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.

NAO