Statistical Methods

The course content is delivered through guided theoretical exposition, introducing the fundamental concepts of probability, random variables, distributions, sampling, estimation, and hypothesis testing. This is complemented by practical problem-solving sessions, applying methods to mechanical engineering contexts and interpreting results.

The methodology emphasises active and collaborative learning, encouraging discussion of solutions and critical analysis of errors. In-class tasks integrate the essential use of Artificial Intelligence tools, prompting students to validate results, identify limitations, and reflect on the ethical implications of statistical analysis.

The course fosters the development of the mathematical competences proposed by Niss: thinking, reasoning, modelling, problem-solving, and communicating mathematically, bridging theoretical rigour with practical application to real engineering problems.

By the end of this course unit, students should be able to:

Think and reason mathematically about random phenomena, using probability concepts (events, independence, conditional probability, fundamental theorems) to interpret situations in mechanical engineering.

Model mathematically discrete and continuous random variables, selecting and applying appropriate probability distributions (Bernoulli, Binomial, Poisson, Normal, Exponential, t-Student, Chi-squared), and analyse bivariate relationships (independence, covariance, correlation).

Represent and manipulate mathematical entities through probability and distribution functions, sample statistics, and sampling distributions, translating data into formal structures.

Pose and solve mathematical problems of point and interval estimation, constructing confidence intervals for means and variances, and conducting parametric hypothesis tests, combining mathematical rigour with the interpretation of results in applied contexts.

Communicate in, with and about statistics, presenting reasoning, conclusions and uncertainties clearly in technical reports and collaborative settings.

Make use of digital and Artificial Intelligence tools critically and responsibly, exploring statistical simulation, validating results, and reflecting on ethical implications in data analysis.

1-Probabilities

Introduction. Random experience, space for results, events. Probability definition. Conditional probability. Independent events. Total probability theorem. Bayes’ theorem.

2-Random Variables and Discrete Probability Distributions

Introduction. Discrete random variables: Definition; Probability function; Distribution function; Location and dispersion parameters. Special discrete distributions: Bernoulli distribution; Binomial Distribution; Hypergeometric Distribution; Poisson distribution. Discrete bidimensional random variables: Definition; Joint probability and distribution functions; Marginal probability function; Conditioned probability function; Independence from random variables; Covariance and linear correlation coefficient.

3-Random Variables and Continuous Probability Distributions

Definition; Probability density function; Distribution function; Location and dispersion parameters. Special continuous distributions: Brief reference to Uniform and Exponential Distributions; Normal Distribution; Chi-square distribution; T-Student distribution.

4-Sampling and Sampling Distributions

Introduction. Random sample. Statistics. Distribution of the Sample Average. Sampling Variance Distribution.

5-Estimation

Fundamental notions of Point and Interval Estimation. Confidence intervals for the mean value and for the population variance.

6-Parametric Hypothesis Tests

Fundamental notions. Tests for the mean value and for the variance of a population.

Assessment Methods (If there is the possibility of doing it in person)

- Continuous/Periodic Evaluation

Continuous assessment consists of three in-class tasks (with critical use of AI) and two individual written tests (without consultation):

In-class tasks (6 points in total):

Task 1 – Probability and events (2 points, 1h, with AI)

Task 2 – Discrete and continuous probability distributions (2 points, 1h, with AI)

Task 3 – Estimation and hypothesis testing (2 points, 1h, with AI)

May be carried out individually or in pairs.

Written tests (14 points in total):

Test 1 – Probability, random variables and discrete distributions (7 points, 2h, without consultation)

Test 2 – Random variables and continuous distributions; Sampling, estimation and hypothesis testing (7 points, 2h, without consultation)

Individual and conducted during the semester.

Passing requirements:

Each written test requires a minimum score of 3.5 points.

In-class tasks must include a record of the AI tool used, the prompt, and the date, and will be assessed on their critical and ethical analysis.

If a student achieves the minimum score in the first test but does not obtain a final grade of 9.5 points or higher, they may retake only the second test during any subsequent exam period.

Partial marks are not rounded; only the final grade is rounded.

A student who does not pass through continuous assessment or does not participate in it may sit the final examination, graded out of 20 points, scheduled according to the ISEC academic calendar. The student passes if they achieve a grade greater than or equal to 9.5 points. This option excludes the marks obtained in the in-class tasks.

Assessment by Final Examination

The examination (theoretical–practical, written) is graded out of 20 points and takes place on the dates set in the ISEC academic calendar. The student passes if they achieve a grade of 9.5 points or higher.

Permitted material and equipment: course unit formula sheet (without personal notes, which the student must bring to the assessment), and a calculator (any model - scientific or graphic).

In addition to being registered for the course unit, the student must also register for the tests and examinations on Moodle and/or Inforestudante (resit exam), strictly within the deadlines announced (via NONIO and Moodle).

Proposed dates for the written tests during the semester:

1st Test: Wednesday of the 8th teaching week

2nd Test: Wednesday of the last teaching week

Assessment Rubrics

In-class tasks (2 points each)
Criteria (total 2 points):

Technical accuracy/validation (0.8 points): correct application of definitions and formulas (e.g. calculation of conditional probabilities, identification of appropriate distributions, correct construction of confidence intervals).

Critical analysis of AI (0.8 points): validation of results obtained with AI tools, identification of errors, simplifications or limitations of applicability, reflection on when to trust or not trust the output.

Clarity and presentation (0.4 points): organised structure, correct statistical and mathematical language, synthesis in 1–2 pages.

Written tests (7 points each)
Criteria (total 7 points):

Correct formulation/modelling (2.0 points): clear definition of events, variables, parameters, hypotheses, and relevant statistics.

Step-by-step resolution (3.5 points): appropriate application of methods (e.g. Bayes’ theorem; calculation of expectation and variance; correct use of distribution tables and other formulas; construction of intervals; execution of parametric tests).

Interpretation and analysis (1.5 points): interpretation of results in the context of the problem, explanation of confidence level or values obtained, connection to mechanical engineering practice.

Alternative Assessment Method (If face-to-face assessment is not possible)

If face-to-face assessments are not possible, the process will remain the same but will take place on the Moodle platform and ZOOM or TEAMS or any other platform in use by the institution.

In addition to being registered for the course unit, the student must also register for the tests and examinations on Moodle and/or Inforestudante (resit exam), strictly within the deadlines announced (via NONIO and Moodle). During the assessments, the student must have a camera and microphone available, ready to be switched on whenever requested by the invigilator (via Zoom). Failure to comply with these rules will result in exclusion from that assessment.

Important Note for all assessment methodologies:
The lecturer reserves the right to complement the assessment with an oral examination, of which students will be notified at least 48 hours in advance.

Grades above 18.0 points may be subject to an oral defence.

NAO