Probability and Statistics

EMMS098S5 (30 credits)

Aims

This module, will provide you with a basic knowledge of mathematical probability theory and the techniques of statistical inference that are used for analysing data. For students specialising in statistics, it provides a foundation for further modules in statistics and applied probability. The module also introduces the use of a statistical computer package, nowadays an essential tool for all who do statistical work at any serious level.

Teaching and Assessment

Teaching for this module will take place throughout the year, with eight evenings in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term. There will be a mix of lectures and computer sessions.

Of the final course mark, 80% is based on a three-hour unseen exam in June and the other 20% is from assessed coursework.

Coursework will consist of short, problem based assignments. You will have around three weeks to complete each one.

Outline syllabus

An introduction to the statistical computer package Minitab. Descriptive statistics and graphical methods.

The elements of mathematical probability theory. Bayes’ Theorem and its applications. Discrete random variables and probability distributions, including the binomial and Poisson distributions. Mean and variance of probability distributions. Continuous random variables and probability distributions, including the normal distributions.

Populations, random samples and sampling distributions, including the  t and chi-square distributions. Estimation, confidence intervals and hypothesis testing with special reference to samples from normal distributions and estimates of proportions. Chi-square tests of goodness of fit. Two-way contingency tables. An introduction to non-parametric methods.

Learning Outcomes

On successful completion of this module you will be expected to be able to:

• use a statistical computer package to summarize and illustrate data through the use of summary statistics and graphical techniques;
• carry out basic calculations of probabilities based upon the use of the mathematical theory of probability;
• use Bayes’ Theorem to update probabilities of hypotheses in the light of observed data;
• recognize situations where the use of the binomial and Poisson distributions is appropriate and carry out the corresponding calculations of probabilities (i) by direct calculation, (ii) by the use of statistical tables, or (iii) by the use of a statistical package;
• carry out calculations of probabilities for normally distributed random variables (i) by the use of statistical tables or (ii) by the use of a statistical package;
• find estimates and construct confidence intervals for unknown parameter values (i) by the use of statistical tables or (ii) by the use of a statistical package, especially for data from normal distributions and for proportions;
• formulate statistical hypotheses, compute appropriate test statistics, evaluate their significance, (i) by the use of statistical tables or (ii) by the use of a statistical package, and draw conclusions, especially for data from normal distributions and for proportions;
• carry out chi-square tests for data from contingency tables (i) by the use of statistical tables or (ii) by the use of a statistical package, and draw conclusions;
• report clearly and simply the results of statistical analyses in a way that may be understood by non-specialists;
• edit the output from a statistical computer package and incorporate extracts into a word-processed report.

Recommended Texts

• Geoffrey Clarke & Dennis Cooke, A Basic Course in Statistics (5th edition) Hodder Arnold, 2004.
• Sheldon M Ross, Introductory Statistics (3rd edition) Academic Press, 2010.
• William Mendenhall, Robert J Beaver and Barbara M Beaver, Introduction to Probability and Statistics (13th edition) Brooks Cole, 2009.

Statistical Tables

The following tables are recommended and will be provided in the examination: New Cambridge Statistical Tables (Second Edition), Lindley, DV and WF Scott, Cambridge University Press, 1995.