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Introduction To Statistical Quality Control 7th Edition Douglas C Montgomery - Solutions

(a) Thirty observations on the oxide thickness of individual silicon wafers are shown in table 6E.23. Use these data to set up a control chart on oxide thickness and a moving range chart. Does the process exhibit statistical control? Does oxide thickness follow a normal distribution?(b) Following
The waiting time for treatment in a €œminute-clinic€ located in a drugstore is monitored using control charts for individuals and the moving range. Table 6E.24 contains 30 successive measurements on waiting time.(a) Set up individual and moving range control charts using this
The waiting time data in Exercise 6.66 may not be normally distributed. Transform these data using a natural log transformation. Plot the transformed data on a normal probability plot and discuss your findings. Set up individual and moving range control charts using the transformed data. Plot the
Thirty observations on concentration (in g/l) of the active ingredient in a liquid cleaner produced in a continuous chemical process are shown in Table 6E.26.(a) A normal probability plot of the concentration data is shown in Figure 6.29. The straight line was fit by eye to pass approximately
In 1879, A.A. Michelson measured the velocity of light in air using a modification of a method proposed by the French physicist Foucault. Twenty of these measurements are in table 6E.27 (the value reported is in kilometers per second and has 299,000 subtracted from it). Use these data to set up
The data shown in Table 6E.2 are x and R values for 24 samples of size n = 5 taken from a process producing bearings. The measurements are made on the inside diameter of the bearing, with only the last three decimals recorded (i.e., 34.5 should be 0.50345).(a) Set up x and R charts on this process.
Michelson actually made 100 measurements on the velocity of light in five trials of 20 observations each. The second set of 20 measurements is shown in Table 6E.28(a) Plot these new measurements on the control charts constructed in Exercise 6.69. Are these new measurements in statistical control?
The uniformity of a silicon wafer following an etching process is determined by measuring the layer thickness at several locations and expressing uniformity as the range of the thicknesses. Table 6E.29 presents uniformity determinations for 30 consecutive wafers processed through the etching
The purity of a chemical product is measured on each batch. Purity determination for 20 successive batches are shown in Table 6E.30.(a) Is purity normally distributed? (b) Is the process in statistical process control? (c) Estimate the process mean and standard deviation.
Reconsider the situation in Exercise 6.61. Construct an individuals control chart using the median of the span-two moving ranges to estimate variability. Compare this control chart to the one constructed in Exercise 6.61 and discuss.
Reconsider the hardness measurements in Exercise 6.62. Construct an individuals control chart using the median of the span-two moving ranges to estimate variability. Compare this control chart to the one constructed in Exercise 6.62 and discuss
Reconsider the polymer viscosity data in Exercise 6.63. Use the median of the span-two moving ranges to estimate σ and set up the individuals control chart. Compare this chart to the one originally constructed using the average moving range method to estimate σ.
Use all 60 observations on oxide thickness. (a) Set up an individuals control chart with σ estimated by the average moving range method. (b) Set up an individuals control chart with σ estimated by the median moving range method. (c) Compare and discuss the two control charts.
Consider the individuals measurement data shown inTable6E.31.(a) Estimate Ïƒ using the average of the moving ranges of span two. (b) Estimate Ïƒ using s/c4. (c) Estimate Ïƒ using the median of the span-two moving ranges. (d) Estimate Ïƒ using the average of
The vane heights for 20 of the castings from Figure 6.25 are shown in table 6E.32. Construct the €œbetween/within€ control charts for these process data using a range chart to monitor the within-castings vane height. Compare these to the control charts shown in Figure 6.27.
The diameter of the casting in Figure 6.25 is also an important quality characteristic. A coordinate measuring machine is used to measure the diameter of each casting at five different locations. Data for 20 casting are shown in the Table 6E.33.(a) Set up x and R charts for this process, assuming
A high-level voltage power supply should have a nominal output voltage of 350 V. A sample of four units is selected each day and tested for process-control purposes. The data shown in Table 6E.3 give the difference between the observed reading on each unit and the nominal voltage times ten; that
In the semiconductor industry, the production of microcircuits involves many steps. The wafer fabrication process typically builds these microcircuits on silicon wafers, and there are many microcircuits per wafer. Each production lot consists of between 16 and 48 wafers. Some processing steps treat
Consider the situation described in Exercise 6.80. A critical dimension (measured in ïm) is of interest to the process engineer. Suppose that five fixed positions are used on each wafer (position 1 is the center) and that two consecutive wafers are selected from each batch. The data
The data shown in Table 6E.4 are the deviations from nominal diameter for holes drilled din a carbon-fiber composite material used in aerospace manufacturing. The values reported are deviations from nominal in ten-thousandths of an inch.(a) Set up x and R charts on the process. Is the process in
A financial services company monitors loan applications. Every day 50 applications are assessed for the accuracy of the information on the form. Results for 20 days arewhere Di is the number of loans on the ith day that re determined to have at least one error. What are the center line and control
The data in Table 7E.5 represent the results of inspecting all units of a personal computer produced for the last 10 days. Does the process appear to be in control?
Based on the data in Table 7E.8 if an np chart is to be established, what would you recommend as the center line and control limits? Assume that n = 500.
A control chart indicates that the current process fraction nonconforming is 0.02. If 50 items are inspected each day, what is the probability of detecting a shift in the fraction nonconforming to 0.04 on the first day after the shift? By the end of the third day following the shift?
Diodes used on printed circuit boards are produced in lots of size 1000. We wish to control the process producing these diodes by taking samples of size 64 from each lot. If the nominal value of the fraction nonconforming is p = 0.10, determine the parameters of the appropriate control chart. To
A control chart for the number of nonconforming piston rings is maintained on a forging process with np = 16.0. A sample of size 100 is taken each day and analyzed.(a) What is the probability that a shift in the process average to np = 20.0 will be detected on the first day following the shift?
A control chart for the fraction nonconforming is to be established using a center line of p = 0.10. What sample size is required if we wish to detect a shift in the process fraction nonconforming to 0.20 with probability 0.50?
Do points that plot below the lower control limit on a fraction nonconforming control chart (assuming that the LCL > 0) always mean that there has been an improvement in process quality? Discuss your answer in the context of a specific situation.
A process is controlled with a fraction nonconforming control chart with three-sigma limits, n = 100, UCL = 0.161, center line = 0.080, and LCL = 0.(a) Find the equivalent control chart for the number nonconforming.(b) Use the Poisson approximation to the binomial to find the probability of a
A process is controlled with a fraction nonconforming control chart with three-sigma limits, n = 100, UCL = 0.161, center line = 0.080, and LCL = 0. (a) Find the equivalent control chart for the number nonconforming. (b) Use the Poisson approximation to the binomial to find the probability of a
A process is being controlled with a fraction nonconforming control chart. The process average has been shown to be 0.07. Three-sigma control limits are used, and the procedure calls for taking daily samples of 400 items. (a) Calculate the upper and lower control limits. (b) If the process average
In designing a fraction nonconforming chart with center line at p = 0.20 and three-sigma control limits, what is the sample size required to yield a positive lower control limit? What is the value of n necessary to give a probability of 0.50 of detecting a shift in the process to 0.26?
A control chart is used to control the fraction nonconforming for a plastic part manufactured in an injection molding process. Ten subgroups yield the data in Table 7E.9.(a) Set up a control chart for the number nonconforming in samples of n = 100. (b) For the chart established in part (a), what is $A control chart is used to control the fraction nonconforming$
A control chart for fraction nonconforming indicates that the current process average is 0.03. The sample size is constant at 200 units. (a) Find the three-sigma control limits for the control chart. (b) What is the probability that a shift in the process average to 0.08 will be detected on the
(a) A control chart for the number nonconforming is to be established, based on samples of size 400. To start the control chart, 30 samples were selected and the number nonconforming in each sample determined, yielding. (b) Suppose the process average fraction nonconforming shifted to 0.15. What is
A fraction nonconforming control chart with center line 0.10, UCL = 0.19, and LCL = 0.01 is used to control a process. (a) If three-sigma limits are used, find the sample size for the control chart. (b) Use the Poisson approximation to the binomial to find the probability of type I error. Using
Consider the control chart designed in Exercise 7.25. Find the average run length to detect a shift to a fraction nonconforming of 0.15.
Consider the control chart in Exercise 7.26. Find the average run length if the process fraction nonconforming shifts to 0.20.
Tale 7E.1 contains data on examination of medical insurance claims. Every day 50 claims were examined.(a) Set up the fraction nonconforming control chart for this process. Plot the preliminary data in Table 7E.1 on the chart. Is the process in statistical control? (b) Assume that assignable causes
Analyze the data in Exercise 7.29 using an average sample size.In Exercise 7.29
Construct a standardized control chart for the data in Exercise 7.29.In Exercise 7.29
Note that in Exercise 7.29 there are only four different sample sizes; n = 100, 150, 200, and 250. Prepare a control chart that has a set of limits for each possible sample size and show how it could be used as an alternative to the variable-width control limit method used in Exercise 7.29. How
A process has an in-control fraction nonconforming of p = 0.02. What sample size would be required for the fraction nonconforming control chart if it is desired to have a probability of at least one nonconforming unit in the sample to be at least 0.95?
A process has an in-control fraction nonconforming of p = 0.01. What sample size would be required for the fraction nonconforming control chart if it is desired to have a probability of at least one nonconforming unit in the sample to be at least 0.9?
A process has an in-control fraction nonconforming of p = 0.01. The sample size is n = 300. What is the probability of detecting a shift to an out-of-control fraction nonconforming of p = 0.05 on the first sample following the shift?
A banking center has instituted a process improvement program to reduce and hopefully eliminate errors in their check processing operations. The current error rate is 0.01. The initial objective is to cut the current error rate in half. What sample size would be necessary to monitor this process
A fraction nonconforming control chart has center line 0.01, UCL = 0.0399, LCL = 0, and n = 100. If three-sigma limits are used, find the smallest sample size that would yield a positive lower control limit.
A fraction nonconforming control chart with n = 400 has the following parameters: UCL = 0.0809, Center line = 0.0500, LCL = 0.0191. (a) Find the width of the control limits in standard deviation units. (b) What would be the corresponding parameters for an equivalent control chart based on the
The fraction nonconforming control chart in Exercise 7.3 has an LCL of zero. Assume that the revised control chart in part (b) of that exercise has a reliable estimate of the process fraction nonconforming. What sample size should be used if you want to ensure that the LCL > 0?
A fraction nonconforming control chart with n = 400 has the following parameters: UCL = 0.0962; Center line = 0.0500; LCL = 0.0038. (a) Find the width of the control limits in standard deviation units. (b) Suppose the process fraction nonconforming shifts to 0.15. What is the probability of
A fraction nonconforming control chart is to be established with a center line of 0.01 and two-sigma control limits. (a) How large should the sample size be if the lower control limit is to be nonzero? (b) How large should the sample size be if we wish the probability of detecting a shift to 0.04
The following fraction nonconforming control chart with n = 100 is used to control a process: UCL = 0.0750; Center line = 0.0400; LCL = 0.0050 (a) Use the Poisson approximation to the binomial to find the probability of a type I error. Pr{type I error} (b) Use the Poisson approximation to the
A process that produces bearing housings is controlled with a fraction nonconforming control chart, using sample size n= 100 and a center line p = 0.02.(a) Find the three-sigma limits for this chart.(b) Analyze the ten new samples (n = 100) shown in Table 7E.11 for statistical control. What
Consider an np chart with k-sigma control limits. Derive a general formula for determining the minimum sample size to ensure that the chart has a positive lower control limit.
Construct a standardized control chart for the data in Exercise 7.11.In Exercise 7.11
A paper mill uses a control chart to monitor the imperfection nin finished rolls of paper. Production output is inspected for 20 days, and the resulting data are shown in Table 7E.13. Use these data to set up a control chart for nonconformities per roll of paper. Does the process appear to be in
The commercial loan operation of a financial institution has a standard for processing new loan applications in 24 hours. Table 7E.2 shows the number of applications processed each day for the last 20 days and the number of applications that required more than 24 hours to complete.(a) Set up the
Consider the paper-making process in Exercise 7.49. Set up a u chart based on an average sample size to control this process.In Exercise 7.49
Consider the paper-making process in Exercise 7.49. Set up a standardized u chart for this process.In Exercise 7.49
The number of nonconformities found on final inspection of a tape deck is shown in Table 7E.14. Can you conclude that the process is in statistical control? What center line and control limits would you recommend for controlling future production?Utilize a c chart based on # of nonconformities per
The data in Table 7E.15 represent the number of nonconformities per 1,000 meters in telephone cable. From analysis of these data, would you conclude that the process is in statistical control? What control procedure would you recommend for future production?Utilize a c chart based on # of
Consider the data in Exercise 7.52. Suppose we wish to define a new inspection unit of four tape decks. (a) What are the center line and control limits for a control chart for monitoring future production based on the total number of defects in the new inspection unit? The new inspection unit is n
Consider the data in Exercise 7.53. Suppose a new inspection unit is defined as 2,500 m of wire. (a) What are the center line and control limits for a control chart for monitoring future production based on the total number of nonconformities in the new inspection unit? (b) What are the center line
An automobile manufacturer wishes to control the number of nonconformities in a subassembly area producing manual transmissions. The inspection unit is defined as four transmissions, and data from 16 samples (each of size 4) are shown in Table 7E.16.(a) Set up a control chart for nonconformities
Find the three-sigma control limits for (a) a c chart with process average equal to four nonconformities. (b) a u chart with c = 4 and n = 4.
Find 0.900 and 0.100 probability limits for a c chart when the process average is equal to 16 nonconformities.
Find the three-sigma control limits for (a) a c chart with process average equal to nine nonconformities. (b) a u chart with c = 16 and n = 4.
Reconsider the loan application data in Table 7E.2. Set up the fraction nonconforming control chart for this process. Use the average sample size control limit approach. Plot the preliminary data in Table 7E.2 on the chart. Is the process in statistical control? Compare this control chart to the
Find 0.980 and 0.020 probability limits for a control chart for nonconformities per unit when u = 6.0 and n = 3. u chart with u = 6.0 and n = 3. c = u n = 18. Find limits such that Pr{D UCL} = 0.980 and Pr{D < LCL} = 0.020. Using the Poisson distribution to find Pr{D x | c = 18}
Find 0.975 and 0.025 probability limits for a control chart for nonconformities when c = 7.6. Using the cumulative Poisson distribution:
A control chart for nonconformities per unit uses 0.95 and 0.05 probability limits. The center line is at u = 1.4. Determine the control limits if the sample size is n = 10. Using the cumulative Poisson distribution with c = u n = 1.4(10) = 14, to find Pr{D ≤ x | c = 14}:
The number of workmanship nonconformities observed in the final inspection of disk-drive assemblies has been tabulated as shown in Table 7E.17. Does the process appear to be in control?
Most corporations use external accounting and auditing firms for performing audits on their financial records. In medium to large businesses there may be a very large number of accounts to audit, so auditors often use a technique called audit sampling, in which a random sample of accounts are
A metropolitan police agency is studying the incidence of drivers operating their vehicles without the minimum liability insurance required by law. The data are collected from drivers who have been stopped by an officer for a traffic law violation and a traffic summons issued. Data from three
A control chart for nonconformities is to be constructed with c 2.0, LCL = 0, and UCL such that the probability of appoint plotting outside control limits when c = 2.0 is only 0.005. (a) Find the UCL. (b) What is the type I error probability of the process is assumed to be out of control only when
A textile mill wishes to establish a control procedure on flaws in towels it manufactures. Using an inspection unit of 50 units, past inspection data show that 100 previous units had 850 total flaws. What type of control chart is appropriate? Design the control chart such that it has two-sided
The manufacturer wishes to set up a control chart at the final inspection station for a gas water heater. Defects in workmanship and visual quality features are checked in this inspection. For the past 22 working days, 176 water heaters were inspected and a total of 924 nonconformities
Assembled portable television sets are subjected to a final inspection for surface defects. A total procedure is established based on the requirement that if the average number of nonconformities per units is 4.0, the probability of concluding that the process is in control will be 0.99. There is
Reconsider the loan application data in Table 7E.2. Set up the fraction nonconforming control chart for this process. Use the standardized control chart approach. Plot the preliminary data in Table 7E.2 on the chart. Is the process in statistical control? Compare this control chart to the one based
A control chart is to be established on a process producing refrigerators. The inspection unit is one refrigerator, and a common chart for nonconformities is to be used. As preliminary data, 16 non Use a c chart for nonconformities with an inspection unit n = 1 refrigerator.Conformities were
Consider the situation described in Exercise 7.70. (c) = 0.533c (a) Find two-sigma control limits and compare these with the control limits found in art (a) of exercise 7.70. b) Find the a-risk for the control chart with two-sigma control limits and compare with the results of part (b) of Exercise
A control chart for nonconformities is to be established in conjunction with final inspection of a radio. The inspection unit is to be a group of ten radios. The average number of nonconformities per radio had, in the past, been 0.5. Find three-sigma control limits for a c chart based on this size
A control chart for nonconformities is maintained on a process producing desk calculators. The inspection unit is defined as two calculators. The average number of nonconformities per machine when the process is in control is estimated to be two. u = average # nonconformities/calculator =2 (a)
A production line assembles electric clocks. The average number of nonconformities per clock is estimated to be 0.75. The quality engineer wishes to establish a c chart for this operation, using an inspection unit of six clocks. Find the three-sigma limits for this chart. 1 inspection unit = 6
Suppose that we wish to design a control chart for nonconformities per unit with L-sigma limits. Find the minimum sample size that would result in a positive lower control limit for this chart. c: nonconformities per unit; L: sigma control limits
Kittlitz (1999) presents data on homicides in Waco, Texas, for the years 1980-1989 (data taken from the Waco Tribune-Herald, December 29, 1989). There were 29 homicides in 1989. Table 7E.20 gives the dates of the 1989 homicides and the number of days between each homicide.(a) Plot the
What practical difficulties could be encountered in monitoring time-between-events data?
A paper by R.N. Rodriguez (€œHealth Care Applications of Statistical Process Control: Examples Using the SAS® System,€ SAS Users Group International: Proceedings of the 21st Annual Conference, 1996) illustrated several informative applications of control charts to the
Reconsider the insurance claim data in Table 7E.1. Set up an np control chart for this data and plot the data from Table 7E.1 on this chart. Compare this to the fraction nonconforming control chart in Exercise 7.3.In Table 7e.1
A paper by R.N. Rodriguez (€œHealth Care Applications of Statistical Process Control: Examples Using the SAS® System,€ SAS Users Group International: Proceedings of the 21st Annual Conference, 1996) illustrated several informative applications of control charts to the
Kaminsky et al. (1992) present data on the number of orders per truck at a distribution center. Some of these data are sown in Table 7E.24.(a) Set up a c chart for the number of orders per truck. Is the process in control? (b) Set up a t chart for the number of orders per truck, assuming a
A process is in statistical control with x =20 and s =1.2. Specifications are at LSL = 16 and USL = 24. (a) Estimate the process capability with an appropriate process capability ratio. (b) Items that are produced below the lower specification limit must be scrapped, while items that are above the
A process is in statistical control with x ï€½ 75 and s ï€½ 2.0 . The process specifications are at 80 ï‚± 8. The sample size is n= 5.(a) Estimate the potential capability. (b) Estimate the actual capability. (c) How much could process fallout be
Consider the two processes shown in Table 8E.1 (the sample size n = 5).Specifications are at 100 ± 10. Calculate Cp, Cpk, and Cpm and interpret these ratios. Which process would you prefer to use?
Suppose that 20 of the parts manufactured by the processes in Exercise 8.11 were assembled so that their dimensions were additive; that is, x  x1  x2 Specifications on x are 2,000  200. Would you prefer to produce the parts using process A or process B? Why? Do the capability ratios
The weights of nominal 1-kg containers of a concentrated chemical ingredient are shown in Table 8E.2. Prepare a normal probability plot of the data and estimate process capability. Does this conclusion depend on process stability?
Consider the package weight data in Exercise 8.13. Suppose there is a lower specification at 0.985 kg. Calculate an appropriate process capability ratio for this material. What percentage of the packages produced by this process is estimated to be below the specification limit?
Table 8E.3 presents data on the cycle time (in hours) to process small loan applications. Prepare a normal probability plot of these data. The loan agency has a promised decision time to potential customers of 24 hours. Based on the data in the table and the normal probability plot, what proportion
Table 8E.4 presents data on the waiting time in minutes) to see a nurse or physician in a hospital emergency department. Prepare a normal probability plot of these data. The hospital has a policy of seeing all patients initially within ten minutes of arrival.(a) Prepare a normal probability plot of
A process is in statistical control with x = 202.5 and s = 2.0. Specifications are at LSL = 196 and USL = 206. (a) Estimate the process capability with an appropriate process capability ratio. (b) What is the potential capability of this process? (c) Items that are produced below the lower

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