Hypothesis Testing

184 Pages·2015·4.071 MB·English

M140
Introducing statistics
Book 3
Hypothesis testing
This publication forms part of the Open University module M140 Introducing statistics. Details of
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First published 2013. Second edition 2015.
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ISBN 978 1 4730 0307 1
2.1
Contents
Contents
Unit 6 Truancy 1
Introduction 3
1 Clarifying the question 5
1.1 The question to be clarified 7
1.2 Analysing the data 8
Exercises on Section 1 14
2 Probability 15
2.1 Measuring chance 15
2.2 Adding probabilities 18
2.3 Multiplying probabilities 24
Exercises on Section 2 32
3 Probability distributions from random samples 33
3.1 Counting combinations 33
3.2 Probabilities of combinations 37
Exercises on Section 3 42
4 Testing hypotheses I 43
4.1 Tackling the problem 44
4.2 The sign test 52
Exercises on Section 4 55
5 Testing hypotheses II 56
5.1 Significance probabilities: p-values 56
5.2 The sign test with ties 59
5.3 Conclusions and reservations 62
Exercises on Section 5 63
6 Computer work: probabilities and the sign test 64
Summary 64
Learning outcomes 65
Solutions to activities 66
Solutions to exercises 82
Acknowledgements 85
Contents
Unit 7 Factors affecting reading 87
Introduction 89
1 Clarifying the question 90
1.1 The question to be clarified 90
1.2 The data to be used 91
1.3 Setting up the hypotheses 94
Exercises on Section 1 97
2 Sampling distributions revisited 98
Exercises on Section 2 106
3 Normal distributions 108
3.1 Normal distributions: location and spread 109
3.2 Normal distributions: relating means, standard
deviations and plots 114
3.3 The standard normal distribution 120
Exercises on Section 3 125
4 Sampling distributions re-revisited 126
Exercises on Section 4 132
5 The one-sample z-test 132
5.1 The z-test with the standard deviation assumed to be
known 133
5.2 The z-test with unknown standard deviation 138
Exercises on Section 5 142
6 The two-sample z-test 144
Exercises on Section 6 151
7 Computer work: one-sample z-tests 153
8 Conclusions and reservations 153
8.1 When to use the z-test 155
8.2 Limitations in stating conclusions 156
Summary 157
Learning outcomes 158
Solutions to activities 159
Solutions to exercises 169
Acknowledgements 178
Index 179
Unit 6
Truancy

Introduction
Introduction
In Unit 6 we address the following question:
How often do pupils truant?
The topic of truancy has been chosen for this unit because it is an example
that can be understood without technical background and which enables
key statistical ideas to be illustrated and generalised. Statistics can enable
informed discussion and decision-making to take place – decisions should
be based on the careful analysis of reliable data. This is a module in
statistics, not in education, but an aim in this unit and Units 7 to 9 is to
demonstrate that statistics has an important role to play in the
interpretation of data on education.
The emphasis in this unit is on using statistical techniques to reach
conclusions; however, background factors which may account for the
conclusions are also discussed. This will give you the opportunity to think
about the issues involved, perhaps with particular reference to your own,
or your friends’, experiences.
Research shows that while the number of surfers
is somewhat constant over the year, there is a
sudden increase in casual bathers over summer.....
It is worth mentioning that approaches to education in the UK change
regularly. This module was written in 2012–13, and if you are studying
M140 some years later, the system may have a different structure, and
different issues may be important. However, the same statistical
techniques will be valid, and many of the questions will still be relevant.
3
Unit 6 Truancy
Schools in England
The school system in England appears to be in a continual state of
change. In the 1960s and 1970s, most grammar schools and secondary
modern schools were replaced by comprehensive schools. Some City
Technology Colleges started in the late 1980s and 1990s – they
typically specialised in technology, science and mathematics, and
often forged close links with businesses and industry.
More recently, many schools have been changing their status to
become academies, thus gaining greater independence, particularly in
respect of budget control. In 2010 there were only 200 academies, but
the number had risen to 1635 by March 2012. Also, free schools were
introduced in 2011 as an extension of the Academies Programme.
These schools are funded by the taxpayer, but they are not controlled
by the local authority and may be set up by parents, teachers,
charities and businesses.
Unit 6, along with Units 7, 8 and 9, is a little more mathematical than
earlier units. Do not worry if you find them difficult. The important thing
is to ensure that you understand the results and how to apply them. In
some cases, explanations are given of how results are derived. These are
provided as you may find them interesting and they could increase your
understanding. However, you will not be expected to reproduce these
explanations in detail.
This unit contains six sections. In Section 1, we consider what we mean by
the question How often do pupils truant?. How are we going to measure
truancy? Which pupils are we talking about? We shall narrow the
question down until we have a more specific one that we are able to
answer. In Sections 2 and 3 we introduce the concept of probability, which
is fundamental to the ideas of statistical inference. In Sections 4 and 5 we
describe a strategy for drawing inferences about a population from a
sample, and introduce a particular method, known as the sign test, that
follows this strategy. Section 6 directs you to the Computer Book, where
you will use Minitab to calculate certain probabilities and perform a sign
test.
The unit introduces a lot of new concepts and ideas, some of which you
may find difficult at first. However, you will find that these concepts occur
many times in the remainder of the module, so you will have plenty of
opportunity to become familiar with them.
4
1 Clarifying the question
Writing down numbers
In doing calculations, you have been encouraged to retain full
accuracy in intermediate steps, perhaps writing down numbers in full
to demonstrate that you are doing so. This is laborious and makes
errors in copying down a number more likely. In the remaining units
of this module, you need not do so. As far as you can, you should still
use your calculator memory to store numbers to the calculator’s
accuracy, but you need not write down all numbers in full (usually
about five significant figures will be enough, enabling you to focus on
the most important part of a number).
1 Clarifying the question
In Section 1 we consider what is meant by the question
How often do pupils truant?
Notice that this question refers implicitly to whole populations: for
example, all schools in a particular area. We are usually not directly
interested in how the children in one particular school behaved. However,
it is often impossible, or at least not feasible, to collect data from the
whole population. Instead we select a random sample of data. It might be
a random sample of schools or of children. The sample is analysed by the Random sampling was discussed
methods we learned in earlier units, and we then need to decide how the in Subsection 1.2 of Unit 4.
results obtained from the sample apply to the whole population.
Statistical inference makes inferences about a population on the
basis of data drawn from that population.
The above question about truancy may well have arisen from more general
questions, such as:
Why do some pupils learn very little? Are we using good ways of
teaching? Does the quality of my child’s education depend on where
I live?
However, these latter questions can only be tackled if they are first made
more precise. Hence, rather than simply posing a question, we will often
need to clarify it, and we may need to clarify it more than once as we
learn more about the problem. In earlier units we used the modelling
diagram shown in Figure 1 as a framework for how we explore and
summarise batches of data.
5
Unit 6 Truancy
1. 2.
POSE COLLECT
question data
4. 3.
INTERPRET ANALYSE
results data
Figure 1 Modelling diagram
From now on we shall use a slightly modified form of this diagram in which
• the differing roles of populations and samples are identified
• the first box is changed from pose question to clarify question.
The modified modelling diagram is given in Figure 2.
1. 2.
CLARIFY COLLECT
question data from
about population random sample
4. 3.
INTERPRET ANALYSE
results data
for population from sample
Figure 2 Modified modelling diagram
The processes involved in statistical inference are very important. While
we are concentrating on them, we will be concerned particularly with
stages 3 and 4 of the modelling diagram. However, this does not mean that
we can ignore the other stages. It is always important to ensure that the
question under discussion is defined carefully and precisely. Also, although
in this unit we shall not be particularly concerned with how the data were
collected, we do need to know the form of the data, as this affects the way
the sample is analysed.
6