# Module Directory

## MA207-5-AU-CO:Statistics II

PLEASE NOTE: This module is inactive. Visit the Module Directory to view modules and variants offered during the current academic year.

The details
2019/20
Mathematical Sciences
Colchester Campus
Autumn
Inactive
Thursday 03 October 2019
Saturday 14 December 2019
15
04 October 2018

Requisites for this module
MA108 or EC252 or IA155
(none)
(none)
(none)

MA304, MA317, MA318, MA319, MA322

(none)

## Module description

After a brief period of revision, this module looks closely at the relations between the principal continuous distributions, using moment generating functions to determine some properties. The module then
considers confidence intervals and hypothesis tests in the context of the mean and variance of a distribution. After examining bivariate distributions the module ends by considering the principal methods for
estimation of unknown parameters. The module uses the R software environment for statistical computing and graphics.

On completion of the module students should be able to:
-Use tables of the t-, F-, and chi-squared distributions.
- Determine a moment generating function (mgf) and understand the
importance of the one-to-one correspondence between an mgf and a pdf.
- Understand the notions of Type I and Type II errors.
- Carry out hypothesis tests concerning means and variances.
- Determine confidence intervals for means, variances and differences
between means.
- Handle bivariate distributions, understanding the relations between
joint, marginal and conditional distributions.
- Determine maximum likelihood and least squares estimates of unknown
parameters.
- Determine maximum likelihood estimates of unknown parameters.
- Use R for the data analysis examples of the module.

Syllabus

REVISION

Descriptive statistics using R. Events and their probabilities. Conditional probability. Independence.
Discrete random variables: probability mass functions; Bernoulli trials; binomial distribution; hypergeometric distribution.
Continuous random variables: probability density functions (pdf); cumulative distribution function (cdf); normal distribution.
Use of tables. Standardization. Linear combinations of normal variables.

CONTINUOUS DISTRIBUTIONS

Gamma distribution family. Chi-squared (relation to normal). Exponential (relation to Poisson).
The t-distribution (Use of tables, relation to normal and chi-squared distributions).
The F-distribution (Use of tables, relation to t- and chi-squared distributions).
Moment Generating Functions.
Additivity of normal and chi-squared distributions.
Basic law of large numbers. Proof of central limit theorem.

CONFIDENCE INTERVALS AND HYPOTHESIS TESTS

Distributions of the sample mean and of the difference between means.
Null and alternative hypotheses. p-values. Type I and II errors. Power curve.
One-sided and two-sided tests for a population mean (variance known and unknown).
Distributions of the sample variance and of a ratio of two variances.
Confidence interval for mean (known or normal and unknown) .
Confidence interval for a difference between means (known variances, or normal distribution with common variance).
Confidence interval for a variance.

BIVARIATE DISTRIBUTIONS

Bivariate discrete random variables; joint distributions; marginal distributions; independence; covariance and correlation;
Bivariate continuous distributions;
Bivariate normal distribution.

ESTIMATION - MAXIMUM LIKELIHOOD AND LEAST SQUARES

The method of maximum likelihood.

## Module aims

No information available.

## Module learning outcomes

No information available.

## Module information

Available to Socrates /IP students spending all relevant terms at Essex.

## Learning and teaching methods

The module has 27 contact hours in total. 2 lectures each week for 9 weeks, 1 lab in weeks 2-4 and 1 class in weeks 5-9 and 11, during the autumn term. In the summer term 3 revision lectures are given. A project is undertaken in (about) 5-person groups.

## Bibliography

This module does not appear to have a published bibliography.

Coursework Exam
20% 80%

### Reassessment

Coursework Exam
20% 80%
Module supervisor and teaching staff
Dr Aris Perperoglou, email aperpe@essex.ac.uk
Miss Claire Watts, Department Manager, Tel. 01206 873040, email cmwatts@essex.ac.uk

Availability
Yes
Yes
No

## External examiner

No external examiner information available for this module.
Resources
Available via Moodle
Of 39 hours, 0 (0%) hours available to students:
6 hours not recorded due to service coverage or fault;
33 hours not recorded due to opt-out by lecturer(s).

Further information
Mathematical Sciences

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