Aims

This module builds on the foundations of statistical inference from MAS1616. Students will know about the distinction between a population and a sample. They will know about the use of estimators calculated from random samples as a means of learning about properties of the population. They will be able to describe the role of likelihood methods in the derivation of estimators and their properties.

Students will also learn about the Bayesian approach to statistical inference. Students will be able to explain the distinctive features of Bayesian methodology, understand the role of prior distributions and compute posterior distributions in simple cases.

Module summary
Statistics aims to learn about populations on the basis of samples drawn from them. Statistical inference is concerned both with estimating parameters and also with quantifying the associated sampling variation. This module covers fundamental notions of a standard error, confidence interval and hypothesis tests, in the context of both discrete and continuous variables. The use of Bayes’ Theorem to compute posterior distributions from given priors and likelihoods will be described, with particular emphasis given to the case of conjugate distributions.

Outline Of Syllabus

Review of basic concepts of statistical inference; Properties of sampling distributions, including standard errors and confidence intervals. Central Limit Theorem. Maximum likelihood estimators and their properties. Likelihood ratio test. Introduction to hypothesis tests: rejection regions, type I and type II errors, power and significance level. Illustrations using one- and two-sample hypothesis tests for means and for proportions. Contingency table analysis.

Sufficiency. Inference for populations using random samples and conjugate priors, including posterior estimates and highest density intervals: inference for the mean of a normal distribution with known variance; inference for parameters in other commonly used distributions. Sequential use of Bayes' Theorem. Parameter constraints. Asymptotic posterior distribution.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Module drop-in sessions allow students to receive learning support in areas where they may need additional guidance.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.