MATH 11400 Statistics 1 2008-09
7. Confidence Intervals
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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:
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Construct an exact confidence interval for the population mean, with a given confidence level, based on a
simple random sample from a Normal distribution.
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Construct an exact confidence interval for the population variance, with a given confidence level, based on a
simple random sample from a Normal distribution.
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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.
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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.
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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.
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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.
Suggested Reading
| Rice | | Chapter 3 | | Sections 7.3.3 | | The Normal approximation |
| | Chapter 8 | | Sections 8.5.3 | | Confidence intervals for maximum likelihood estimates |
| | Chapter 10 | | Sections 10.4.6 | | Estimating variability of location estimates by the bootstrap |
Handouts and Problem Sheets
Copies of Handouts, Problem Sheets and Solution Sheets for the unit
will be made available each week on the Statistics 1 course pages on
Blackboard.
Questions - set this week
PROBLEM SHEET 7 -- Questions 1, 2, 3
Interesting links
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.
Return to the Statistics1 information page
Dr E J Collins,
Department of Mathematics,
University of Bristol, Bristol, BS8 1TW, UK
Email: E.J.Collins@bristol.ac.uk
Telephone: +44 (0) 117 928 7977; Fax: +44 (0) 117 928 7999