Cambridge International AS & A Level Mathematics: Probability & Statistics 2

186 Pages·2018·29.257 MB·English

by  Jayne Kranat

Cambridge International
AS & A Level Mathematics:
Probability & Statistics 2
Coursebook

Copyright Material - Review Only - Not for Redistribution
Copyright Material - Review Only - Not for Redistribution
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Sreries Editor: Julian Gilbey
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Probag bility & Statistics 2
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International Education bears no responsibility for the example answers to questions
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e Contents
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Contents w
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Series introdumction R vi
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How to- use this book s viii
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Ackpnowledgements P x
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1.1 Introduction to hypothesis testing 3
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1.3 Type I and Type II erro rs 17
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End-of-chapter review exgercise 1 23
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2 The Poisson dbistribution e 25
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2.1 Introductaion to the Poisson distribution 27
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2.2 Adapting the Poisson distribution for different intservals 33
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2.3 Th e Poisson distribution as an approximatione to the binomial distribution 36
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2.4 Using the normal distribution as an approximation to the Poisson distribution 40
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2.5 Hypothesis testing with the Poisson dtistribution 43
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End-of-chapter review exercise 2 r 47
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e 3 Linear combinations of rniandom variables p 50
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3.1 Expectation and varian ce 51
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3.2 Sum and difference ogf independent random variables 56
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3.3 Working with nroirmal distributions vi 61
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3.4 Linear combminations of Poisson distributions R 64
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End-of-chapter review exercise 3 - 68
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Cross -topic review exercise 1 e 70
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4C Continuous random variables y 74
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4.2 Finding the median and other percentiles of a continuous random variable 83
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End-of-chapter review exerc ise 4 94
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Cambridge International AS & A Levele Mathematics: Probability & Statistics 2
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5 Sampling w 99
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5.1 Introduction to sarmipling vi 102
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5.2 The distribution of sample means 106
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End-of-chapter raeview exercise 5 118
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6 Estim a- tion es 120
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6.1 Unpbiased estimates of population mean and vPariance 122
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6.2C Hypothesis testing of the population meany 127
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6.3 Confidence intervals for population mean 131
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vi 6.4 Confidence intervals for populativon proportion y 138
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Cross-topic review exercise 2 w 147
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Practice exam-styble paper e 153
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The standard normal distribution function - 155
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Answerys 157
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GlCossary y 171
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Cambridge International AS & A Levele Mathematics: Probability & Statistics 2
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Series introg duction
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Cambridge Internaational AS & A Level Mathematics can be a life-changing course. On the one hand, it is a
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facilitating subject: there are many university courses that esither require an A Level or equivalent qualification in
mathematic-s or prefer applicants who have it. On the othser hand, it will help you to learn to think more precisely
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and logically, while also encouraging creativity. Doinrg mathematics can be like doing art: just as an artist needs to
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mastoer her tools (use of the paintbrush, for example) and understand theoretical ideas (perspective, colour wheels

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and so on), so does a mathematician (using tools such as algebra and calculus, which you will learn about in this
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wcourse). But this is only the technical side: thse joy in art comes through creativity, when the artist uses her tools
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e to express ideas in novel ways. Mathemateics is very similar: the tools are needed, but the deep joy in the subject

vi comes through solving problems. v y
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You might wonder what a math ematical ‘problem’ is. This is a very good question, and many people have offered
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different answers. You mightg like to write down your own thoughts on thisw question, and reflect on how they
change as you progress thrdough this course. One possible idea is that a meathematical problem is a mathematical
question that you do norti immediately know how to answer. (If you dovi know how to answer it immediately, then
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we might call it an ‘emxercise’ instead.) Such a problem will take timRe to answer: you may have to try different
approaches, usinga different tools or ideas, on your own or with others, until you finally discover a way into it. This
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may take minuCtes, hours, days or weeks to achieve, and yousr sense of achievement may well grow with the effort it
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In adodition to the mathematical tools that you will learn in this course, the problem-solving skills that you
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will develop will also help you throughout life, whatever you end up doing. It is very common to be faced with
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wproblems, be it in science, engineering, mathsematics, accountancy, law or beyond, and having the confidence to
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e systematically work your way through theem will be very useful.

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R This series of Cambridge InternatiUonal AS & A Level Mathematics coursebookso, written for the Cambridge
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Assessment International Educa tion syllabus for examination from 2020, will support you both to learn the
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mathematics required for thegse examinations and to develop your mathemwatical problem-solving skills. The new
examinations may well incdlude more unfamiliar questions than in the paest, and having these skills will allow you
to approach such questiroins with curiosity and confidence. vi
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In addition to proablem solving, there are two other key concep ts that Cambridge Assessment International
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Education have introduced in this syllabus: namely commusnication and mathematical modelling. These appear
in various f-o rms throughout the coursebooks. s
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Comomunication in speech, writing and drawing lies at the heart of what it is to be human, and this is no less

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true in mathematics. While there is a temptatiotn to think of mathematics as only existing in a dry, written form
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win textbooks, nothing could be further from sthe truth: mathematical communication comes in many forms, and
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e discussing mathematical ideas with colleaegues is a major part of every mathematician’s working life. As you study

evi this course, you will work on many pnriobvlems. Exploring them or struggling with thpemy together with a classmate
R will help you both to develop your Uunderstanding and thinking, as well as improvoing your (mathematical)
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communication skills. And bein g able to convince someone that your reasoning is correct, initially verbally and
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then in writing, forms the hegart of the mathematical skill of ‘proof’. w
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e Series introduction
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Mathematical modelling igs where mathematics meets the ‘real world’. Thwere are many situations where people need
to make predictions or tdo understand what is happening in the world,e and mathematics frequently provides tools
to assist with this. Marithematicians will look at the real world situavtiion and attempt to capture the key aspects
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of it in the form omf equations, thereby building a model of realiRty. They will use this model to make predictions,
and where posasible test these against reality. If necessary, th ey will then attempt to improve the model in order
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to make bettCer predictions. Examples include weather prse diction and climate change modelling, forensic science
(to under-s tand what happened at an accident or crime sscene), modelling population change in the human, animal
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and plyant kingdoms, modelling aircraft and ship brehaviour, modelling financial markets and many others. In this
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coourse, we will be developing tools which are vital for modelling many of these situations.

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vi ■ Explore activities: These activities avre designed to offer problems for classroom usey. They require thought and
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R deliberation: some introduce aU new idea, others will extend your thinking, whiole others can support consolidation.
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The activities are often best approached by working in small groups and then sharing your ideas with each other
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and the class, as they areg not generally routine in nature. This is one of twhe ways in which you can develop problem-
solving skills and confiddence in handling unfamiliar questions. e
■ Questions labelledb rais P , M or PS: These are questions with a epavirticular emphasis on ‘Proof’, ‘Modelling’ or
‘Problem solvinmg’. They are designed to support you in preparRing for the new style of examination. They may or
may not be haarder than other questions in the exercise.
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■ The language of the explanatory sections makes much msore use of the words ‘we’, ‘us’ and ‘our’ than in previous
cour se-b ooks. This language invites and encourages eyosu to be an active participant rather than an observer, simply
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fopllowing instructions (‘you do this, then you doP trhat’). It is also the way that professional mathematicians usually
owrite about mathematics. The new examinatio ns may well present you with unfamiliar questions, and if you are vii
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used to being active in your mathematics, ytou will stand a better chance of being able to successfully handle such
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e At various points in the books, thenire are also web links to relevant Undergroundp Mathematics resources,
R which can be found on the free uUndergroundmathematics.org website. Undergroound Mathematics has the aim
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of producing engaging, rich m aterials for all students of Cambridge International AS & A Level Mathematics
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and similar qualifications.g These high-quality resources have the potential to simultaneously develop your
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mathematical thinking skills and your fluency in techniques, so we do encourage you to make good use of them.
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We wish you everym success as you embark on this course. R
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Julian Gilbey s
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London , 2018 e
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Past exam paper questions throu ghout are reproduced by permission of Cambridge Assessment International Education.
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Cambridge Assessment Intergnational Education bears no responsibility for the example answers to questions taken from its
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past question papers which are contained in this publication.
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The questions, exammple answers, marks awarded and/or comments thatR appear in this book were written by the author(s). In
examination, thea way marks would be awarded to answers like thes-e may be different.
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How to use this book C

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Throughout this book you will notice particular features that are designed to help your learning. This section
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provides a brieCf overview of these features.
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■In this cohapter you will learn how to: PREREQUISITEKNOW(cid:18)EDGE
uCnderstand the distinction between a sample and a population y
w ■■■ uexspe lraainnd wohmy nau smambeprlsin agn md eatphporde cmiaatey tbhee u nnescaetsissiftayc tfoorr ychoosing random sampsleist PWrohbeareb iilti tcyo m&e Ss tfartoimstics 1, Chapter 8 CnWoarhlmcauta lyal otdeui spsthrrooibbuuladtbi obilenit .aiebsl eu tsoin dgo the 1C hTeXchke~ yroNaun(rd2 s2ok,mi4l l2vs)a. rFiainbdle:
e recognise that a sample mean can be regarded as a random variable, aned urse the facts that a P(X<27)
vi σ2 v y b P(20<Xø23)
e Learning objectives indicate the imponritant p 2 TXhe~ raNn(d3o5m,1 2v)a.r Fiaibnlde:
R U o a P(X>32)
concepts within each chapter and help you to C b P(Xø30)

navigate through the coursebooke.
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d Prerequisitee knowledge exercises identify prior learning
ri that you nveied to have covered before starting the chapter.
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m Try thRe questions to identify any areas that you need to
KEY POINT 1.2
a rev iew before continuing with the chapter.
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Data in a stemC-and- Key point boxes contain s
leaf diagr-a m are a summary of the most sp
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ordereyd in rows of important methods, facts r
equapl widths. P WORKED EXAMPLE 5.1
o and formulae.
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it a SVhaorw(X t(h1a))t f=or3 s5a.mples of size 1 drawn from a fair six-sided die numbered 1,2,3,4,5 and 6, E(X(1))=313 and
w s 12
r b Work out E(X(2)) and Var(X(2)).
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R evi te Kadreiesy tlr etiaebrrunmtiinosg na. rTiesh aiem yPp aooriserts ahoningt hd tileisgrtmhritbse uidnet i iotnhn oe.Ur taonnipgivec bthoaldt y. oThue aA nsVEwa(eXrr(X(1)()1)==) =2611×1=2613×21+612+×6122×+61 3×+613C2+×o461×6+1p+4y25× ×6161+6×61 Yata opburleo m.baaby iclihtyo odsies ttroib durtaiown
glossary contains clear definitgions of these key terms. +52×61+62w×61 −3212
d 91 49 3e5
= − =
x bri b E(X(2))=21E6e(Xv)4+i21E12(X)=E(X)=321 Yalogueb craan, auss ey oexup heacvtea tfioounn d
m Var(X(2)R)=212Var(X)+212Var(X)=21Var(X)=21×1325=3254 E(X) and Var(X).
EEXXLOLORRE E5 .5.4 a
(cid:27) -
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Use Microsoft Excel, or the random number function on your calculator, to produce s
five random sin-gl e digit numbers. Work out the mean of the five random numbers s Worked examples provide stnep-by-step approaches to
Repeat this p rocess eight times. e µ µ
y answering questions. The left side shows a fully worked
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Plot the mpean values obtained as a frequency graph, together with the results from eaPch
memboer of your teaching group. What do you notice about the shape of your graph? solution, while the right side contains a commentary

WChat do you think would happen to the shape of the graph if you repeated thyis eFFTTx pFF00lRRaWWinAARRiDDng each step in the working.
process many more times? t
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What do you think may happen if you had plotted the means of 20 single digit
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e numbers chosen at random instead of 5 random numbers? Try it aend see.

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R e EEXXEERRCCII 55AA U ni TIP op
Explore boxes contain enrichment activities for extension C
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work. These activities promote greoup work and peer- A variable is d enoted
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to-peer discussion, and are idntegnded to deepen your by an uppeer-case letter Tip nboxes contain helpful
understanding of a concerpit. (Answers to the Explore and itsv ipossible values guidance about calculating
questions are provided ibn the Teacher’s Resource.) by thee same lower-case or checking your answers.
m leRtter.
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