Statistics 1

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7. Confidence Intervals

Aims

In previous sections we have seen how use the observations in a simple random sample from a given population to estimate a population parameter or some other population quantity of interest. However, we have also seen that different samples would give different estimates, so we know that our estimate cannot be 'exactly' correct. In this section we derive procedures for reporting the accuracy of our estimate by constructing a confidence interval - an interval of values around the estimate which has a pre-set level of probability of containing the true value of the parameter or other quantity being estimated.

Objectives

The following objectives will help you to assess how well you have mastered the relevant material. By the end of this section you should be able to:

Construct an exact confidence interval for the population mean, with a given confidence level, based on a simple random sample from a Normal distribution.

Construct an exact confidence interval for the population variance, with a given confidence level, based on a simple random sample from a Normal distribution.

Recall and explain the assumptions under which the standard formulae for confidence intervals are applicable and be aware of how the validity of these assumptions might be explored using Exploratory Data Analysis.

Explain how the length of a confidence interval for a population mean depends qualitatively on the required confidence level and on the size of the simple random sample.

Construct an approximate confidence interval for a population mean, with a given confidence level, based on the mean of a simple random sample from the underlying population distribution.

Construct an approximate confidence interval for a proportion, with a given confidence level, based on the mean of a simple random sample from a Bernoulli distribution.

Handouts and Problem Sheets

Copies of Handouts, Problem Sheets and Solution Sheets for the unit will be made available each week here.

Handout for Section 7 | Problem sheet 8 | Problem sheet 9 | Annex to problem sheet 9 | Solution sheet 8 | Solution sheet 9
NB Sheet 9, question 3 is based on the material in Section 7.7, which we skipped.

Copyright notice

All material in these pages is copyright of the University unless explicitly stated otherwise. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only, and not for distribution to anyone else.

Please also note that material from previous years' delivery of this unit is not necessarily a reliable indicator of what will be covered or examined this year.

Questions - set in week 9

PROBLEM SHEET 8 -- Questions 2, 3, 4

Questions - set in week 10

PROBLEM SHEET 9 -- Questions 1, 2, 5, 6

Interesting links

R demos - the function I used in lectures 14 and 15 to visualise some aspects of confidence intervals.

Rice Virtual Lab in Statistics
The Rice site has some nice confidence interval applet, including one which allows you to visualise sampling from a fixed population with a mean of 50 and a standard deviation of 10. The applet simulates a sequence of 100 simple random samples and plots the corresponding 95% and 99% confidence intervals, one for each sample, allowing you to see exactly which intervals do (and which don't) contain the true parameter.

Several of the other sites we have met earlier have confidence interval applets. The Rossman-Chance site has an applet like the Rice site, which simulates a set of intervals showing which contain the true value. The Vestac site, under its Statistical Tests section, has one or two simple applets visualising confidence intervals for means and variances. The California State University, San Bernardino site has an applet which simulates finding confidence intervals for the mean of a normal random variable.

Note that I have no control over the content or availability of these external web pages. The links may be slow to load, or may sometimes fail altogether - please email me to report if a link goes down. Similarly applets may be slow to load or run, but beware that you may experience problems if you try to exit them before they have finished loading.

Professor Peter Green, School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK.
Email link Telephone: +44 (0)117 928 7967; Fax: +44 (0)117 928 7999