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CORK INSTITUTE OF TECHNOLOGY INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2010
Module Title: Statistics for Engineering 301 C/A
Module Code: STAT
School: Mechanical and Process Engineering Civil and Structural Engineering
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 3 Bachelor of Engineering (Honours) in Biomedical Engineering – Stage 3 Bachelor of Engineering (Honours) in Civil and Structural Engineering – Stage 3
Programme Code: EMECH_8_Y EBIOM_8_Y CSTRU_8_Y
External Examiner(s): Mr. J. Reilly Internal Examiner(s): Mr. D. O’Hare
Instructions: Answer any three questions. All questions carry equal marks.
Duration: 2 HOURS
Sitting: Autumn 2010
Requirements for this examination:
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
- (a) A certain manufacturing process produces 5% defective parts. Twenty percent of all parts are produced by machine A. There is a 10% probability that a part is defective given that it was produced by machine A. Draw a probability tree to assist in answering the following questions: (i) What is the probability that a tested part is defective and was produced by machine A? (ii) Of the product that is not manufactured on machine A, what percentage is defective? (6 marks) (b) A discrete random variable X is distributed according to the following probability function: , 0 , 1 , 2 , 3 , 4. 2 P ( x ) kx x (i) Find the value of k. (ii) Establish the values of E(X) and V(X), the mean and variance of X. (9 marks) (c) A particular part used in automatic transmissions has, on average, 0.8 burrs per part. Burrs occur randomly in accordance with the pattern of a Poisson distribution. (i) What is the probability that a part will have no more than one burr? (ii) If ten such parts are examined, what is the probability that the total number of burrs observed exceeds 8? (5 marks) (d) A quality control engineer wants to check whether, in accordance with specifications, 95% of the electronic components shipped by his company are in good working order. To this end, he randomly selects 20 from each large batch ready to be shipped and passes the batch if the selected components are all in good working order; otherwise, each of the components in the batch is checked. Find the probability that the QC engineer will commit the error of holding a batch for further inspection even though 95% of the components are in good working order. (5 marks)
- (a) Write an explanatory note on the term sampling distribution of x_._ Make reference to any relevant relationships between this distribution and the population from which the samples are taken. (4 marks) (b) A machine produces metal rods used in an automobile suspension system. A random sample of 12 rods is selected and the diameter is measured in each case. The resulting data are: 8.24 8.18 8.19 8. 8.23 8.24 8.26 8. 8.22 8.21 8.27 8. Assuming that rod diameters are normally distributed, find 95% confidence intervals for the mean and the variance of rod diameters. (9 marks)
- (c) Researchers are experimenting with the use of microprocessors to help reduce fuel and power consumption in furnaces used to process magnetic ore. A particular system is designed to maintain gas flow through the machine in such a way as to ensure that sufficient heat is available to raise the raw ore pellets to 1300˚C. A study is conducted to compare the temperature setting needed to accomplish this using the computerised system to that needed using the conventional method. It is thought that the computerised system will result in a lower average required setting. Computerised Conventional n 1 = 12 n 2 = 14 x 1 = 733^0 C x 2 = 822^0 C s 1 = 100 C s 2 = 160 C
A comparison of variances using Minitab produced the following output.
Test for Equal Variances 95% Bonferroni confidence intervals for standard deviations Sample N Lower StDev Upper 1 14 11.1053 16 27. 2 12 6.7622 10 18.
F-Test (Normal Distribution) Test statistic = 2.56, p-value = 0.
What implication does this have for the test on mean values? Carry out this test, and state your conclusions. (12 marks)
- (a) Prior to a quality-improvement program, a random sample of 200 units contained 26 that required rework. After the implementation of the program, a further sample of 200 units contained 12 requiring rework. Has there been a significant reduction in the proportion of units requiring rework? (7 marks) (b) A random sample of 500 one-litre containers of antifreeze is taken from a production process that uses two filling machines. The contents are categorised as under-filled, acceptable and over-filled. The results are summarised in the table below. Test the claim that the category of filling is independent of the machine. Under-filled Acceptable Over-filled Machine A 11 205 14 Machine B 47 181 42 (8 marks)
(c) Corrosion of steel reinforcing bars is an important durability problem for reinforced concrete structures. Carbonation of concrete results from a chemical reaction that lowers the pH value by enough to initiate corrosion of the bar. Data on x = carbonation depth (mm) and y = strength (MPa) for a sample of core specimens taken from a particular building are shown below.
x 8.0 15.0 16.5 20.0 20.0 27.5 30.0 30.0 35. y 22.8 27.2 23.7 17.1 21.5 18.6 16.1 23.4 13. (i) Find the value of the correlation coefficient, and test its significance. (ii) Plot the data on a scatter diagram. Is the pattern consistent with your answer in part (i)? Justify your answer. (10 marks)
