MATHEMATICS 01-510-LW Business Statistics Martin Huard Fall 008 1. Suppose the statistics x = 14.1 and R= 17.5 are obtained from 50 samples of size 9. a) What is the centerline and the control limits for an x chart? b) What is the centerline and the control limits for an R chart?. If, when a production process is under control, it has been determined that µ= 76.1 and σ =.07, what is the centerline and the control limits, if the process is monitored by taking samples of size 8 for a) an x chart? b) an σ chart?. In order to ensure that a manufacturing process is under control, 50 samples of size 100 are taken and the number of defectives in each sample is counted. If the mean sample proportion is 0.01, what is the center line and the control limits for a p chart? 4. In an attempt to monitor customer satisfaction, an air line recorded the number of complaints received per day for the last 50 days. If in the past the average number of complaints was 7 per day, what is the center line and the control limits for a c chart? 5. Each day the quality control department at a candy factory takes a random sample of 5 chocolate bars and measures their weight, to see if the process is in control. Here are the results for the past 1 days. Day x R 1 6.8 65.8 65. 64.6 65.8 65.04.0 64.8 64. 65.4 64 64.7 1.4 64.1 65.6 64. 6.5 65.1 64.50.1 4 64.6 65.8 65.9 6.6 65.7 65.1. 5 64.7 66.4 65.5 64.7 65.4 65.4 6 65.7 64.6 64.4 66.1 64.8 65.1 1.7 7 66. 65.1 66.0 64. 65.7 65.46.1 8 65.4 66. 64.9 61.4 65.6 64.70 4.8 9 66.7 66.0 65. 65.9 65.6 65.88 1.5 10 6.4 67.0 66.4 66. 68.7 66.14 6. 11 66.6 6.9 66.4 66. 67.8 65.98 4.9 1 68. 65.7 67.5 66. 6. 66.0 6.0 a) Fill in the missing numbers in the last two columns. b) Make an x chart. Is the process in control? c) Make an R chart. Is the process in control?
6. A coffee machine fills a cup of coffee to a fixed volume. When the machine is in control, the mean volume is 50 ml with a standard deviation of.1 ml. To see whether the machine is in control, samples of 4 cups were taken each day for the past 0 days and their volume measured. Here are the results. Day x R 1 51.7 4.1 48.0 48. 47.75 8.6 48.9 45.7 5.7 4. 47.8 10.5 5.8 50.4 4. 49.5 49.00 11.5 4 5.5 47.6 49.8 56. 51.55 8.7 5 51.7 44.1 50.9 47.7 48.60 7.6 6 48.7 48.4 5.9 51.5 50.8 4.5 7 48. 50. 47.5 51.0 49.5.5 8 45.7 5.0 49.6 5.9 50.05 7. 9 57.6 60.9 56.9 54.8 57.55 6.1 10 64.7 54.0 59.4 57. 58.85 10.7 11 59.1 56.6 57. 61.4 58.58 4.8 1 54.6 60.5 55.5 6.0 58.15 7.4 1 55.0 54.8 67.4 54.7 57.98 1.7 14 6.6 58.9 60.1 55.9 59.8 6.7 15 55. 5.6 60. 60.4 57.5 6.8 16 57.8 6. 57. 59.9 59.0 4.9 17 61.9 60.5 55.4 57.9 58.9 6.5 18 55.1 59. 58.0 60.0 58.10 4.9 19 55.9 51.6 57. 54.1 54.70 5.6 0 56.0 5.6 59.8 5.0 55.60 6.8 a) Make an x chart using the known µ and σ. Is the process in control? b) Make an σ chart using the known µ and σ. Is the process in control? 7. The Canadian Math Delivery Service (CMDS) guarantees delivery of small packages by 9:00 AM the next working day anywhere in Canada. To monitor the efficiency of their system, CMDS took random samples of 00 packages during the last 15 working days and observed the following number of packages which were delivered after the deadline. Is the process in control? 4 1 6 9 7 19 5 6 8 10 6 8. The number of near misses recorded for the past 0 months at Zola airport are 5,, 1,, 4, 1,,, 0, 1,,,, 4,, 5,, 7,, and 7. Is the process in control? Fall 008 Martin Huard
9. A company that produces CD s for record company has been receiving complaints that a significant number of CD s are defective. To remedy the problem, the company decided to take random samples of 50 CD every hour and check them to see if they are defective or not. Here are the results from the first 40 hours. Is the process out of control? 6 1 17 17 10 16 16 11 5 6 10 14 14 5 11 11 14 15 11 10 11 1 7 15 9 7 10 14 1 17 9 14 16 7 0 1 18 11 10 10. The following data represents the number fires per week that were signaled to the Quebec City Fire Department for the past 5 weeks. Was the number of fires out of control at any time during the last 5 weeks? 9 4 1 10 5 10 1 10 1 15 10 16 9 11 1 9 15 7 1 10 14 9 7 6 11. A new industrial oven has been installed at Great Bakery. In order to test the distribution of the temperature inside the oven, an employee read the temperature in 7 different places inside the oven every 15 minutes for the first four hours. Here are the results (in degrees celcius). Time x R 8:00 A.M. 96 44 418 95 85 96 8 40.1 61 8:15 A.M. 94 89 401 78 405 409 404 97.1 1 8:0 A.M. 409 400 411 408 414 409 408 408.4 14 8:45 A.M. 401 407 86 415 6 400 84 5 9:00 A.M. 75 41 77 98 76 95 91 89.1 7 9:15 A.M. 407 401 7 418 95 81 47 401.7 64 9:0 A.M. 406 40 68 94 66 40 418 97.9 9:45 A.M. 81 4 97 90 407 414 97 401. 4 10:00 A.M. 96 97 97 96 96 97 95 10:15 A.M. 87 40 6 95 40 8 86 88.4 40 10:0 A.M. 408 99 85 41 95 9 90 97.6 8 10:45 A.M. 99 400 404 88 417 79 416 400.4 8 11:00 A.M. 87 416 71 414 418 40 77 400.4 49 11:15 A.M. 41 4 416 401 41 411 411 41.7 11:0 A.M. 409 404 401 89 405 407 96 401.6 11:45 A.M. 88 417 91 95 95 97 45 40.6 47 a) Fill in the missing information in the table. b) Make an x chart. Is the process in control? c) Make an R chart. Is the process in control? Fall 008 Martin Huard
ANSWERS 1. a) Centerline: 14.1 UCL: 149.0 LCL : 19. LCL 17. b) Centerline: 17.5 UCL: 1.78 LCL : 7.98 LCL.. a) Centerline: 76.1 UCL: 86.1 LCL : 69.65 LCL 66.1 b) Centerline: 8.74 UCL: 16.9 LCL :.71 LCL 1.19. Centerline: 0.01 UCL: 0.0640 LCL : 0 LCL 0 4. Centerline: 7 UCL: 14.94 LCL : 1.71 LCL 0 5. a) x = 64.64 R 5 = 1.7 b) x = 65. grams c) R=.07 grams In control Out of control (signal ) x Mean Weight 67.1 66.5 65. 64.1 6.6 6.48 5.4.07 1.0 6. a) Out of control Signal 64.4 59.6 50 40.4 5.6 b) In control 14.57 11.84 6.8.1 7. Out of Control signal 1 0.0667 0.0546 61 00 Fall 008 Martin Huard 4
8. Out of Control Signal 8.8 6.6.10 9. In control 0.0867 0.074 0.0467 0 67 10. Out of Control Signal 1 0.0 16.80 10.6.9 0.70 11. a) x 4 = 9.7, x 9 = 96., R 7 = 64, R 15 = 0 b) In Control c) Out of Control Signal 1 and (although only signal is of concern) x 415.5 410. 7.5 61.7 99.5 8. 88.9 8.5 17.7.9 Fall 008 Martin Huard 5