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8. Hypothesis Tests
Aims
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Objectives
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Reading
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Handouts & Problem Sheets
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Questions
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Links
Return to the Statistics 1 course information page
Aims
A hypothesis test is a procedure for evaluating the evidence for or against
two contrasting statements about the value taken by one (or more) population parameters,
based on sample data.
We will focus on the case when the data is in the form of a simple random sample
from a single Normal population and the parameter of interest is the population mean.
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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Recall the definition of the following terms: null hypothesis, alternative hypothesis,
p-value, significance level, critical region, type I and type II error, power.
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Perform standard hypothesis tests on the value of the population mean,
based on a simple random sample from a Normal distribution
with either known or unknown variance.
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Starting with an informal problem description,
formulate appropriate statements of any model assumptions
and of the null and alternative hypotheses of interest.
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In standard cases, identify an appropriate test statistic
and state its distribution under the null hypothesis.
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For each of the standard types of alternative hypothesis,
identify the set of values of the test statistic that are at least as extreme as
a given observed value.
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In standard cases, calculate the p-value corresponding to a given alternative hypothesis
and a given observed value of the test statistic.
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In standard cases, identify the form of the critical region for a test with a given
significance level for each of the standard types of alternative hypothesis.
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In standard cases, calcuate the probability of a type II error for a test with a given
significance level.
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In standard cases, calcuate the power against a given simple alternative hypothesis
for a test with a given significance level.
Suggested Reading
| Rice | | Chapter 9 | | Sections 9.1-9.5
| | Hypothesis Testing and Assessing Goodness of Fit |
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 8
| Handout for Section 8.7
| Problem sheet 9
| Annex to problem sheet 9
| Problem sheet 10
| Solution sheet 9
| Solution sheet 10
NB Sheet 9, question 3 is based on the material in Section 7.7, which we skipped.
Copyright notice
© University of Bristol 2011
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 10
PROBLEM SHEET 9 -- Questions 1, 2, 5, 6
Questions - set in week 11
PROBLEM SHEET 10 -- Questions 1, 5, 7, 8
Interesting links
R demos - the function I used in lecture 17 to visualise some basic ideas of testing hypotheses.
The Vestac site, under its Statistical Tests section,
has a simple applet visualising a one sample hypothesis test and another illustrating the concepts of
type I and type II error and power.
The
California State University, San Bernardino
site has an applet which simulates a series of hypothesis of tests for the value of the parameter p in
a Bernoulli random variable, and can look at the effect of changing alpha,
changing the form of the hypotheses, and making p different from its null value.
Finally, the Statistical Java
site has some nice applets showing the effect of varying the sample size, the alternative
hypothersis and the size of the test,
which can be found by following the menus Statistical Theory->Hypothesis Tests.
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.
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