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.

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.

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. Central Limit theorem and applications.

4-Sampling and Sampling Distributions

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

5-Estimation

Point estimation. Estimators and estimates.
Estimator properties. Interval estimation. Confidence interval for the expected value. Confidence interval for the variance of a normal distribution. Confidence interval for the difference of expected values and for the quotient of variances of two normal distributions.

6-Parametric Hypothesis Tests

Introduction, notions and methodology. Testing the expected value. Testing the variance of a normal
distribution. Testing the proportion. Testing the difference of expected values and the quotient of variances
of normal distributions.

The student can choose between continuous evaluation or evaluation by final exam.
• Continuous evaluation - consists of a writen test and an assessment of regular class atendance. The
student can get approval if the sum of the previous components is equal to or greater than 9.5 points.
The writen test will be graded with 12 points and will happen in the last week of classes.
The evaluation of regular class atendance will be materialized through problems proposed during the
semester for a total of 8 points. The dates of realization of these problems may not be announced in
advance to students.
A minimum score of 50% is required for each assessment component. Students who do not meet this
requirement must take a final exam.
• Final Exam - consists of conducting an exam quoted for 20 points, which will evaluate all chapters of the
program contents. Students who have not passed the continuous assessment or who have not chosen
continuous assessment and those who are allowed to improve their rating may access the exam.

NAO