Section 1 Statistical Process Control (SPC)

Operations Management, 12e Supplement 6 Statistical Process Control Section 1 Statistical Process Control (SPC) 1) Some degree of variability is present in almost all processes. Answer: TRUE Diff: 1 Key Term: Natural variations Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 2) The purpose of process control is to detect when natural causes of variation are present. Answer: FALSE Diff: 2 Key Term: Statistical Process Control (SPC) Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 3) A normal distribution is generally described by its two parameters: the mean and the range. Answer: FALSE Diff: 1 Key Term: Central limit theorem Objective: LO S6.2 Explain the role of the central limit theorem in SPC Learning Outcome: Apply basic statistical process control (SPC) methods 4) A process is said to be in statistical control when assignable causes are the only sources of variation. Answer: FALSE Diff: 2 Key Term: Statistical Process Control (SPC) Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 5) Mistakes stemming from workers' inadequate training represent an assignable cause of variation. Answer: TRUE Diff: 1 Key Term: Assignable variation Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 6) Averages of small samples, not individual measurements, are generally used in statistical process control. Answer: TRUE Diff: 2 Key Term: Statistical Process Control (SPC) Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 1 7) The x-bar chart indicates that a gain or loss of uniformity has occurred in dispersion of a production process. Answer: FALSE Diff: 2 Key Term: x-bar chart Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 8) The Central Limit Theorem states that when the sample size increases, the distribution of the sample means will approach the normal distribution. Answer: TRUE Diff: 2 Key Term: Central limit theorem Objective: LO S6.2 Explain the role of the central limit theorem in SPC Learning Outcome: Apply basic statistical process control (SPC) methods 9) In statistical process control, the range is often used as a substitute for the standard deviation. Answer: TRUE Diff: 2 Key Term: Control charts Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 10) If the process average is in control, then the process range must also be in control. Answer: FALSE Diff: 2 Key Term: Control charts Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 11) A process range chart illustrates the amount of dispersion within the samples. Answer: TRUE Diff: 2 Key Term: R-chart Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 12) Mean charts and range charts complement one another, one detecting shifts in process average, the other detecting shifts in process dispersion. Answer: TRUE Diff: 2 Key Term: Control charts Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 13) An x-bar chart is used when we are sampling attributes. Answer: FALSE Diff: 1 Key Term: x-bar chart Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 2 14) To measure the voltage of batteries, one would sample by attributes. Answer: FALSE Diff: 1 Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 15) A p-chart is appropriate to plot the number of typographic errors per page of text. Answer: FALSE Diff: 2 Key Term: p-chart Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 16) A c-chart is appropriate to plot the number of flaws in a bolt of fabric. Answer: TRUE Diff: 2 Key Term: c-chart Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 17) The x-bar chart, like the c-chart, is based on the exponential distribution. Answer: FALSE Diff: 3 Key Term: Control charts Learning Outcome: Apply basic statistical process control (SPC) methods 18) If a sample of items is taken and the mean of the sample is outside the control limits, the process is: A) likely out of control and the cause should be investigated. B) in control, but not capable of producing within the established control limits. C) within the established control limits with only natural causes of variation. D) monitored closely to see if the next sample mean will also fall outside the control limits. E) producing high quality products. Answer: A Diff: 2 Key Term: Control charts Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 19) The causes of variation in statistical process control are: A) cycles, trends, seasonality, and random variations. B) producer's causes and consumer's causes. C) mean and range. D) natural causes and assignable causes. E) Type I and Type II. Answer: D Diff: 2 Key Term: Statistical Process Control (SPC) Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 3 20) Natural variations: A) affect almost every production process. B) are the many sources of variation that occur when a process is under control. C) when grouped, form a pattern, or distribution. D) are tolerated, within limits, when a process is under control. E) All of the above are true. Answer: E Diff: 2 Key Term: Natural variations Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 21) Natural variations: A) are variations that are to be identified and investigated. B) are variations that can be traced to a specific cause. C) are the same as assignable variations. D) lead to occasional false findings that processes are out of control. E) play no role in statistical process control. Answer: D Diff: 2 Key Term: Natural variations AACSB: Reflective thinking Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 22) Assignable variation: A) is a sign that a process is under control. B) is to be identified and investigated. C) is the same as random variation. D) is variation that cannot be traced to a specific cause. E) leads to a steep OC curve. Answer: B Diff: 2 Key Term: Assignable variation Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 23) Assignable causes: A) are not as important as natural causes. B) are within the limits of a control chart. C) depend on the inspector assigned to the job. D) are also referred to as "chance" causes. E) are causes of variation that can be identified and investigated. Answer: E Diff: 2 Key Term: Assignable variation Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 4 24) Control charts for variables are based on data that come from: A) acceptance sampling. B) individual items. C) averages of small samples. D) averages of large samples. E) the entire lot. Answer: C Diff: 2 Key Term: Control charts Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 25) The purpose of an x-bar chart is to determine whether there has been a: A) change in the dispersion of the process output. B) change in the percent defective in a sample. C) change in the central tendency of the process output. D) change in the number of defects in a sample. E) change in the AOQ. Answer: C Diff: 2 Key Term: x-bar chart Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 26) The number of defects after a hotel room cleaning (sheets not straight, smears on mirror, missed debris on carpet, etc.) should be measured using what type of control chart? A) x-bar chart B) R-chart C) p-chart D) c-chart E) either x-bar chart or R chart Answer: D Diff: 2 Key Term: c-chart AACSB: Reflective thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 5 27) The number of late insurance claim payouts per 100 should be measured with what type of control chart? A) x-bar chart B) R-chart C) p-chart D) c-chart E) either p-chart or c-chart Answer: C Diff: 2 Key Term: p-chart AACSB: Reflective thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 28) The upper and lower limits for diving ring diameters made by John's Swimming Co. are 40 and 39 cm., respectively. John took 11 samples with the following average diameters (39, 39.1, 39.2, 39.3, 39.4, 39.5 39.6, 39.7, 39.8, 39.9, 40). Is the process in control? A) Yes, no diameters exceeded the control limits. B) No, some diameters exceeded the control limits. C) No, there is a distinguishable pattern to the samples. D) No, the range is not in control. E) There is not enough information to make a decision. Answer: C Diff: 3 Key Term: x-bar chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 29) Red Top Cab Company receives multiple complaints per day about driver behavior. Over 9 days the owner recorded the number of calls to be 3, 0, 8, 9, 6, 7, 4, 9, and 8. What is the upper control limit for the 3-sigma c-chart? A) 13.35 B) 8.45 C) 24.00 D) 0.00 E) 9.03 Answer: A Diff: 2 Key Term: c-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 6 30) A process that is assumed to be in control with limits of 89 ± 2 had sample averages for the x-bar chart of the following: 87.1, 87, 87.2, 89, 90, 88.5, 89.5, and 88. Is the process in control? A) Yes. B) No, one or more averages exceeded the limits. C) Not enough information to tell. D) No, there is a distinguishable trend. E) No, two or more consecutive points are very near the lower (or upper) limit. Answer: E Diff: 2 Key Term: x-bar chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 31) Ten samples of a process measuring the number of returns per 100 receipts were taken for a local retail store. The number of returns were 10, 9, 11, 7, 3, 12, 8, 4, 6, and 11. Find the standard deviation of the sampling distribution for the p-bar chart. A) There is not enough information to answer the question. B) .081 C) 8.1 D) .0273 E) .0863 Answer: D Diff: 2 Key Term: p-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 32) An x-bar control chart was examined and no data points fell outside of the limits. Can this process be considered in control? A) Not yet, there could be a pattern to the points. B) Not yet, the R-chart must be checked. C) Not yet, the number of samples must be known. D) Yes. E) Both A and B Answer: E Diff: 2 Key Term: x-bar chart AACSB: Reflective thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 7 33) Statistical process control charts: A) display the measurements on every item being produced. B) display upper and lower limits for process variables or attributes and signal when a process is no longer in control. C) indicate to the process operator the average outgoing quality of each lot. D) indicate to the operator the true quality of material leaving the process. E) are a graphic way of classifying problems by their level of importance, often referred to as the 80-20 rule. Answer: B Diff: 2 Key Term: Control charts Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 34) The Central Limit Theorem: A) is the theoretical foundation of the c-chart. B) states that the average of assignable variations is zero. C) allows managers to use the normal distribution as the basis for building some control charts. D) states that the average range can be used as a proxy for the standard deviation. E) controls the steepness of an operating characteristic curve. Answer: C Diff: 3 Key Term: Central limit theorem Objective: LO S6.2 Explain the role of the central limit theorem in SPC Learning Outcome: Apply basic statistical process control (SPC) methods 35) For a 3-sigma x-bar chart where the process standard deviation is known, the upper control limit: below the mean of sample means for a E) cannot be calculated unless the average range is known. Answer: D Diff: 2 Key Term: x-bar chart Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 36) Up to three standard deviations above or below the centerline is the amount of variation that statistical process control allows for: A) Type I errors. B) about 95.5% variation. C) natural variation. D) all types of variation. E) assignable variation. Answer: C Diff: 2 Key Term: Natural variations Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 8 37) A manager wants to build 3-sigma x-bar control limits for a process. The target value for the mean of the process is 10 units, and the standard deviation of the process is 6. If samples of size 9 are to be taken, what will be the upper and lower control limits, respectively? A) -8 and 28 B) 16 and 4 C) 12 and 8 D) 4 and 16 E) 8 and 12 Answer: B Diff: 2 Key Term: x-bar chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 38) Jars of pickles are sampled and weighed. Sample measures are plotted on control charts. The ideal weight should be precisely 11 oz. Which type of chart(s) would you recommend? A) p-chart B) c-chart C) both an x-bar chart and an R-chart D) an x-bar chart, but not an R-chart E) both a p-chart and a c-chart Answer: C Diff: 2 Key Term: Control charts AACSB: Reflective thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 39) If A) 21.8 to 24.2 ounces B) 23 ounces C) 22.70 to 23.30 ounces D) 22.25 to 23.75 ounces E) 22.90 to 23.10 ounces Answer: C x-bar chart? Diff: 2 Key Term: x-bar chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 9 40) The usual purpose of an R-chart is to signal whether there has been a: A) gain or loss in dispersion. B) change in the percent defective in a sample. C) change in the central tendency of the process output. D) change in the number of defects in a sample. E) change in the consumer's risk. Answer: A Diff: 2 Key Term: R-chart Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 41) A manager wishes to build a 3-sigma range chart for a process. The sample size is five, the mean of sample means is 16.01, and the average range is 5.3. From Table S6.1, the appropriate value of D3 is 0, and D4 is 2.115. What are the UCL and LCL, respectively, for this range chart? A) 33.9 and 11.2 B) 33.9 and 0 C) 11.2 and 0 D) 6.3 and 0 E) 31.91 and 0.11 Answer: C Diff: 2 Key Term: R-chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 42) Plots of sample ranges indicate that the most recent value is below the lower control limit. What course of action would you recommend? A) Since there is no obvious pattern in the measurements, variability is in control. B) One value outside the control limits is insufficient to warrant any action. C) Lower than expected dispersion is a desirable condition; there is no reason to investigate. D) The process is out of control; reject the last units produced. E) Variation is not in control; investigate what created this condition. Answer: E Diff: 3 Key Term: R-chart AACSB: Reflective thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 10 43) To set x-bar chart upper and lower control limits, one must know the process central line, which is the: A) average of the sample means. B) total number of defects in the population. C) percent defects in the population. D) size of the population. E) average range. Answer: A Diff: 2 Key Term: x-bar chart Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 44) According to the text, what is the most common choice of limits for control charts? A) ±1 standard deviation B) ±2 standard deviations C) ±3 standard deviations D) ±3 standard deviations for means and ± 2 standard deviations for ranges E) ±6 standard deviations Answer: C Diff: 2 Key Term: Control charts Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 45) Which of the following is true of a p-chart? A) The lower control limit is found by subtracting a fraction from the average number of defects. B) The lower control limit indicates the minimum acceptable number of defects. C) The lower control limit equals D3 times p-bar. D) The lower control limit may be at zero. E) The lower control limit is the same as the lot tolerance percent defective. Answer: D Diff: 2 Key Term: p-chart AACSB: Reflective thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 46) The normal application of a p-chart is in: A) process sampling by variables. B) acceptance sampling by variables. C) process sampling by attributes. D) acceptance sampling by attributes. E) process capability ratio computations. Answer: C Diff: 2 Key Term: p-chart Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 11 47) What is the statistical process chart used to control the number of defects per unit of output? A) x-bar chart B) R-chart C) p-chart D) AOQ chart E) c-chart Answer: E Diff: 2 Key Term: c-chart Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 48) The c-chart signals whether there has been a: A) gain or loss in uniformity. B) change in the number of defects per unit. C) change in the central tendency of the process output. D) change in the percent defective in a sample. E) change in the AOQ. Answer: B Diff: 2 Key Term: c-chart Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 49) The local newspaper receives several complaints per day about typographic errors. Over a seven-day period, the publisher has received calls from readers reporting the following total daily number of errors: 4, 3, 2, 6, 7, 3, and 9. Based on these data alone, what type of control chart(s) should the publisher use? A) p-chart B) c-chart C) x-bar chart D) R-chart E) x-bar chart and R-chart Answer: B Diff: 2 Key Term: c-chart AACSB: Reflective thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 12 50) A manufacturer uses statistical process control to control the quality of the firm's products. Samples of 50 of Product A are taken, and a defective/acceptable decision is made on each unit sampled. For Product B, the number of flaws per unit is counted. What type(s) of control charts should be used? A) p-charts for both A and B B) p-chart for A, c-chart for B C) c-charts for both A and B D) p-chart for A, mean and range charts for B E) c-chart for A, mean and range charts for B Answer: B Diff: 3 Key Term: Control charts AACSB: Reflective thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 51) A nationwide parcel delivery service keeps track of the number of late deliveries (more than 30 minutes past the time promised to clients) per day. They plan on using a control chart to plot their results. Which type of control chart(s) would you recommend? A) both x-bar chart and R-chart B) p-chart C) c-chart D) x-bar chart E) both p-chart and c-chart Answer: C Diff: 2 Key Term: c-chart AACSB: Reflective thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 52) A run test is used: A) to examine variability in acceptance sampling plans. B) in acceptance sampling to establish control. C) to examine points in a control chart to check for natural variability. D) to examine points in a control chart to check for nonrandom variability. E) to test the validity of the Central Limit Theorem. Answer: D Diff: 2 Key Term: Run test Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 53) ________ is variation in a production process that can be traced to specific causes. Answer: Assignable variation Diff: 2 Key Term: Assignable variation Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 13 54) The ________ -chart is the chief way to control attributes. Answer: p Diff: 3 Key Term: p-chart Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 55) If a process has only natural variations, ________ percent of the time the sample averages will fall inside the ±3-sigma control limits. Answer: 99.73 Diff: 2 Key Term: Natural variations Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 56) The ________ is a quality control chart that indicates when changes occur in the central tendency of a production process. Answer: x-bar chart Diff: 2 Key Term: x-bar chart Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 57) The ________ is a quality control chart used to control the number of defects per unit of output. Answer: c-chart Diff: 2 Key Term: c-chart Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 58) What is the basic objective of a process control system? Answer: It is to provide a statistical signal when assignable causes of variation are present. Diff: 2 Key Term: Statistical Process Control (SPC) Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 59) Briefly explain what the Central Limit Theorem has to do with control charts. Answer: It provides the theoretical foundation for x-bar charts. It leads to the usability of the normal distribution in control charts because regardless of the distribution of the population of all parts or services, the distribution of the samples means tends to follow a normal curve as the number of samples increases. Diff: 3 Key Term: Central limit theorem Objective: LO S6.2 Explain the role of the central limit theorem in SPC Learning Outcome: Apply basic statistical process control (SPC) methods 14 60) What are the three possible results (or findings) from the use of control charts? Answer: The results of a control chart can indicate (a) in control and capable, (b) in control but not capable, and (c) out of control. Diff: 2 Key Term: Control charts Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 61) Why do range charts exist? Aren't x-bar charts enough? Answer: Range charts and mean charts perform different functions. The mean chart is used to detect changes in the average of a process. But that average might stay the same while output is getting more scattered. The purpose of the range chart is to detect changes in the uniformity (dispersion) of a process. Diff: 2 Key Term: Control charts Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 62) Examine the Statistical Process Control outputs below. Answer the following questions. a. What is the sample size? b. What is the number of samples? c. What is the mean of sample 8; what is the range of sample 10? d.. Is this process in control? Explain--a simple Yes or No is insufficient. e. What additional steps should the quality assurance team take? Answer: a. The sample size is 4. b. ten samples were taken. c. The mean of sample 8 is 12.175; the range of sample 10 is 0.5. d. The process is not in control while all means are within limits, the range for sample 3 is too large. e. Investigate for assignable cause and eliminate that cause. Diff: 3 AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 15 63) What is the difference between natural and assignable causes of variation? Answer: Natural variations are those variations that are inherent in the process and for which there is no identifiable cause. These variations fall in a natural pattern. Assignable causes are variations beyond those that can be expected to occur because of natural variation. These variations can be traced to a specific cause. Diff: 2 Objective: LO S6.1 Explain the purpose of a control chart Learning Outcome: Apply basic statistical process control (SPC) methods 64) A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that the process standard deviation is two ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appear below. Weight Day Package 1 Package 2 Package 3 Package 4 Monday 23 22 23 24 Tuesday 23 21 19 21 Wednesday 20 19 20 21 Thursday 18 19 20 19 Friday 18 20 22 20 (a) Calculate all sample means and the mean of all sample means. (b) Calculate upper and lower 2-sigma x-bar chart control limits that allow for natural variations. (c) Based on the x-bar chart, is this process in control? Answer: (a) The five sample means are 23, 21, 20, 19, and 20. The mean of all sample means is 20.6 (b) UCL = 20.6 + 2(2/ ) = 22.6; LCL = 20.6 - 2(2/ ) = 18.6 (c) Sample 1 is above the UCL; all others are within limits. The process is out of control. Diff: 3 Key Term: x-bar chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 16 65) A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that when the process is operating as intended, packaging weight is normally distributed with a mean of twenty ounces, and a process standard deviation of two ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appear below. Weight Day Monday Tuesday Wednesday Thursday Friday Package 1 23 23 20 18 18 Package 2 22 21 19 19 20 Package 3 23 19 20 20 22 Package 4 24 21 21 19 20 (a) If he sets an upper control limit of 21 and a lower control limit of 19 around the target value of twenty ounces, the control chart is based on what value of z? (b) With the UCL and LCL of part a, what do you conclude about this process is it in control? Answer: (a) These control limits are one standard error away from the centerline. (b) The mean of sample 1 lies outside the control limits. All other points are on or within the limits. The process is not in control based on the 1-sigma limits, although there's a significant chance with 1-sigma limits that the chart is sending a false positive out-of-control signal. Diff: 2 Key Term: x-bar chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 66) An operator trainee is attempting to monitor a filling process that has an overall average of 705 cc. The average range is 17 cc. If you use a sample size of 6, what are the upper and lower control limits for the x-bar chart and R-chart? Answer: From the table, A2 = 0.483, D4 = 2.004, D3 = 0 UCL = + A2 × = 705 + 0.483 × 17 = 713.211 LCL = - A2 × = 705 - 0.483 × 17 = 696.789 UCLR = D4 × = 2.004 × 17 = 34.068 LCLR = D3 × = 0 × 17 =0 Diff: 2 AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 17 67) The defect rate for a product has historically been about 1.6%. What are the upper and lower control chart limits for a p-chart, if you wish to use a sample size of 100 and 3-sigma limits? Answer: UCLp = + 3 = 0.016 + 3 . = .0536 UCLp = - 3 = 0.016 - 3 . = -0.0216, or zero Diff: 2 Key Term: p-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 68) A small, independent amusement park collects data on the number of cars with out-of-state license plates. The sample size is fixed at n=25 each day. Data from the previous 10 days indicate the following number of out-of-state license plates: Day 1 2 3 4 5 6 7 8 9 10 Out-of-state Plates 6 4 5 7 8 3 4 5 3 11 (a) Calculate the overall proportion of "tourists" (cars with out-of-state plates) and the standard deviation of proportions. (b) (c) Is the process under control? Explain. Answer: (a) p-bar is 56/250 = 0.224; the standard deviation of proportions is the square root of (.224)(.776)/ 25 = 0.0834 (b) UCL = .224 + (3)(0.834) = .4742; LCL = .224 -(3)(.0834) which is negative, so the LCL = 0 (c) The largest percentage of tourists (day 10) is 11/25 = .44, which is still below the UCL. Thus, all the points are within the control limits, so the process is under control. Diff: 2 Key Term: p-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 18 69) Cartons of Plaster of Paris are supposed to weigh exactly 32 oz. Inspectors want to develop process control charts. They take ten samples of six boxes each and weigh them. Based on the following data, compute the lower and upper control limits and determine whether the process is in control. Sample 1 2 3 4 5 6 7 8 9 10 Mean 33.8 34.6 34.7 34.1 34.2 34.3 33.9 34.1 34.2 34.4 Range 1.1 0.3 0.4 0.7 0.3 0.4 0.5 0.8 0.4 0.3 Answer: n = 6; overall mean = 34.23; = 0.52. Upper control limit 34.48116 1.04208 Center line 34.23 0.52 Lower control limit 33.97884 0 The mean values for samples 1, 2, 3, and 7 fall outside the control limits on the x-bar chart and sample 1 falls outside the upper limit on the R-chart. Therefore, the process is out of control. Diff: 2 19 AACSB: Analytical thinking Objective: LO S6.3 Build x -charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 70) McDaniel Shipyards wants to develop a control chart to assess the quality of its steel plate. They take ten sheets of 1" steel plate and compute the number of cosmetic flaws on each roll. Each sheet is 20' by 100'. Based on the following data, develop limits for the 3-sigma control chart, plot the control chart, and determine whether the process is in control. Sheet 1 2 3 4 5 6 7 8 9 10 Number of flaws 1 1 2 0 1 5 0 2 0 2 Answer: Total units sampled Total defects Defect rate, c-bar Standard deviation z value 10 14 1.4 1.183216 3 Upper Control Limit Center Line Lower Control Limit 4.949648 1.4 0 Sample six is above the control limits; therefore, the process is out of control. 20 Diff: 2 Key Term: c-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 71) The mean and stand control chart, a sample size of 16 will be used. Calculate the standard deviation of the sample means. Answer: Sigma x-bar = 9/ = 2.25 Diff: 2 Key Term: x-bar chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods -sigma control limits for the x-bar chart. Answer: 8.50 to 9.50 ounces Diff: 2 Key Term: x-bar chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 73) A hospital-billing auditor has been inspecting patient bills. While almost all bills contain some errors, the auditor is looking now for large errors (errors in excess of $250). Each day last week, the auditor examined 100 bills and found an average defect rate of 16%. Calculate the upper and lower limits for the billing process for 99.73% confidence. Answer: 0.16 plus or minus (3)(0.03667): LCL = .050 and UCL = 0.270 Diff: 2 Key Term: p-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 74) A local manufacturer supplies you with parts, and you would like to install a quality monitoring system at his factory for these parts. Historically, the defect rate for these parts has been 1.25 percent (You've observed this from your acceptance sampling procedures, which you would like to discontinue). Answer: p-bar is 0.0125; the standard error of the proportion is The upper control limit is 0.0125 + (3)(0.00786) = 0.03608; the lower control limit is 0.0125 - (3)(0.00786) which is negative, so the LCL is 0. Diff: 2 Key Term: p-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 21 = 0.00786 75) Repeated sampling of a certain process shows the average of all sample ranges to be 1.0 cm. The sample size has been constant at n = 5. What are the 3-sigma control limits for this R-chart? Answer: LCL = 0, and UCL = (1.0)(2.115) = 2.115 Diff: 2 Key Term: R-chart AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 76) A woodworker is concerned about the quality of the finished appearance of her work. In sampling units of a split-willow hand-woven basket, she has found the following number of finish defects in ten units sampled: 4, 0, 3, 1, 2, 0, 1, 2, 0, 2. a. Calculate the average number of defects per basket. b. If 3-sigma control limits are used, calculate the lower control limit and upper control limit. Answer: (a) 1.5; (b) LCL = 0, and UCL = 5.2. Diff: 2 Key Term: c-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 77) The width of a bronze bar is intended to be one-eighth of an inch (0.125 inches). Inspection samples contain five bars each. The average range of these samples is 0.01 inches. What are the upper and lower control limits for the x-bar and R-chart for this process, using 3-sigma limits? Answer: x-bar chart: LCL = .119; UCL =.131. R-chart: LCL = 0.0; UCL =.021 Diff: 2 AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 78) A part that connects two levels should have a distance between the two holes of 4". It has been determined that x-bar and R-charts should be set up to determine if the process is in statistical control. The following ten samples of size four were collected. Calculate the control limits, plot the control charts, and determine if the process is in control. Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 Sample 9 Sample 10 Mean 4.01 3.98 4.00 3.99 4.03 3.97 4.02 3.99 3.98 4.01 Range 0.04 0.06 0.02 0.05 0.06 0.02 0.02 0.04 0.05 0.06 22 Answer: x-doublebar value R bar Upper control limit Center line Lower control limit X-bar 3.998 0.042 4.029 3.998 3.967 Range 0.096 0.042 0 The process is out of control because of sample 5 on the x-bar chart. Diff: 3 AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 23 79) Ten samples of size four were taken from a process, and their weights measured. The sample averages and sample ranges are in the following table. Construct and plot an x-bar and R-chart using these data. Is the process in control? Sample 1 2 3 4 5 6 7 8 9 10 Mean 20.01 19.98 20.25 19.90 20.35 19.23 20.01 19.98 20.56 19.97 Range 0.45 0.67 0.30 0.30 0.36 0.49 0.53 0.40 0.95 0.79 Answer: x-doublebar value X-bar 20.024 R bar 0.524 Upper control limit Center line Lower control limit 20.406 20.024 19.642 Range 1.196 0.524 0 24 The x-bar chart is out of control because samples 6 and 9 are outside of the control limit, and therefore the process is out of control. Diff: 3 AACSB: Analytical thinking Objective: LO S6.3 Build x-charts and R-charts Learning Outcome: Apply basic statistical process control (SPC) methods 25 80) Larry's boat shop wants to monitor the number of blemishes in the paint of each boat. Construct a 3sigma c-chart to determine if their paint process is in control using the following data. Sample Number 1 2 3 4 5 6 7 8 9 10 Number of Defects 3 4 2 1 3 2 1 4 2 3 Answer: Total units sampled Total defects Defect rate, c-bar Standard deviation z value 10 25 2.5 1.581 3 Upper Control Limit Center Line Lower Control Limit 7.243 2.5 0 The process is in control. Diff: 2 Key Term: c-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 26 81) A bank's manager has videotaped 20 different teller transactions to observe the number of mistakes being made. Ten transactions had no mistakes, five had one mistake and five had two mistakes. Compute proper control limits at the 90% confidence level. Answer: A c-chart should be used, and from Table S6.2, the z-value = 1.65 The mean c-bar = [10(0) + 5(1) + 5(2)]/20 = 0.75 UCLc = 0.75 + 1.65 = 2.18 LCLc = 0.75 - 1.65 = -0.68 (or 0) Diff: 2 Key Term: c-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 82) A department chair wants to monitor the percentage of failing students in classes in her department. Each class had an enrollment of 50 students last spring. The number of failing students in the 10 classes offered that term were 1, 4, 2, 0, 0, 0, 0, 0, 0, and 3, respectively. Compute the control limits for a p-chart at the 95% confidence level. Is the process in control? Answer: From Table S6.2, the z-value = 1.96. The mean p-bar = [1 + 4 + 2 + 0 + 0 + 0 + 0 + 0 + 0 + 3]/(50 × 10) = 0.02. p= 0.0198 UCLp = 0.02 + 1.96(.0198) = 0.0589 LCLp = 0.02 - 1.96(.0198) = -0.0189 (or 0) Since the percent defects in classes 2 and 10 both exceeded 5.89%, the percentage of failing students is not in statistical control. The department chair should investigate. Diff: 2 Key Term: p-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 83) A city police chief decides to do an annual review of the police department by checking the number of monthly complaints. If the total number of complaints in each of the 12 months were 15, 18, 13, 12, 16, 20, 5, 10, 9, 11, 8, and 3 and the police chief wants a 90% confidence level, are the complaints in control? Answer: From Table S6.2 z = 1.65 c-bar = (15 + 18 + 13 + 12 + 16 + 20 + 5 + 10 + 9 + 11 + 8 +3)/12 = 11.67 complaints UCL= 11.67+ 5.6366 = 17.307 complaints LCL= 11.67 - 5.6366 = 6.033 complaints Since 18 and 20 complaints fall above the UCL, while 5 and 3 complaints fall below the LCL, the complaints are not in control. Diff: 2 Key Term: c-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 27 84) A retail store manager is trying to improve and control the rate at which cashiers sign customers up for store credit cards. Suppose the manager takes 10 samples, each with 100 observations. The p-bar value is found to be .05, and the manager does not want a lower limit below .0064. What z-value would this imply, and how confident can she be that the true lower limit is greater than or equal to .0064? Answer: Sigma-p = sqrt (.05*(1 - .05)/100) = .0217945. Using the equation that LCL = p-bar - z*sigma-p gives .0064 = .05 - z*(.0217945). Solving gives z=2. Using table S6.2 shows that for z = 2 the manager can be 95.45% confident. Diff: 3 Key Term: p-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 85) A retail store manager is trying to improve and control the rate at which cashiers sign customers up for store credit cards. The manager takes several samples of size 50. He finds that each sample of 50 contained 5 credit card signups on average. Find p-bar and 99.73% control limits. Answer: P-bar = 5/50 = .1 or 10%. Sigma-p = sqrt(.1(1-.1)/50) = .042426. z = 3 for the given confidence level. Using the equations for control limits gives UCL= .1 + 3(.042426) = .22728 LCL= .1 - 3(.042426) = -.02728 and since a control limit cannot be negative round up to 0. Diff: 2 Key Term: p-chart AACSB: Analytical thinking Objective: LO S6.5 Build p-charts and c-charts Learning Outcome: Apply basic statistical process control (SPC) methods 28