Acceptance Sampling Plans and Control Charts for Quality Control in Manufacturing - Prof. , Exams of Systems Engineering

Download Acceptance Sampling Plans and Control Charts for Quality Control in Manufacturing - Prof. and more Exams Systems Engineering in PDF only on Docsity! ISE 4404 Statistical Quality Control Fall 2008 Old Test File Tests 1996-99 [ home | syllabus | test schedule | homework sets | homework solutions] [ tests 00-06 | tests 96-99 | tests 88-94] [ Probability Tables | Numerical Approximations | Documents ] Fall 1999 Test I Problem 1 (30 points) An item by item sequential sampling (SPRT) acceptance sampling plan by attributes is to be used to monitor the quality of incoming components from a supplier. The O. C. function objectives for the plan are p1=0.0125, p2=0.06, =0.08, and =0.04. a. Identify the sampling plan that meets the stated objectives. b. State how the plan is operated. c. For the plan identified in part (a), compute the smallest number of observations for which having 1 non-conforming item implies lot acceptance. The do the same for 2 and for 3 non-conforming items. d. Compute the probability of acceptance and the value of the ASN under the plan of part (a) for lots of incoming components that come from a population that has proportion non- conforming p=0.015. e. For the plan in (a), the coordinate point (81,3) is in the rejection region. What is the probability that sampling from a lot from a population having p=0.04 terminates at that point. Fall 1999 Test II Problem 1 (28 points) Suppose you work for a firm that receives two shipments per day of an important component of its product. The component is expected to meet the specifications U=28.03 mm and L=26.97 mm and  is know to be 0.2 mm. The acceptance sampling plan that is to be used to evaluate incoming lots of the component is to have =0.05 when p=p2=0.045 and  approximately equal to 0.075 when p=p1=0.011. a. What is the minimum proportion non-conforming achieveable under the defined specifications? b. Identify the proceedure 1 sampling plan that meets the indicated requirements. c. Compute the adjusted value of the producer's risk parameter. d. If  were unknown, what would be the proceedure 1 sampling plan that would be used for this case? What would be the adjusted value of the producer's risk for this plan? Problem 2 (20 points) A variables acceptance sampling plan with respect to the proportion non-conforming is needed for a product for which the standard deviation in the quality measure is unknown and the quality specification is defined by U=262.5 gm. Lots of N=3000 are to be evaluated and the sampling plan objectives are to have =0.05 when p=p2=0.042 and  approximately 0.075 when p=p1=0.008. a. Identify the proceedure 1 sampling plan that most closely meets thatstated objectives. b. Identify the corresponding proceedure 2 plan. c. Compute the adjusted value of  for each proceedure. d. If a recent sample displayed a sample mean of 253.44 and a sample standard devistion of 4.23, what decision is indicated by each of the two proceedures for the lot from which that sample was taken? Problem 3 (28 points) A variables acceptance sampling plan relative to the mean tensile strength is needed for a product for which 1=425 kg/cm2 and 2=410 kg/cm2 The plan should have  approximately equal to 0.075 and =0.04. The best available estimate for the unknown standard deviation in tensile strength is 6.5 kg/cm2. a. Identify the sampling plan that meets the stated objectives. b. State how the plan is operated. c. Compute the adjusted valus of the producer's risk parameter. d. Compute the probability of acceptance for lots from a population having =420 kg/cm2. e. Suppose a recent sample displayed a sample mean of 417.92 kg/cm2 and a sample standard deviation of 6.84 kg/cm2. What decision was made for the lot from which that sample was taken? Problem 4 (24 points) Suppose you work for a firm that purchases daily lots of 2800 copies of a component of its product and that product design requirements imply a desirable unit height of 1 = 18 cm while 2 = 17.2 cm. The component height is known to display a standard deviation of 0.5 cm. For this component a new type of acceptance sampling plan is to be used - a sort of double sampling plan. The plan is to be operated as follows: Step 1 A random sample of n items is to be selected and each item is to be numbered. Step 2 Each of the odd numbered items is inspected and its quality is to be measured. If the average of these quality readings exceeds or equals a criterion value, say , the lot is accepted and no further inspection is performed. If this condition is not satisfied, go to step 3. Step 3 A second sample average is computed using the even numbered items of the sample. If this average equals or exceeds , the lot is accepted and otherwise the lot is rejected. It has been decided to have =0.04 and  approximately equal to 0.075. a. Construct the O. C. function for this type of sampling plan. b. Analyze the O. C. function to obtain expressions for the sample size and criterion values. c. Solve your expressions for the values of n and  that meet the stated objectives. d. Compute the adjusted value of the producer’s risk parameter. Fall 1999 Test III Problem 3 (30 points) e) For the plan in (a), the coordinate point (39,3) is in the rejection region. What is the probability that sampling from a lot from a population having p=0.02 terminates at that point. Fall 1998 - Test II Problem 1 (30 points) Suppose you work for a firm that purchases daily lots of N=2400 copies of a component of its product and that product design requirements imply a quality specification of U=14.25 cm for component height which has =0.05 cm. An acceptance sampling plan with respect to the proportion non-conforming is to be used to evaluate incoming components and is to have =0.04 when p=p2=0.045 and  approximately 0.08 when p=p1=0.008. a. Determine the procedure 2 sampling plan that meets the indicated needs and state how the plan is operated. b. Compute the adjusted value of the producer’s risk parameter. c. If it were decided to use a single sample attributes plan with the smallest appropriate value of c, what plan would be used and what producer’s risk probability would be realized? d. If the plan is defined relative to the mean height, what are the values of 1 and 2? e. State an algebraic relationship between k, the criterion distance when sampling with respect to the proportion non-conforming, and x a , the acceptance criterion when sampling with respect to the mean. f. Suppose that in a recent sample taken using the plan in part a, the sample mean was 14.14 cm. What estimate of the proportion non-conforming does this value imply and what decision would have been taken concerning the lot from which the sample came? Problem 2 (30 Points) A variables acceptance sampling plan with respect to the proportion non-conforming is needed for evaluation of a product that is purchased in lots of N=3600 and has U=25.04 mm, L=23.96 mm, and =0.20 mm. The plan objectives are to have =0.04 when p=p2=0.06 and  approximately equal to 0.08 when p=p1=0.012. a. For the defined quality specifications, what is the minimum attainable proportion non- conforming? b. Identify the proceedure 1 sampling plan that meets the indictaed requirements and state how it is operated. c. Compute the adjusted value of the producer’s risk parameter. d. Suppose that the standard deviation were not known. What is the corresponding proceedure 1 acceptance sampling plan that should be used? e. What is the adjusted value of  for the plan of part d? Problem 3 (20 points) A variables acceptance sampling plan relative to the mean quality is needed for a produsct for which 1 = 7.0 cm, 2l = 7.06 cm, 2u = 6.94 cm and  = 0.04 cm. Identify the plan that will have =0.08 and  approximately equal to 0.04. Compute the adjusted value of  and the value of the O. C. function for a population having mean quality 7.02 cm. Problem 4 (20 points) Identify the variables acceptance sampling plan relative to the mean quality that has =0.04 and  approximately equal to 0.08 for a product having 1 = 24 kg/cm2, 2 = 23.6 kg/cm2, and for which the best available estimate of the unknown standard deviation is  ^ = 0.25 kg/cm2. Compute the adjusted value of the producer’s risk for the plan. If a recent lot of the product yielded a sample for which s=0.022 and x =23.82, what decision was made relative to the lot? Fall 1998 - Test III Problem 2 (30 points) In an effort to construct an attributes control chart for a production process, hourly samples of output product have been obtained over the most recent 48 production hours. If the data obtained displayed 25 values of p in excess of the mean, what are the 0.05 and 0.005 probability limits for the numbers of runs in the data set? The performance objectives for the chart are to have an in-control ARL of at least 125 and an out of control ARL of at most 2.5 when p increases to 1.8 times the in-control value. If the in-control process average is 0.015, what control limits should be used? Problem 3 (25 points) Suppose you work for a firm that uses a p type control chart to monitor a process that generates a component sold in lots of N=3600 to an appliance manufacturer. When the production process is in control, the component has a thickness tolerance of   2.43x and the component population is considered to be of desirable quality when it meets these specifications. The p chart is operated with a sample size of n=256, a lower control limit of 0.0 and kp set so that the false signal (false alarm) rate is no greater than once per 160 samples. The customer for the component uses a double sampling plan with n2=n1 to evaluate incoming lots. That plan is designed using the in-control value of the proportion non- conforming as p1 and six times that value as p2 and is based on having =0.10 and  approximately equal to 0.05. a. Identify the center line and control limits for the p chart. b. If the production process experiences a change that causes a 120% increase in that proportion non-conforming, what is the probability that the control chart signals the change within three samples following the shift? c. Assuming that of the acceptance sampling plans that meet the quality objectives, the one with the larger value of c1 is used, identify the acceptance sampling plan and the adjusted value of the producer’s risk parameter. d. Determine the acceptance probability and the ASN under complete inspection the acceptance sampling plan if it is used to evaluate lots generated when the process is out-of-control as described in (b). Problem 4 (25 points) A manufacturer of home appliances purchases a component of one of its products in lots of N=3600. Product design implies a need to confirm compliance with both variability and location of the key component dimension so for each arriving lot the following acceptance sampling scheme has been defined. Step 1: A variables sampling plan relative to the product standard deviation is used to verify that component variability is satisfactory. This plan has 1 = 0.5 cm, 2 = 1.15 cm,  = 0.04, and  approximately equal to 0.075. Step 2: For lots found to have satisfactory variability, it is assumed that  is known and is equal to 1 and an SPRT is used to verify that the mean quality is acceptable. This plan has =0.8, =0.04, 1=24.25 cm, and 2= 23.75 cm. a. Identify the acceptance sampling plan that is used to evaluate component variability, state how the plan is implemented, and compute the adjusted value of the producer’s risk parameter for the plan. b. Determine the SPRT that is used to monitor mean quality and state how the plan is used. c. For the SPRT of part (b), compute the Pa(=24.10) and ASN(=24.10). Extra credit (8 points) Suppose the supplier of the component experiences a change in the process that generates the component so that lots sent actually have =0.70 and =24.05, what is the net probability that a lot passes the two tests and is accepted. (Hint: Changing  while holding the criterion lines fixed implies that the sampling error probabilities for the SPRT change.) Fall 1998 - Test IV Problem 1 ( 30 points) Suppose you work for a firm that uses a coordinated quality control system in which purchased machining blanks are inspected for dimensional compliance relative to the specification limits of U=18.77 mm and L=18.23 mm (x is known to be 0.10 mm). The acceptance sampling plan is defined to have =0.05 when p=p2=0.04 and  approximately equal to 0.08 when p=p1=0.01. Accepted lots of N=3600 blanks are cut using a machining process that when it is in control, damages 0.2% of all pieces handled. This process is monitored using a p type control chart having LCLp=0.0 and designed to yield an in-control ARL( p )  200 and ARL(p=3 p )  1.75. a. Assuming the use of Procedure 1, identify the acceptance sampling plan that is to be used and the adjusted value of the producer’s risk parameter. b. What value of the mean of the population of blanks corresponds to p=p2? c. What are the center line and control limits for the p chart? Problem 2 (30 points) X and s type control charts are to be used to monitor a production process for which the product specs are U=33.5 and L=30.5. The charts are to be constructed so that LCLs=0 and: AR xL ( x )  125 AR xL ( x +1.4x)  1.8 ARLs(xc)  170 a. What are the values for the chart parameters xk , ks and n? b. If a sequence of samples were taken and found to have x =32.0 and s =0.0552, what are the control limits for the two charts? c. What is the process capability index kpC when the process is in control? d. If the process should experience a change the result of which is a decrease in the mean of 1.5 x and a simultaneous increase of 60% in x, what is the probability that at least one of the two control charts will signal the change within three samples after the change. Problem 3 (20 points) It has been suggested that the p type (attributes) control chart could be replaced by an alternate form defined for the number of non-conforming items observed in a sample. This would work as follows: At each sampling time, a sample of n items would be inspected and classified as conforming or non-conforming. For the ith item inspected, xi = 0 if it is conforming and xi=1 if it is non-conforming. The total number of items found to be non-conforming, say W, where: Step 4 A new sample of 5 copies of part b only is inspected. If that second set of part b pieces displays an average thickness that is  22.325", the lots of both parts are accepted and otherwise they are both rejected. It is known that a=0.09" and b=0.12" and sampling error probabilities of =0.10 when both parts have their least tolerable mean thickness and  approximately 0.05 when both parts have mean thickness on target. The target values for the two parts are a1= 0.72" and b1= 2.28". Values of a1= 0.80" and b1= 2.40" have also been identified. a. State an expression for the O. C. function for the sampling plan. b. Use the expression to obtain appropriate values for n and T. b. Compute the adjusted value of the producer's risk parameter. Fall 1997 - Test III Problem 2 (24 points) Suppose you work for a firm that wishes to use a coordinated quality control system. Consequently, lots of N=3600 incoming work pieces are subjected to sampling inspection to assure uniformity as reflected in an acceptable standard deviation in hardness. Accepted lots are input to a production process that is monitored using a p type control chart. a. For the acceptance sampling plan, management has specified that  =0.075 when  =  2=0.72 and that  be approximately 0.075 when  = 1=0.4. What sampling plan should be used and what is the adjusted value of the producer's risk parameter. b. The effectiveness of the production process is a direct function of workpiece hardness so that all pieces with hardness values in the interval +/- 2.345  are conforming and all others are non-conforming. The p chart for the process is to have kp=2.51 and n=225. What are the values for the center line and the control limits of the chart? c. For the control limits computed in part b, what is the probability that a process change that results in a 50% increase in p will be recognized within 3 sample periods of the process shift? Problem 4 (24 points) During the Crimean War, the Turkish munitions manufacturing facility outside of Constantinople employed a p type control chart to monitor quality on its rifle bullet line which produced bullets for the Sardinian and British allies. Initially management specified that kp=3.0 was to be used and as a result the control limits were UCLp =0.025 and LCLp=0.0. After the chart was in use, an IE computed ARL(0.02) and found it to be 4.21 which was considered to be too high. a. What was the sample size used for the chart? b. What was the value of p for the process? Since the ARL(0.02) was too high, the IE was asked to select a revised value of kp and an increased sample size so that using the same value of p as the in control value, the chart would have LCLp =0.0, ARL( p ) 200, and ARL(0.02) 2.25. c. What revised values of kp and n should the IE have recommended? d. What value of UCLp would then be obtained? Fall 1997 - Test IV Problem 1(22 points) Suppose you work for a firm that produces fasteners from metal blanks that are purchased in lots of N=4000. The blanks are heated, formed, notched and trimmed on a single pass line that when it is in control damages 1.0% of the blanks regardless of their initial quality. Management has defined a coordinated quality control strategy in which each of the 8 incoming lots each day is subjected to a double sample acceptance sampling plan with curtailment. The plan is designed to have n2 =n1, =0.10 when p= p2 =0.05 and  near 0.05 when p= p1 =0.0075. Of the sampling plans that meet these objectives, the one with the larger value of c1 is used. The fastener production line is monitored with a p-type control chart for which the in-control state is defined assuming an input quality level of p1. The chart is constructed to have ARL( p ) 150 and ARL(1.9 p ) 3.5. Finished fasteners are collected into lots of 1200 and shipped without further inspection. a. What acceptance sampling plan is used for the blanks? b. What are the values of the center line and UCL p for the control chart? (Hint: Assume from the start that LCLp=0.0.) c. Suppose that the supplier's process experiences an undetected shift to p=2p1. What is the acceptance probability for the acceptance sampling plan and assuming your firm's process settings are unchanged, what is the ARL for the control chart when it is applied to the product generated using the supplier's poorer quality material? Problem 2 (26 points) Variables control charts of the X and R type are used to monitor disc brake caliper thickness. If the charts are designed so that when the process is in control, ARL( = x ) 200.0 and when it is out of control due to a shift only in the mean to a value of x +0.75x, the chart has ARL( x +0.75x) 5. For the R chart, the control limits are to be set so that ARL(1.5 x)3.2. a. What are the parameters of the X chart? b. What value of kR should be used? c. If a sequence of samples were taken and found to have x =6.00 mm and R =0.0507 mm, what are the control limits for the two control charts? d. If a hydraulic leak in the caliper press results in a shift in the mean thickness to x +1.2x and a simultaneous increase of 80% in the process standard deviation, what is the signal probability for the combined use of the two charts on any sample following the change? Problem 3 (32 points) Suppose you work for a firm that is planning to use X and s type control charts to monitor a process which generates a product for which the specification limits are L=11.80 and U=12.20. The X chart is to be defined so that when the process is in control ARL( = x )160.0. Also, for the out of control state in which x is unchanged but the process mean has shifted to a value of x -1.6x, chart performance should provide ARL( x -1.6x) 1.6. In the case of the s chart, the control limits are to be set so that ARL(1.6x) 1.3. a. What are the parameters of the X chart? b. What value of ks should be used? c. If a sequence of samples were taken and found to have x =12.00 and s =0.0737, what are the control limits for the two control charts? d. What are the proportion non-conforming and the process capability index, Cpk, when the process is in control? e. If the process should experience a change with the consequence that the mean increases to 12.048 and x increases by 50%, what is the probability that at least one of the two control charts signals within three samples after the process change? Problem 4(20 points) Suppose you are working for a firm that plans to implement exponentially weighted moving average control charts using =0.25. Suppose further that the limiting value of the false alarm probability is to be 0.008 and that this value is to be used to select kT. In addition, the sample size is to be selected so that the limiting value of the out-of-control signal probability is 0.90 when the process mean shifts to a value of  +0.75 x. What values should be used for kT and n? If previous data analysis has indicated that ̂ =225.0 and x̂ =5.0, what are the limiting values of the control limits for the chart? [ home | syllabus | test schedule | homework sets | homework solutions] [ tests 00-06 | tests 96-99 | tests 88-94] [ Probability Tables | Numerical Approximations | Documents ] Fall 1996 - Test I Problem 1 (32 points) An item by item sequential sampling (SPRT) acceptance sampling plan by attributes is to be used to monitor the quality of lots of incoming items from a supplier. The O. C. function objectives for the plan are p1=0.008, p2=0.042,  =0.08, and  =0.04. a. Identify the sampling plan that meets the stated objectives and state how it is operated. b. For the plan defined in (a), compute the minimum sample size for acceptance and the minimum sample size for rejection. c. For the plan defined in (a), compute the probability of acceptance and the ASN for lots from a population having a proportion nonconforming of p=0.012. d. For the plan defined in (a), compute the probability that when evaluating lots from a process having proportion nonconforming of p=0.012, the inspection process terminates at the point (139, 1) which is in the acceptance region and at the point (20, 2) which is in the rejection region. Problem 2 (30 points) Consider a firm that is planning to implement a double sample acceptance sampling plan using n2=2n1 and complete inspection. The O. C. function objectives are to have  =0.10 when p2=0.042 and  approximately equal to 0.05 when p1=0.008. a. Identify two candidate sampling plans that meet these requirements. b. For the plan in part (a) having the larger value of c1, compute the adjusted value of and the ASN(p1 ) under complete inspection. c. For the plan in part (a) having the larger value of c1, what is the conditional probability of acceptance for lots from a population having p=p1, given that c1+2 nonconforming items are found on the first sample? d. Suppose the plan in part (a) with the larger value of c1 were used with curtailed inspection. What is the probability that curtailment occurs upon the inspection of the 51st item of the second sample during the inspection of lots having proportion nonconforming p2? e. (4 points extra credit) Suppose the plan in part (a) with the larger value of c1 were used with curtailed inspection and that in a recent lot c1+1 nonconforming items were found in the first sample. What is the probability that curtailment occurred at all during the inspection of the second sample if the lot had proportion nonconforming p2? Problem 3 (18 points) Suppose you work for a firm that is planning to implement a single sampling acceptance sampling plan with  =0.05 when the proportion nonconforming is p2=0.075 and  approximately 0.05 when the proportion nonconforming is p1=0.010. Administrative considerations dictate that a value of c=2 be used. process damages 1.3% of all parts regardless of their quality. When the rate of damage to incoming parts remains unchanged, the process is considered to be in control. The objectives for the control chart are to have ARL( p )150 and ARL(1.5 p )8. What are the control limits for the p chart? If the supplier of the input work pieces reduces the variation in thickness so that x is 0.18 mm, what is the change in each of the signal probabilities? If the work piece supplier informed you of the reduction in x and you adjusted your control chart for the change, what would be the change in inspection volume for the control chart? Problem 4 (25 points) Suppose you are planning to construct attributes control charts for your firm and that you have inspected samples from a process every 4 hours during the most recent week (176 hours) of production. The resulting sample proportions nonconforming display p =0.015 and 21 of the sample values exceed the mean value. a. What are the 0.05 and the 0.005 probability limits for the number of runs above and below the mean for this data? b. If your calculations indicate that p= 0.00608, what sample size was used to compute the sample proportions nonconforming? c. Suppose that instead of constructing a control chart for p, the sample proportion non conforming, you simply defined a control chart for the number of sample items found to be nonconforming. Of course, to do this you would use the Binomial distribution to define the chart (and would approximate Binomial probabilities with Poisson). Assuming that the lower control of the chart is to be set to zero, state the equations for the control limits and the O. C. function for your chart. d. For the plan identified in part c, determine the upper control limit in order to have ARL(2 p )5. Then compute the ARL( p ). Fall 1996 - Test IV Problem 1 (25 points)Suppose you work for a firm that is planning to use X and s type control charts to monitor a process which generates a product for which the specification limits are L=11.099 and U =2.901. Hourly samples of n=5 have been taken from the process in order to construct the charts. The resulting 60 samples display x =12.0 and s =0.329 and it is considered that these values represent the in control behavior of the process. a. What are the process capability index, Cpk, and the proportion nonconforming when the process is in control? b. While a sample size of n=5 was used to identify the process behavior, this sample size is not necessarily best for the continuing monitoring of the process. Parameters for the X chart are to be selected as follows. kx is to be chosen so that ARL( = x )200.0 and the sample size is to be set so that ARL( = x +1.3 x ) 2.5. What are the control limits for the X chart? c. For the s chart, ks is to be selected so that the in-control ARL is 150. What are the control limits for the s chart? d. For the process out of control state corresponding to a decrease in the mean to a value of x 0.9x and a simultaneous 50% increase in the process standard deviation, what is the proportion nonconforming? What is the probability that the combined control charts signal the out-of-control situation on any sample after the occurrence of a shift to this state? Problem 2 (35 points) In order to construct X and R type control charts, 24 samples of 6 items each have been inspected and the following data have been obtained (note that x =27.477): 27.683 27.686 27.613 27.357 27.699 27.584 27.171 27.434 27.075 27.717 27.376 27.975 26.984 27.778 27.103 27.225 27.254 27.465 27.738 27.241 27.486 27.419 27.255 27.294 27.820 27.070 27.160 27.706 27.514 27.360 27.199 27.825 27.767 27.446 27.436 27.336 27.327 27.517 27.482 27.541 26.957 27.289 27.661 27.121 27.846 27.349 27.167 27.159 27.667 27.063 27.179 27.539 27.979 27.187 27.540 27.940 27.529 27.422 27.401 27.703 27.545 27.272 27.649 27.326 27.739 27.613 27.528 27.726 27.215 27.693 27.946 27.666 27.872 26.844 27.715 27.700 27.528 27.017 27.625 27.370 27.121 27.332 27.536 27.636 27.453 27.483 27.558 27.264 27.738 27.631 27.394 27.371 27.778 27.457 27.381 27.957 27.274 27.177 27.706 27.672 28.110 27.465 27.376 27.942 27.501 27.520 27.738 27.512 27.218 27.316 27.335 26.967 27.312 27.736 27.633 27.563 27.838 28.068 27.685 27.630 27.389 27.588 27.565 27.455 27.228 27.162 27.338 27.325 27.186 27.406 27.285 27.468 27.952 27.403 27.307 27.364 28.174 27.800 27.092 27.311 27.675 27.422 26.952 27.579 The value of kx is to be selected so that ARL( = x )83 1/3 and the ongoing application of the chart may use a different sample size. In particular, the sample size is to be selected so that ARL(= x 1.25x )3 1/3. Once these parameters are selected, the value of kR is to be set to 2.60. a. What are the values of the center line and the control limits of the R chart? b. What are the values of the center line and the control limits of the X chart? c. If the process is in control, what is the combined ARL for the two charts together? d. If there is a change in the process such that the mean shifts to x +0.80x and the process standard deviation increases by 60%, what is the probability that combined use of the two charts yields a signal within three samples following the change? e. If the primary customer for the product accepts your estimate for and evaluates lots of N=2000 units of the product using a  known variables acceptance sampling plan to control for the mean with  =0.04 when = x x or = x +x and  approximately equal to 0.075, what is the sampling plan used and what is the probability of acceptance for lots of product generated when the process is out of control? Problem 3 (24 points) Suppose you are working for a firm and have been asked to develop a variables type control chart using an SPRT format. In particular, assume that on the basis of historical data, the process standard deviation for the quality measure of interest is believed to be known to be x=0.40. The sample statistic to be monitored and accumulated at each point in time is the sample mean for which the sample size is m. As long as the accumulated sample information falls with the decision interval, the process is considered to be in control. If the sampling process terminates in the SPRT acceptance region, the process is considered to have remained in control and the accumulation of the sample statistic is restarted. If the sampling process terminates in the SPRT rejection region, this is interpreted as a signal. a. Being sure to explain your reasoning fully, suggest algebraic expressions for the incontrol and out of control ARL for the chart. b. Assuming  =0.005,  =0.075 and m=4, compute the decision limits for the case in which the incontrol mean is 1=20.0 and the outofcontrol mean values are 2u=20.25 and 2l=19.75. Give an explanation of how the decision limits are to be applied. c. For the plan defined in (b), use the equations defined in (a) to compute the incontrol and out of control ARL values. Problem 4 (16 points) Suppose you are working for a firm that plans to implement exponentially weighted moving average control charts using  =0.20 and samples of n=5. What value of kT should be used if the limiting value of the Type I sampling error probability is set at 0.0076? If this value of k T is used and if previous data analysis has indicated that x =8.0 and x=0.08, what are the control limits for the chart? Extra Credit (4 points) Identify the person (full name) who: developed the t distribution developed the SPRT originally defined control charts constructed the standardized range distribution [ home | syllabus | test schedule | homework sets | homework solutions] [ tests 00-06 | tests 96-99 | tests 88-94] [ Probability Tables | Numerical Approximations | Documents ]