MA20033: Statistical Inference 1



Monday 15:15 (ARTS L.T.) and Thursday 12:15 (5W 2.3) in weeks 1-11, 15.


Monday 16:15 (8W 1.33) for Group 1 in weeks 2, 4, 6, 8, 10.

Monday 17:15 (3WN 3.8) for Group 2 in weeks 2, 4, 6, 8, 10.

Wednesday 10:15 (3W 3.9) for Group 3 in weeks 2, 4, 6, 8, 10.
Thursday 10:15 (8W 1.32) for Group 4 in weeks 2, 4, 6, 8, 10.


Monday 16:15 (1W 2.25) for Group 1 in weeks 3, 5, 7, 9.

Monday 17:15 (1W 2.25) for Group 2 in weeks 3, 5, 7, 9.

Wednesday 10:15 (1W 2.25) for Group 3 in weeks 3, 5, 7, 9.
Thursday 10:15 (1W 2.25) for Group 4 in weeks 3, 5, 7, 9.

Lecturer: Simon Shaw; at


Groups 1 and 4-

Simon Shaw; at

Group 2-

Vangelis Evangelou; e.evangelou at

Group 3-

Wenyang Zhang; w.zhang at

Credits: 6
Level: Intermediate
Assessment: EX75CW25 (There will be three pieces of assessed coursework set in the Practical Classes of Weeks 5, 7 and 9).
Other work: There will be fortnightly question sheets. A schedule of when work will be set and handed in is available as either pdf or postscript.
Requisites: Before taking this unit you must take MA10032.

Aims & Learning Objectives:
Aims: Introduce classical estimation and hypothesis-testing principles.
Ability to perform standard estimation procedures and tests on normal data. Ability to carry out goodness-of-fit tests, analyse contingency tables, and carry out non-parametric tests. Ability to use R to calculate estimates, carry out hypothesis tests and compute confidence intervals.


Point estimation: Maximum-likelihood estimation, including computational aspects; further properties of estimators, including mean square error, efficiency and consistency; robust methods of estimation such as the median and trimmed mean. Confidence intervals. Hypothesis testing: Size and power of tests; Neyman-Pearson lemma. One-sided and two-sided tests. Distributions related to the normal: t, chi-square and F distributions. Interference for normal data: Tests and confidence intervals for normal means and variances, one-sample problems, paired and unpaired two-sample problems. Contingency tables and goodness-of-fit tests. Non-parametric methods: Sign test, signed rank test, Mann-Whitney U-test. Examples of all the above, including case studies in R.

Some useful books

We won't follow a book as such but useful references include:

  1. J.A. Rice, Mathematical Statistics and Data Analysis, Third Edition, 2007. 512.75 RIC
  2. L.J. Bain and Engelhardt, M., Introduction to probability and mathematical statistics, 1992. 512.75 BAI
  3. C. Chatfield, Statistics for technology: a course in applied statistics, Third Edition, 1983. 512.760.5 CHA
  4. S.Chatterjee, Handcock, M.S. and Simonoff, J.S., A casebook for a first course in statistics and data analysis, 1995. 512.76 CHA

The rather dishevelled appearance of my copy of Rice indicates it as a personal favourite. The interested student may find other books around the 512.75 and 512.76 shelfmarks.

For help with programming in R, a recommended reference is:

  1. Peter Dalgaard, Introductory statistics with R, 2002. 512.764 DAL

The book is also available electronically from the University of Bath library by clicking here for the first edition or here for the second edition.

2009/10 Course summary and material

Lecture 1 (05 Oct 09): Introduction, §1 Point estimation: introduction, estimators and estimates.
Lecture 2 (08 Oct 09): Sampling distribution, maximum likelihood estimation.

Lecture 3 (12 Oct 09): Log-likelihood, examples of mle for Poisson and Normal distribution. Question Sheet One handed out.
Lecture 4 (15 Oct 09): §2 Evaluating point estimates: bias.
Lecture 5 (19 Oct 09): Mean square error, relative efficiency.
Lecture 6 (22 Oct 09): Consistency, robustness, measures of location.
Lecture 7 (26 Oct 09): Trimmed mean, §3 Interval estimation: principle of interval estimation. Question Sheet Two handed out.
Lecture 8 (29 Oct 09): Pivot, confidence interval, confidence interval for normal mean (variance known).
Lecture 9 (02 Nov 09): Chi-squared distribution, confidence interval for normal variance.

Lecture 10 (05 Nov 09): t-distribution, confidence interval for normal mean §4 Hypothesis testing: null and alternative hypothesis.

Lecture 11 (09 Nov 09): Critical region, Type I and Type II errors, significance level. Question Sheet Three handed out.
Lecture 12 (12 Nov 09): Power of a test, Neyman-Pearson lemma, examples of use of Neyman-Pearson lemma (normal mean, exponential).
Lecture 13 (16 Nov 09):
One-sided alternative hypotheses, uniformly most powerful test.
Lecture 14 (19 Nov 09):
Two-sided alternative hypotheses, duality between hypothesis test and confidence interval.
Lecture 15 (23 Nov 09): Power function, §5 Inference for normal data: investigating the variance in one-sample problems. Question Sheet Four handed out.
Lecture 16 (26 Nov 09): Investigating µ in one-sample problems (known variance).

Lecture 17 (30 Nov 09): p-value for one and two-sided tests.
Lecture 18 (03 Dec 09): Investigating µ in one-sample problems (unknown variance): t-tests.
Lecture 19 (07 Dec 09): Comparing paired samples. Question Sheet Five handed out.
Lecture 20 (10 Dec 09): Unpaired data, F-distribution, investigating the variance for unpaired data.
Lecture 21 (14 Dec 09): Investigating the means for unpaired data (known and unknown variances), pooled sample variance.
Lecture 22 (17 Dec 09): §6 Goodness-of-fit tests: multinomial distribution, Pearson's chi-square statistic. Question Sheet Six handed out.

Forthcoming material

The course is now completed.

Omitted material

In previous years, the course has contained 24 lectures, the contents of these lectures are shown below.
Questions on these topics may appear on past papers (in particular 2004/05 3(c)(i) and (iii)) but will not appear in the 2009/10 paper.

Lecture 23: §7 Non-parametric methods: Signed-rank test aka Wilcoxon matched-pairs test.

Lecture 24: Mann-Whitney U-test. 

Lecture notes: pdf or postscript

Handouts - Adding independent normals:
pdf or postscript

Tables -

normal: pdf

chi-squared: pdf

t-distribution: pdf

F-distribution: pdf

(thanks to Ruth Salway)

Homework -

There will be fortnightly question sheets. Question sheets and full solution sheets will be made available.

Practicals -

Practical Sheet One: Set in Week 3 practicals

 text file of functions: txt
Practical Sheet Two: Set in Week 5 practicals Assessed coursework, due in Week 6 tutorials
 text file of functions: txt Coursework Cover Sheet: pdf

Practical Sheet Three: Set in Week 7 practicals Assessed coursework, due in Week 8 tutorials
Practical Sheet Four: Set in Week 9 practicals Assessed coursework, due in Week 10 tutorials

Exams -



pdf pdf pdf pdf pdf
Solutions: pdf

Last revision:




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