ISyE-ME512_Exam 2B

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University of Wisconsin, Madison *

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512

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Mechanical Engineering

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Dec 6, 2023

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Exam #2B: ISyE/ME 512 Inspection, Quality Control and Reliability Spring 2023 Monday, 4/3/2023 NOTES: 1. One-page note sheet (double sided) is allowed. Calculators are allowed. Cell phone and computer are not allowed during the exam. 2. Write your answer on this exam paper. DO NOT use your own paper. If you need more space, write on the back side of this exam paper. 3. Show your steps. Answers with only final result will NOT earn full mark of that problem. 4. The test is worth 100 points. The assigned points for each question are given beside it. 5. There are 15 pages in this exam. Please show the steps and procedures in your solution in order to get full points. Name(print): Student ID: Signature, Date: =============================================================== ISyE/ME 512_Exam #2 Cover Page
Question 1: An X-bar control chart with a sample size of n=4 is used to monitor the process mean. Two decision rules are given here: Rule 1: If a single point falls outside the 3-sigma control limits, conclude that the process is out of control. Rule 2: If one or more of the next five samples yield values fall between 2-sigma and 3-sigma control limits. a) What is the Type I error using rule 1 alone? b) What is the Type I error using rule 2 alone? c) What is the overall Type I error based on the two rules, given that the two rules are independent? d) If the process has a mean shift of 1.5 σ , what is the Type II error using Rule 1? ISyE/ME 512_Exam #2 1 of 15
(Additional page for Question 1) ISyE/ME 512_Exam #2 2 of 15
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Question 2: Samples of n = 6 items each are taken from a process at regular intervals. A quality characteristic is measured, and ´ x and R values are calculated for each sample. After 50 samples, we have i = 1 50 ´ x i = 2000 , i = i 50 R i = 200 Assume that the quality characteristics is normally distributed. a) Compute the control limits for the ´ x and R control charts. b) All points on both control charts fall between the control limits computed in a). What are the natural tolerance limits of the process? c) If the specification limits are 41 ± 5.0, what are your conclusions regarding the ability of the process to produce items within these specifications? d) Assuming that if an item exceeds the upper specification limit it can be reworked, and if it is below the lower specification limit it must be scrapped, what percentage scrap and rework is the process producing? ISyE/ME 512_Exam #2 3 of 15
(Additional page for Question 2) ISyE/ME 512_Exam #2 4 of 15
Question 3: A process is being monitored with a fraction nonconforming control chart. The process average has been shown to be 0.081. 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 should suddenly shift to 0.1, what is the probability that the shift would be detected on the first subsequent sample? c) What is the probability that the shift in b) would be detected on the first or second sample taken after the shift? ISyE/ME 512_Exam #2 5 of 15
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(Additional answer page for Question 3) ISyE/ME 512_Exam #2 6 of 15
Question 4: ( 20 points ) 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 were reported. a) What type of control chart would you recommend here, and how would you use it? b) Using two water heaters as the inspection unit, calculate the center line and control limits that are consistent with the past 22 days of inspection data. c) What is the probability of type I error for the control chart in b)? ISyE/ME 512_Exam #2 7 of 15
(Additional answer page for Question 4) ISyE/ME 512_Exam #2 8 of 15
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Question 5. ( 20 points ) Control charts for ´ x and s have been maintained on a process and have exhibited statistical control. The sample size is n = 4. The control chart parameters are as follows: ´ xChart s Chart UCL = 710 UCL = 18.08 CL = 700 CL = 7.979 LCL = 690 LCL = 0 a) Estimate the process parameters μ and σ. b) If the specifications are at 705 ± 15, and the process output is normally distributed, estimate the fraction nonconforming. c) For the ´ x chart, find the probability of a type I error, assuming σ is constant. d) Suppose the process mean shifts to 693 and the standard deviation simultaneously shifts to 12. Find the probability of detecting this shift on the ´ x chart on the first subsequent sample, and find the average run length. ISyE/ME 512_Exam #2 9 of 15
(Additional answer page for Question 5) Appendix ISyE/ME 512_Exam #2 10 of 15
ISyE/ME 512_Exam #2 11 of 15
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