Sample Mean Range

Transcription

1 Lesson 14 Statistical Process Control Homework Solved Problem #2: see textbook Solved Problem #4: see textbook Solved Problem #5: see textbook Solved Problem #6: see textbook (manual problem) #1: Checkout time at a supermarket is monitored using a range and mean chart. Six samples which contain 20 observations per sample have been collected and the sample means and sample ranges have been computed as shown below. Sample Mean Range Perform the manual calculations necessary to answer the following questions. a. What is the sample size? b. Is the variability of the checkout time known or unknown? c. Which chart is used to analyze the checkout time variability? d. Which chart is used to analyze the checkout time? e. Should the range or mean chart be analyzed first? Explain your answer. Analyze the range chart by answering the following questions. f. What is the grand range? g. Calculate the centerline, upper and lower control limits for the range chart? 1

2 h. Plot the range chart showing the centerline, upper and lower control limits. i. Are the sample ranges within the control limits? j. Count the number of A/B and U/D runs for the range chart. k. What is the expected number of A/B runs? l. What is the variability of the expected number of A/B runs? m. What is the test statistic for the expected number of runs? n. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? Why? o. What is the expected number of U/D runs? p. What is the variability of the expected number of A/B runs? q. What is the test statistic for the expected number of runs? 2

3 r. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? Why? s. Is the variability of the checkout times in control (i.e is the range chart in control)? Why? Analyze the mean chart by answering the following questions. t. What is the grand mean? u. Calculate the centerline, upper and lower control limits for the mean chart? v. Plot the mean chart showing the centerline, upper and lower control limits. w. Are the sample means within the control limits? x. Count the number of A/B and U/D runs for the mean chart. y. What is the expected number of A/B runs? z. What is the variability of the expected number of A/B runs? aa. What is the test statistic for the expected number of runs? 3

4 bb. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? Why? cc. What is the expected number of U/D runs? dd. What is the variability of the expected number of A/B runs? ee. What is the test statistic for the expected number of runs? ff. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? Why? gg. Are the average checkout times in control (i.e. is the mean chart in control)? Why? Based on your analysis is the checkout process in control? Why? #2: The Bayfield Mud Company supplies railroad boxcars of 50 pound bags of mud treating agents to the Wet-Land Drilling Company. Mud treating agents are used to control the ph and other chemical properties of the cone during oil drilling operations. Wet-Land has complained to Bayfield that it s most recent shipment of bags was underweight by 5%. This is a problem which Wet-Land needs corrected because the use of under weight bags may result in poor chemical control during drilling which may hurt drilling efficiency resulting in serious economic consequences. Afraid of losing a long time customer, Bayfield immediately began investigating their production process. Management suspected that the causes of the problem were their recently added third shift and the fact that all three of the shifts were under pressure to increase output to meet increasing demand for the product. Their quality control staff began taking random samples of the output of 3 bags per hour. They observed the process for 36 hours beginning at 6am. 4

5 To avoid manually entering the data into the templates it can be copied and pasted from Data Sets on the Lesson Page. Use copy, paste special, values to transfer the data to the appropriate SPC template worksheet.. Answer the following questions. a. What is the sample size? Sample Time Bag 1 Bag 2 Bag 3 1 6am am am am am am noon pm pm pm pm pm pm pm pm pm pm pm pm am am am am am am am am am am am noon pm pm pm pm pm b. Is the variability of the checkout time known or unknown? 5

6 c. Which chart is used to analyze the variability in the bag filling process? d. Which chart is used to analyze the amount of fill in each bag? e. Should the range or mean chart be analyzed first? Explain your answer. Analyze the range chart by answering the following questions. f. What is the grand range? g. Calculate the centerline, upper and lower control limits for the range chart? 3σ Control Chart Settings UCL Centerline LCL h. Plot the range chart showing the centerline, upper and lower control limits. i. Are the sample ranges within the control limits? j. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? k. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? Why? l. Is the variability of the checkout times in control (i.e is the range chart in control)? Why? 6

7 Analyze the mean chart by answering the following questions. m. What is the grand mean? n. Calculate the centerline, upper and lower control limits for the mean chart? 3σ Control Chart Settings UCL Centerline LCL o. Plot the mean chart showing the centerline, upper and lower control limits. p. Are the sample means within the control limits? q. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? r. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? s. Are the average checkout times in control (i.e. is the mean chart in control)? Why? Based on your analysis is the bag filling process in control? Why? What should management do? Based on the mean control chart, provide an explanation of what may be occurring. 7

8 #3: Optical scanners are used in all supermarkets to speed the checkout process. Whenever the scanner fails to read the bar code on the product, the cashier is required to manually punch the code into the register. Obviously, unreadable bar codes slow the checkout process. SPC is used to determine whether the scanner is working properly. Once a day at the check out counter, a sample of 500 scans is taken and the number of times the scanner is unable to read the bar code is determined. The sampling process is performed automatically by the cash register. To avoid manually entering the data into the templates it can be copied and pasted from Data Sets on the Lesson Page. Use copy, paste special, values to transfer the data to the appropriate SPC template worksheet. Answer the following questions. a. What is the average sample size? Sample Sample Size Incorrect b. How many samples were taken? 8

9 c. Which chart should be used to determine whether the scanning process is in control? Explain your answer. d. What is the estimate of the expected proportion of scans which are unable to read the bar code?. e. Calculate the centerline, upper and lower control limits for the P chart? 3σ Control Chart Settings UCL Centerline LCL f. Plot the P chart showing the centerline, upper and lower control limits. g. Are the sample proportions within the control limits? h. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? i. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? j. Is the optical scanning process in control? Why? What should management do? 9

10 #4: Given the following control chart. Control Chart UCL, Centerline, LCL, Mean Known Variation Answer the following questions. a. How many observations are in the control chart? b. Count the number of A/B and U/D runs for the control chart. c. What is the expected number of A/B runs? d. What is the variability of the expected number of A/B runs? e. What is the test statistic for the expected number of runs? 10

11 f. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? Why? Are three too few, just the right amount, or too many runs? g. What is the expected number of U/D runs? h. What is the variability of the expected number of A/B runs? i. What is the test statistic for the expected number of runs? j. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? Why? k. Is the process in control? Why? #5: An appliance manufacturer wants to contract with a repair shop to handle authorize repairs in Indianapolis. The company has an acceptable range of repair time of 50 minutes to 90 minutes. Two vendors have submitted bids for the work: A and B. In test trials, vendor A had a mean repair time 74 minutes with a standard deviation of 4.0 minutes, and vendor B had a repair time of 72 minutes with a standard deviation of 5.1 minutes. a. What is the tolerance range the manufacturer will accept? Answer the following questions assuming the appliance manufacture wants the repair shop to meet 3-sigma quality control. b. What is the process capability for vendor A? Vendor B? c. Which company meets the appliance manufacturer s tolerance specifications? Why? 11

12 d. Assuming the appliance manufacturer will only choose one vendor and all other factors are equal (e.g. price, delivery time, etc), which vendor should the company choose? Why? Answer the following questions assuming the appliance manufacture wants the repair shop to meet 6-sigma quality control. e. What is the process variability a vendor should have if their average repair time is 70 minutes? Why? 12