MATH 1730Introduction to Statistics 1
Department: StatisticsSemester:1No. of credits: 10 Level: 1
Prerequisites:A-level Mathematics, or equivalent.
Programmes of Study:MMath; Mathematical Studies; Joint Honours (Science); Joint Honours (Arts); Mathematics with Finance.

Aims: To provide an introduction to statistics.

Objectives: On completion of this module, students should be able to:

a)demonstrate foundation skills in statistical methods, including;

b)descriptive statistics and methods of statistical inference.

Methods of teaching: Hours: Lectures: 20Tutorials: 0 Practicals: 3Other Hours: 5 Examples Classes (joint with MATH 1750). Monitoring of progress: Marked exercises and assessed practicals.
Outline Syllabus: Summarising data, graphs and summary statistics; probability and random variables, discrete and continuous; normal distribution; independent identically distributed random variables; confidence intervals and hypothesis tests for means.

Detailed Syllabus:

  1. Introduction. Applications. Types of data. Populations and samples. Frequency distributions. Histograms.
  2. Measures of location. Measures of spread. Interpreting the standard deviation. Quartiles. Sample moments.
  3. Probability and random variables. Probability rules. Independence. Random variables. Mean and variance of a discrete random variable.
  4. Discrete distributions. Binomial and Poisson distributions. Poisson approximation to the Binomial. The Geometric distribution. Probability generating functions.
  5. Continuous random variables. Cumulative distribution function. Probability density function. Mean and variance of a continuous random variable. Population Moments.
  6. Continuous distributions. Exponential distribution. The Normal distribution. Use of tables. Normal approximation to Binomial.
  7. Statistical Inference. Iid random variables. Point estimation. Sampling distribution of the sample mean. Central limit theorem. Interval estimation. Confidence intervals for mean (variance known and unknown).
  8. Hypothesis testing for means. p-values. Tests concerning means. z-test. t-test.

Booklist:

  1. F. Daly, D. J. Hand, M. C. Jones, A. D. Lunn, K. J. McConway, Elements of Statistics, Addison-Wesley, 1995*.
  2. D. G. Rees, Foundations of Statistics, Chapman and Hall, 1987.

Informal Description: The subject of Statistics plays an increasingly important role in all our lives. [.1]Questions such as Does this drug work ?, Will this candidate be elected ?, Is product A of better quality than product B ?, Will this flood defence work ?, can all be answered by statistical analysis. This course provides an introduction to the essential elements of Statistics. We shall first consider descriptive statistics for data summary, including graphical and numerical techniques. The key ideas of probability and random variables are then discussed, including the Binomial, Poisson, Exponential and Normal distributions. Finally, in order to answer questions such as those above, we introduce statistical inference and simple hypothesis testing.
Assessment: 80% 2 hour written examination at end of semester, 20% coursework.

MATH 1740 Introduction to Statistics 2
Department: StatisticsSemester:2No. of credits: 10 Level: 1
Prerequisites:MATH 1730, or appropriate A-level in Mathematics and Statistics, or equivalent. Programmes of Study: MMath; Mathematical Studies; Mathematics with Finance, Joint Honours (Science); Joint Honours (Arts).

Aims: To consider hypothesis tests and the relationships between variables, to analyse count data and to introduce quality control.

Objectives: On completion of this module, students should be able to:

a)carry out appropriate hypothesis tests on the means of one or two populations;

b)understand and carry out simple least squares linear regression;

c)carry out inference on proportions;

d)carry out chi-squared tests;

e)use control charts for mean and range.

Methods of teaching: Hours: Lectures: 20Tutorials: 0 Practicals: 3Other Hours: 5 Examples Classes (joint with MATH 1830). Monitoring of progress: Marked exercises and assessed practicals.

Outline Syllabus: Inference concerning two normal populations; several random variables; regression; attribute data; chi-squared tests; quality control.

Detailed Syllabus:

  1. Revision of single population hypothesis tests on the mean.
  2. Inference for two populations. Two independent samples. Paired samples. Comparing independent vs paired cases.
  3. Several random variables. Sample covariance and correlation. Several discrete random variables. Continuous bivariate distributions. Properties of expectations, population covariance, correlation. Bivariate normal distribution. Linear combinations of random variables.
  4. Regression. Least squares regression. Inference concerning slope. Confidence interval for mean, predicted value. MINITAB for regression.
  5. Attribute data. Hypothesis tests for a population proportion. Large sample confidence interval for a population proportion. Comparing two proportions.
  6. Chi-squared tests. Single sample classified into two or more groups. Fitting distributions, for example binomial, Poisson, normal. Goodness of fit tests. Contingency tables.
  7. Quality control. Control charts for mean and range of data. Statistical process control.

Booklist:

1. D. G. Rees, Foundation of Statistics, Chapman & Hall, 1987.

2. F. Daly, D. J. Hand, M. C. Jones, A. D. Lunn and K. J. McConway, Elements of Statistics, Addison-Wesley, 1995.

Informal Description: This module builds on the ideas introduced in MATH 1730 by focusing on relationships between variables, including techniques for handling data arising from both related and independent samples. Inferential methods are used to compare the means of two populations, e.g. to compare the average wages of males and females doing similar occupations. Where two variables are related, the nature and strength of the relationship can be examined by regression procedures. The course also includes techniques relevant to the analysis of count data, tests concerning proportions, and concludes with an introduction to quality control methods used to continuously monitor a production process.

Assessment: 80% 2 hour written examination at end of semester, 20% coursework.
MATH 1750Introduction to Probability
Department:StatisticsLecturer:

Semester:1No. of credits:10 Level:1
Prerequisites:A-level Mathematics, or equivalent.

Programmes of Study: Joint Honours (Science); Joint Honours (Arts).

Aims: To understand the basic rules of probability theory and their application to sampling; the Bernoulli trial model and quality control.

Objectives: On completion of this module, students should be able to:

a)state and use the basic rules of probability;

b)recognise problems involving and use the rules of counting to evaluate probabilities in equally likely outcome sample spaces;

c)describe simple random sampling, random sampling numbers and use them for simulation;

d)describe and use the Bernoulli trial model and its connection to the Binomial, geometric and negative binomial distributions;

e)apply the rules of probability in acceptance sampling quality control;

f)use a statistical package, such as Minitab, to simulate simple stochastic systems.

Methods of teaching: Hours:Lectures: 20Tutorials: 0 Practicals: 3Other Hours: 5 Examples Classes (joint with MATH 1730). Monitoring of progress: Marked exercises and assessed practicals.

Outline Syllabus: Probability axioms and rules, conditional probability; sampling with and without replacement, simulation; Bernoulli trial model; applications to quality control.

Detailed Syllabus:

  1. Introduction. Venn diagrams. Set Theory (1 lecture).
  2. Probability. Probability axioms. Probability rules. Independence. Conditional probabilities. Bayes Theorem (6 lectures).
  3. Permutations and combinations. Equally likely outcomes. Methods of counting. Permutations and combinations. Probability examples (3 lectures).
  4. Sampling with replacement. Random sampling with replacement from a finite population. Random sampling numbers and simple simulation of a stochastic model (2 lectures).
  5. Bernoulli trial model. Bernoulli trials and their connection with random sampling. Binomial distribution. Geometric distribution. Negative binomial distribution as time to kth event (4 lectures).
  6. Sampling without replacement. Random sampling without replacement. Hypergeometric distribution (1 lecture).

Quality control. Monitoring number of defectives in a sample. Poisson as limit of binomial probabilities. Acceptance sampling. OC curves (3 lectures).

Booklist:

  1. G. M. Clarke and D. Cooke, A Basic Course in Statistics, 3rd edition, Arnold, 1992*.
  2. J. A. Rice, Mathematical Statistics and Data Analysis, 2nd edition, Duxbury Press, 1995.
  3. S. Ross, A First Course in Probability, Macmillan, 1976.
  4. R. L. Scheaffer, Introduction to Probability and its Applications, PWS-Kent, 1990.

Informal Description: “Probability is basically common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct.” So said Laplace. In the modern scientific and technological world, it is even more important to understand probabilistic argument. This module introduces the tools needed for probability and goes on to use them in simple situations related to repeated experiments with applications to quality control.

Assessment: 80% 2 hour written examination at end of semester, 20% coursework

MATH 1830Statistics through Applications
Department:StatisticsLecturer:

Semester:2No. of credits:10

Level:1

Prerequisites:MATH 1730, or equivalent.

Programmes of Study:Mathematics with Finance, Joint Honours (Science); Joint Honours (Arts).

Aims: To introduce students to methods for analysing data using statistical packages and computer software; to introduce group work.

Objectives: On completion of this module, students should be able to:

a)choose appropriate methods for simple data analysis;

b)use computer software and a statistical package.

Methods of teaching:

Hours:Lectures: 9Tutorials: 5 Practicals: 2Other Hours: 0.

Group Presentations: Students will do one group presentation on completion of a 2 week project.

(Note: Students will be expected to use teaching software in their own time without supervision Communication and additional information will be through the internet).
Monitoring of progress:Reports and presentations.

Outline Syllabus: Techniques for data analysis using statistical packages and computer software; solving problems using statistical modelling techniques; case studies, group projects and presentations.

Detailed Syllabus:

  1. Techniques for data analysis using statistical packages and computer software. The material covered will include topics from simple EDA, contingency tables, correlation and regression, paired and independent samples t-tests, nonparametric tests, probability plots, random walk and Poisson process.
  1. Solving problems using statistical modelling techniques. A series of problems will be used to drive the models and techniques.
  1. Case studies, group projects and presentations.

Booklist:

  1. F. Daly, D. J. Hand, M. C. Jones, A. D. Lunn, K. J. McConway, Elements of Statistics, Addison-Wesley, 1995.
  2. Bowman and McCall, Title to be announced, Edward Arnold, 1998.

Informal Description: Statistics is about solving problems. These problems will come from a range of different subjects areas, and the appropriate statistical techniques for the collection and analysis of data relevant to solving them will be introduced, explained and used. Computer software prepared for the STEPS (Statistical Education through Problem Solving) project will be used.

Assessment: 100% individual reports, group reports and presentations.

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