate and plot the A, B, and C zone boundaries on the p-chart. c. Plot the sample proportions on the p-chart. d. Is the process under control? Explain. e. Should the control limits and centerline of part a be used to monitor future process output? Explain.

To construct a p-chart for a manufacturing process, 25 samples of size 200 were drawn from the process. The number of defectives in each sample is listed in time order in the table that follows. a. Calculate the proportion defective in each sample. b. Calculate and plot \(\bar{p}\) and the upper and lower control limits for the p-chart. c. Calculate and plot the A, B, and C zone boundaries on the p-chart. d. Plot the sample proportions on the p-chart and connect them with straight lines. e. Use pattern-analysis Rules 1–4 for detecting the presence of special causes of variation (Figure 13.22) to determine whether the process is out of control.

Monitoring surgery complications. An article on the use of control charts for monitoring the proportion of postoperative complications at a large hospital was published in the International Journal for Quality in Health Care (Oct. 2010). A random sample of surgical procedures was selected each month for 30 consecutive months, and the number of procedures with postoperative complications was recorded. The data are listed in the accompanying table. a. Identify the attribute of interest to the hospital. b. What are the rational subgroups for this study? c. Find the value of \(\bar{p}\) for use in a p-chart. d. Compute the proportion of post-op complications in each month. e. Compute the critical boundaries for the p-chart (i.e., UCL, LCL, Upper A–B boundary, etc.). f. Construct a p-chart for the data. g. Interpret the chart. Does the process appear to be in control? Explain.

Defective micron chips. A manufacturer produces micron chips for personal computers. From past experience, the production manager believes that 1% of the chips are defective. The company collected a sample of the first 1,000 chips manufactured after 4:00 p.m. every other day for a month. The chips were analyzed for defects; then these data and Minitab were used to construct the p-chart shown here. a. From a statistical perspective, is a sample size of 1,000 adequate for constructing the p-chart? Explain. b. Calculate the chart’s upper and lower control limits. c. What does the p-chart suggest about the presence of special causes during the time when the data were collected? d. Critique the rational subgrouping strategy used by the disk manufacturer.

Leaky process pumps. Quality (Feb. 2008) presented a problem that actually occurred at a company that produces process pumps for a variety of industries. The company recently introduced a new pump model and immediately began receiving customer complaints about “leaky pumps.” There were no complaints about the old pump model. For each of the first 13 weeks of production of the new pump, quality-control inspectors tested 500 randomly selected pumps for leaks. The results of the leak tests are summarized by week in the accompanying table. Construct an appropriate control chart for the data. What does the chart indicate about the stability of the process?

Quality of rewritable CDs. A Japanese compact disc (CD) manufacturer has a daily production rate of about 20,000 CD-RW (rewritable disks). Quality is monitored by randomly sampling 200 finished CDs every other hour from the production process and testing them for defects. If one or more defects are discovered, the CD is considered defective and is destroyed. The production process operates 20 hours per day, 7 days a week. The table below reports data for the last 3 days of production a. Construct a p-chart for the CD-RW production process. b. What does it indicate about the stability of the process? Explain. c. What advice can you give the manufacturer to assist them in their search for the special cause(s) of variation that is plaguing the process?

Explain why it is inappropriate to conduct a capability analysis study for a process that is not in statistical control.

Problem 50E Explain the difference between process spread and specification spread.

Problem 45E Monitoring newspaper typesetters. Accurate typesetting is crucial to the production of high-quality newspapers. The editor of the Morristown Daily Tribune, a weekly publication with a circulation of 27,000, has instituted a process for monitoring the performance of typesetters. Each week 100 paragraphs of the paper are randomly sampled and read for accuracy. The number of paragraphs with errors is recorded in the following table for each of the last 30 weeks. a. Construct a p-chart for the process. b. Is the process under statistical control? Explain. c. Should the control limits of part a be used to monitor future process output? Explain. d. Suggest two methods that could be used to facilitate the diagnosis of causes of process variation.

Problem 48E Rubber company tire tests. Goodstone Tire & Rubber Company is interested in monitoring the proportion of defective tires generated by the production process at its Akron, Ohio, production plant. The company’s chief engineer believes that the proportion is about 7%. Because the tires are destroyed during the testing process, the company would like to keep the number of tires tested to a minimum. However, the engineer would also like to use a p-chart with a positive lower control limit. A positive lower control limit makes it possible to determine when the process has generated an unusually small proportion of defectives. Such an occurrence is good news and would signal the engineer to look for causes of the superior performance. That information can be used to improve the production process. Using the sample size formula, the chief engineer recommended that the company randomly sample and test 120 tires from each day’s production. To date, 20 samples have been taken. The data are presented below. Sample Sample Size Defectives 1 120 11 2 120 5 3 120 4 4 120 8 5 120 10 6 120 13 7 120 9 8 120 8 9 120 10 10 120 11 11 120 10 12 120 12 13 120 8 14 120 6 15 120 10 16 120 5 17 120 10 18 120 10 19 120 3 20 120 8 a. Use the sample size formula to show how the chief engineer arrived at the recommended sample size of 120. b. Construct a p-chart for the tire production process. c. What does the chart indicate about the stability of the process? Explain. d. Is it appropriate to use the control limits to monitor future process output? Explain. e. Is the p-chart you constructed in part b capable of signaling hour-to-hour changes in p? Explain.

Describe two different ways to assess the capability of a process.

Find (or estimate) the process spread for each of the following. a. \(\sigma=21\) b. \(\sigma=5.2\) c. s = 110.06 d. s = .0024

Problem 53E For a process that is in control and follows a normal distribution, interpret each of the following Cp values: a. 1.00 b. 1.33 c. .50 d. 2.00

Why is it recommended to use and interpret \(C_p\) in conjunction with a capability analysis diagram rather than in isolation?

Find the specification spread for each of the following: a. USL = 19.65, LSL = 12.45 b. USL = .0010, LSL = .0008 c. USL = 1.43, LSL = 1.27 d. USL = 490, LSL = 486

Upper specification limit of a process. An in-control, centered process that follows a normal distribution has a \(C_p = 2.0\). How many standard deviations away from the process mean is the upper specification limit?

Find the value of \(C_p\) for each of the following situations: a. USL = 1.0065, LSL = 1.0035, s = .0005 b. USL = 22, LSL = 21, s = .2 c. USL = 875, LSL = 870, s = .75

Capability of an in-control process. A process is in control with a normally distributed output distribution with mean 1,000 and standard deviation 100. The USL and LSL for the process are 1,020 and 980, respectively. a. Assuming no changes in the behavior of the process, what percentage of the output will be unacceptable? b. Find and interpret the \(C_p\) value of the process.

Water use at a thermal power plant. Thermal power plants use demineralized (DM) water for steam generation. Since it is costly to replace, power plants must conserve the use of DM water. DM water consumption was monitored at a thermal power plant in India and the results published in Total Quality Management (Feb. 2009). Plant management set the target for DM water consumption at .5%, the upper specification limit at .7%, and the lower specification limit at .1%. Based on data collected for a sample of 182 flow meter measurements, the overall standard deviation of the process was .265%. Use this information to find the capability index for this process. Interpret the result.

Bioreactor production of antibodies. Benchtop bioreactors are used to produce antibodies for anti-cancer drugs. Engineers calibrate bioreactors in order to maximize production. The African Journal of Biotechnology (Dec. 2011) published a study designed to achieve a high percentage of antibody production from a bioreactor. The variable of interest was the natural logarithm of the number of viable cells produced in a bioreactor run. Data were collected for a sample of four bioreactor runs every 6 hours for 20 consecutive time periods. These data (simulated from information provided in the article) are listed in the accompanying table. Engineers have specified the following for the bioreactor runs: target mean = 6.3, LSL = 5.9, and USL = 6.5. Run a complete capability analysis on the data. How would you categorize the performance of the process?

Problem 60E Cereal box filling process. Refer to the data on weights of cereal boxes, Exercise 13.16 (p. 13-31). Assume the specification limits for the weights are USL = 24.2 ounces and LSL = 23.8 ounces. a. Assuming the process is under control, construct a capability analysis diagram for the process. b. Is the process capable? Support your answer with a numerical measure of capability.

Military aircraft bolts. Refer to Exercise 13.20 (p. 13-32) and the data on lengths of bolts used in military aircraft. Management has specified the USL and LSL as 37 cm and 35 cm, respectively. a. Assuming the process is in control, construct a capability analysis diagram for the process. b. Find the percentage of bolts that fall outside the specification limits. c. Find the capability index, \(C_p\). d. Is the process capable? Explain.

New iron-making process. Mining Engineering (Oct. 2004) published a study of a new technology for producing high-quality iron nuggets directly from raw iron ore and coal. For one phase of the study, the percentage change in the carbon content of the produced nuggets was measured at 4-hour intervals for 33 consecutive intervals. The data for the 33 time intervals are listed in the table below. Specifications state that the carbon content should be within \(3.42 \pm 0.3\) percent. a. Construct a capability analysis diagram for the iron-making process. b. Determine the proportion of carbon measurements that fall outside specifications. c. Find the capability index for the process and interpret its value.

Lowering the thickness of an expensive blow-molded container. Refer to the Quality (Mar. 2009) study of a plant produces a high-volume, blow-molded container, Exercise 13.33 (p. 13-40). Recall that the quality manager at the plant wants to lower the average thickness for the expensive layer of material and still meet specifications. Specification limits for individual thickness values are .10 to .30 millimeter. a. Find the standard deviation of the process data. b. Calculate the capability index, \(C_p\), for the process and interpret the result. c. Compare the LCL of the process (from Exercise 13.33) to the LSL. Does this imply that the average thick-ness of the material can be lowered and still meet specifications?

Define quality and list its important dimensions.

What is a process? Give an example of an organizational process.

What is a system? Give an example of a system with which you are familiar and describe its inputs, outputs, and transformation process.

Describe the six major sources of process variation.

Suppose all the output of a process over the last year was measured and found to be within the specification limits required by customers of the process. Should you worry about whether the process is in statistical control? Explain.

Problem 70SE Select a problem, event, or condition whose cause or causes you would like to diagnose. Construct a cause-andeffect diagram that would facilitate your diagnosis.

In estimating a population mean \(\mu\) using a sample mean \(\bar{x}\), why is it likely that \(\bar{x}\ \neq\ \mu\)? Construct a cause-and-effect diagram for the effect \(\bar{x}\ \neq\ \mu\).

Construct a cause-and-effect diagram to help explain why customer waiting time at the drive-in window of a fastfood restaurant is variable.

Problem 74SE Compare and contrast special and common causes of variation.

Explain the difference between control limits and specification limits.

Problem 73SE Processes that are in control are predictable; out-of-control processes are not. Explain.

Should control charts be used to monitor a process that is both in control and capable? Why or why not?

A process is under control and follows a normal distribution with mean 100 and standard deviation 10. In constructing a standard \(\bar{x} \text {-chart }\) for this process, the control limits are set 3 standard deviations from the mean—that is, \(100 \pm 3(10 / \sqrt{n})\). The probability of observing an \(\bar{x}\) outside the control limits is (.00135 + .00135) = .0027. Suppose it is desired to construct a control chart that signals the presence of a potential special cause of variation for less extreme values of \(\bar{x}\). How many standard deviations from the mean should the control limits be set such that the probability of the chart falsely indicating the presence of a special cause of variation is .10 rather than .0027?

Under what circumstances is it appropriate to use \(C_p\) to assess capability?

Lengths of pencils. The length measurements of 20 consecutively produced pencils are recorded in the table below. a. Construct a time series plot. Be sure to connect the plotted points and add a centerline. b. Which type of variation pattern in Figure 13.6 best describes the pattern shown in your plot?

Weight of a product. Consider the time series data for the weight of a manufactured product shown in the next column. a. Construct a time series plot. Be sure to connect the points and add a centerline. b. Which type of variation pattern in Figure 13.6 best describes the pattern revealed by your plot?

Problem 81SE Applying pattern-analysis rules. Use the appropriate pattern-analysis rules to determine whether the process being monitored by the following control chart is under the influence of special causes of variation.

Defective plastic mold. A company that manufactures plastic molded parts believes it is producing an unusually large number of defects. To investigate this suspicion, each shift drew seven random samples of 200 parts, visually inspected each part to determine whether it was defective, and tallied the primary type of defect present (Hart, 1992). These data are presented in the table at the top of the next page. a. From a statistical perspective, are the number of samples and the sample size of 200 adequate for constructing a p-chart for these data? Explain. b. Construct a p-chart for this manufacturing process. c. Should the control limits be used to monitor future process output? Explain. d. Suggest a strategy for identifying the special causes of variation that may be present.

Problem 83SE Robotics clamp gap width. University of Waterloo (Canada) statistician S. H. Steiner applied control chart methodology to the manufacturing of a horseshoe-shaped metal fastener called a robotics clamp (Applied Statistics, Vol. 47, 1998). Users of the clamp were concerned with the width of the gap between the two ends of the fastener. Their preferred target width was .054 inch. An optical measuring device was used to measure the gap width of the fastener during the manufacturing process. The manufacturer sampled five finished clamps every 15 minutes throughout its 16-hour daily production schedule and optically measured the gap. Data for 4 consecutive hours of production are presented in the accompanying table. a. Construct an R-chart from these data. b. Construct an from these data. c. Apply the pattern-analysis rules to the control charts. Does your analysis suggest that special causes of variation are present in the clamp manufacturing process? Which of the six rules led you to your conclusion? d. Should the control limits be used to monitor future process output? Explain.

Package sorting time. AirExpress, an overnight mail service, is concerned about the operating efficiency of the package-sorting departments at its Toledo, Ohio, terminal. The company would like to monitor the time it takes for packages to be put in outgoing delivery bins from the time they are received. The sorting department operates 6 hours per day, from 6:00 p.m. to midnight. The company randomly sampled four packages during each hour of operation during 4 consecutive days. The time for each package to move through the system, in minutes, is given in the table on the next page. a. Construct an \(\bar{x} \text {-chart }\) from these data. For this chart to be meaningful, what assumption must be made about the variation of the process? Why? b. What does the chart suggest about the stability of the package-sorting process? Explain. c. Should the control limits be used to monitor future process output? Explain.

Problem 85SE Waiting times of airline passengers. Officials at Mountain Airlines are interested in monitoring the length of time customers must wait in line to check in at their airport counter in Reno, Nevada. To develop a control chart, five customers were sampled each day for 20 days. The data, in minutes, are presented in the table below. a. Construct an R-chart from these data. b. What does the R-chart suggest about the stability of the process? Explain. c. Explain why the R-chart should be interpreted prior to the . d. Construct an from these data. e. What does the suggest about the stability of the process? Explain. f. Should the control limits for the R-chart and be used to monitor future process output? Explain.

Credit histories with data-entry errors. A company called CRW runs credit checks for a large number of banks and insurance companies. Credit history information is typed into computer files by trained administrative assistants. The company is interested in monitoring the proportion of credit histories that contain one or more data-entry errors. Based on her experience with the data-entry operation, the director of the data processing unit believes that the proportion of histories with data-entry errors is about 6%. CRW audited 150 randomly selected credit histories each day for 20 days. The sample data are presented below. a. Use the sample size formula to show that a sample size of 150 is large enough to prevent the lower control limit of the p-chart they plan to construct from being negative. b. Construct a p-chart for the data-entry process. c. What does the chart indicate about the presence of special causes of variation? Explain. d. Provide an example of a special cause of variation that could potentially affect this process. Do the same for a common cause of variation. e. Should the control limits be used to monitor future credit histories produced by the data-entry operation? Explain.

Waiting times of airline passengers (cont’d). Consider the airline check-in process described in Exercise 13.85. a. Assume the process is under control and construct a capability analysis diagram for the process. Management has specified a USL of 5 minutes. b. Is the process capable? Justify your answer. c. If it is appropriate to estimate and interpret \(C_p\) for this process, do so. If it is not, explain why. d. Why didn’t management provide a LSL?

Defects in graphite shafts. Over the last year, a company that manufactures golf clubs has received numerous complaints about the performance of its graphite shafts and has lost several market share percentage points. In response, the company decided to monitor its shaft production process to identify new opportunities to improve its product. The process involves pultrusion. A fabric is pulled through a thermosetting polymer bath and then through a long heated steel die. As it moves through the die, the shaft is cured. Finally, it is cut to the desired length. Defects that can occur during the process are internal voids, broken strands, gaps between successive layers, and microcracks caused by improper curing. The company’s newly formed quality department sampled 10 consecutive shafts every 30 minutes, and nondestructive testing was used to seek out flaws in the shafts. The data from each 8-hour work shift were combined to form a shift sample of 160 shafts. Data on the proportion of defective shafts for 36 shift samples are presented in the table above. a. Use the appropriate control chart to determine whether the process proportion remains stable over time. b. Does your control chart indicate that both common and special causes of variation are present? Explain. c. Data on the types of flaws identified are given in the table below. [Note: Each defective shaft may have more than one flaw.] To help diagnose the causes of variation in process output, construct a Pareto diagram for the types of shaft defects observed. Which are the “vital few”? The “trivial many”?