Title: PROBABILITY AND STATISTICAL METHODS FOR MODELLING ENGINEERS
Teacher(s): Prof. Maria Richetta
Credits: 3
LEARNING OUTCOMES
After a careful study during the course the students should be able to:
1. Identify the role of statistics in engineering problems.
2. Discuss the methods used by engineers to collect data.
3. Explain the differences between mechanistic and empirical models.
4. Understand and describe sample spaces and events of random experiments with graphs, tables, lists or tree diagrams.
5. Interpret and use the probability of the results to calculate the probabilities of the events. Calculate the probability of joint events and interpret / calculate the conditional probabilities of events.
6. Apply the Bayes theorem.
7. Understand the meanings of a random variable.
8. Select an appropriate discrete / continuous probability distribution. Determine probability, mean, and variance for the presented discrete / continuous probability distributions.
9. Calculate and interpret mean, variance, standard deviation, median and sample interval.
10. Build and interpret normal probability diagrams.
11. Know the general concepts of estimating the parameters of a population or a probability distribution.
12. Explain the properties of point estimators (bias, variance, mean square error).
13. Construct point estimators with moments method and maximum likelihood method.
14. Calculate and explain the precision of the estimation of a parameter.
15. Understand the central limit theorem.
16. Explain the role of normal distribution as a sampling distribution.
17. Build confidence intervals, forecast intervals, tolerance intervals.
18. Structure engineering decision problems as hypothesis tests.
19. Check the hypotheses on the average of a normal distribution using a Z-test or t-test procedure.
20. Test the hypotheses on variance or standard deviation of a normal distribution. Check the hypotheses on a population.
21. Use the P value approach to make decisions in hypothesis tests.
22. Select a sample size for tests on averages, variances and proportions.
23. Explain and use the relationship between confidence intervals and hypothesis testing.
24. Use the chi-square test to test hypotheses about the distribution.
25. Use simple linear regression to build empirical models of technical and scientific data.
26. Understand the use of the least squares method to estimate parameters in a linear regression model.
27. Analyze the residuals to determine if the regression model fits the data or to see if there are violations of the initial hypotheses.
28. Test the statistical hypotheses and construct confidence intervals on the parameters of the regression model.
29. Use the regression model for the prediction of a future observation and construct an appropriate prediction interval on future observation.
30. Use simple transformations to obtain a linear regression model.
31. Apply the correlation model.
32. Finally, discuss how probabilities and probability models are used in engineering and science in general.
KNOWLEDGE AND UNDERSTANDING
Students acquire understanding and knowledge of: 1) fundamental statistical techniques (summary statistics, normal distribution, interval estimation, regression analysis, modelling) and how they relate to the baseline discipline; 2) software statistical techniques; 3) process monitoring by control charts; 4) process optimization by response surface methodology; 5) determining important factors by hypothesis testing; 6) process modelling by, e.g., regression analysis; 7) design of experiments and laboratory recommendation.
The teaching approach provides the foundation for this understanding, in such a way that at the end of the course students have assimilated a complete knowledge of the basic themes.
APPLYING KNOWLEDGE AND UNDERSTANDING
The goals of the course are to help the students to: i) model and simulate basic engineering problems, ii) collect, analyze and present numerical data in general and simulation results in particular, iii) interpret simulation results by means of statistical methods, iv) use statistical principles and concepts, v) develop software for reporting and for graphical presentation, vi) be familiar with basic probability theory and perform estimation, hypothesis testing, simple correlation-/regression analysis, vii) identify, formulate, and solve engineering problems. Such applications of statistics are widespread in all branches of engineering.
MAKING JUDGEMENTS
The training provided for students of the course is hallmarked by the acquisition of a flexible mentality that helps them to extend the knowledge learned to new concepts, enabling them to introduce elements of innovation. These activities encourage students to develop: critical thinking and problem solving; critical analysis; independence of judgement. At the end of the course, students are therefore able to pose, refine and evaluate scientific questions, this being a fundamental objective both educational and cognitive.
COMMUNICATION SKILLS
Students develop the ability to present clearly what they have learned during the course and, in the same way, the additional knowledge gained from practical exercises, classroom exercises and textbooks. They are expected to present their knowledge effectively. These skills, which concern both oral and written presentations, are based on the ability to analyze and integrate the knowledge areas acquired during the course. Students are also encouraged to develop a positive attitude towards teamwork.
The evaluation of the achievement of written and oral communication skills is verified during classroom exercises, practical exercises, tutoring and through written and oral exams at the end of the course.
LEARNING SKILLS
Students, through the introduction of a range of fundamental statistical techniques, learn how to: analyse data, apply statistics in engineering contexts, use appropriate statistical sofware. Furthermore they acquire: numeracy skills, effective Information retrieval and research skills, computer literacy. On these bases they will be able to connect and relate knowledge across various scales, concepts, and representations “in” and “across” domains.
PREREQUISITES
There are no mandatory prerequisites for this course. However, a basic knowledge of Mathematical Methods for Engineering (calculus, algebra, trigonometry, etc.) is assumed.
TOPICS
– The role of Statistics in Engineering: Mechanistic and Empirical Models, Probability and Probability Models.
– Probability: Discrete Random Variables and Probability Distributions; Continuous Random Variables and Probability Distributions.
– Point Estimation of Parameters.
– Random Sampling and data Description, Statistical Intervals for a Single Sample.
– Tests of Hypotheses for a Single Sample.
– Simple Linear Regression and Correlation: Empirical Models.
– Multiple Linear Regression Model.
– The Analysis of Variance (ANOVA): Residual Analysis and Model Checking; The Random Model.
– Design of Experiments with Several factors.
– Statistical Quality Control.
EVALUATION
- Type: written, oral and practical examination.
- Description: A series of written exercises and practical tests offers teachers and students the opportunity to assess progress and understanding of students, during the course, before the final assessment.
The final exam consists of a written and / or practical test and an oral test.
The written / practical test is structured to: i) emphasize concepts and techniques acquired during the course; ii) request an explanation of the candidate’s reasoning; iii) allow sufficient time for most well-prepared students to complete each application; iv) use innovative types of questions that probe the depth of understanding.
The oral exam consists in three theoretical questions (each contributes with 10/30 to the final vote). The exam evaluates the overall preparation of the student, the ability to integrate the knowledge of the different parts of the program, the consequentiality of the reasoning, the analytical ability and the autonomy of judgment. Furthermore, language properties and clarity of presentation are assessed, in compliance with the Dublin descriptors (1. Knowledge and understanding; 2. Ability to apply knowledge and understanding; 3 . Making judgments; 4. Learning skills; 5: Communication skills).
The final vote of the exam is expressed out of thirty and will be obtained through the following graduation system:
Not pass: important deficiencies and / or inaccuracies in the knowledge and understanding of the topics; limited capacity for analysis and synthesis, frequent generalizations and limited critical and judgment skills, the arguments are presented in an inconsistent way and with inappropriate language,
18-20: just sufficient knowledge and understanding of the topics with possible generalizations and imperfections; sufficient capacity for analysis, synthesis and autonomy of judgment, the topics are frequently exposed in an inconsistent way and with inappropriate / technical language,
21-23: Routine knowledge and understanding of topics; ability to analyze and synthesize with sufficiently coherent logical argument and appropriate / technical language
24-26: Fair knowledge and understanding of the topics; good analysis and synthesis skills with rigorously expressed arguments but with a language that is not always appropriate / technical.
27-29: Complete knowledge and understanding of the topics; remarkable abilities of analysis and synthesis. Good autonomy of judgment. Topics exposed rigorously and with appropriate / technical language
30-30L: Excellent level of knowledge and in-depth understanding of the topics. Excellent skills of analysis, synthesis and autonomy of judgment. Arguments expressed in an original way and with appropriate technical language.
ADOPTED TEXTS
– Statistics for Engineers and Scientists, W.Navidi, McGraw-Hill Education 2020.
– Fundamentals of Probability and Statistics for Engineers, T.T. Soong, Jhon Wiley & Sons 2004.
– Probability and Statistics for Engineering and the Sciences, J. Devore, Brooks/Cole 2010.
– Probability and Statistics; John J. Schiller, R. Alu Srinivasan, Murray R Spiegel, 4 th Edition 2013.
– Essential Matlab for Engineers and Scientists; Brian Hahn, 5 th Edition 2012.
BIBLIOGRAPHY
– Applied Statistics and Probability for Engineers, D.C. Montgomery, G.C. Runger, Jhon Wiley & Sons 2003
– A Beginner’s Guide to R , A.F. Zuur, E.N. Ieno, E.H.W.G. Meesters, Springer 2009
– Introductory Statistics with R, P. Dalgaard, Springer 2008
– The R Book, M.J. Crawley, Wiley 2007
– Statistical Methods for Engineers, G. Vining, Thomson Brooks/Cole 2011
– Probability and Statistics, J.L. Devore, Thomson Brooks/Cole 2000
– Data analysis with Matlab, James Braselton, 2014
– Mathematical and Computational Modeling: With Applications in Natural and Social Sciences, Engineering, and the Arts, First Edition. Roderick Melnik. 2015 John Wiley & Sons, Inc.
– Mathematical Modeling with Excel, B. Albright, Jones & Bartlett Learning, 2009.
– Computational Statistics Handbook with MATLAB, W.L. Martinez, A.R. Martinez, Chapman and Hall Book/CRC Press 2015.
– Linear Models with R, J.J.Faraway, Chapman and HallBook/CRC Press 2014.
DELIVERY MODE (Presence/e-learning)
Precence.
TEACHING METHODS
The course is delivered through the following Learning Activities.
1. Attendance of lectures where course material is presented through discussions, worked examples, and demonstrations.
2. Attendance of exercises and practicals where students perform and discuss exercises as part of their formative assessment. These practices help students in consolidating the course material and provide a source of feedback on understanding.
3. Private study to review the course material presented in lectures, read the textbooks, and practice solving conceptual and numerical problems from textbooks, and other sources.
4. Completion of online quizzes and problems that are designed to give students further practice in the application of course material, as well as feedback on their understanding. These also form part of their formative assessment.
5. Application of basic and more advanced statistical methods. Use of the statistical package R, developed through a sequence of computer practicals.