Mathematics for the International Student:Mathematics HL options-Statistics and Probability for IB

208 Pages·2013·5.694 MB·English

by  Catherine Quinn, Peter Blythe, Robert Haese, Michael Haese

HAESE MATHEMATICS
Specialists in mathematics publishing
Mathematics
for the international student
Mathematics HL (Option):
Statistics and Probability
HL Topic 7
FM Topic 3
CCaatthheerriinnee QQuuiinnnn
PPeetteerr BBllyytthhee
RRoobbeerrtt HHaaeessee
MMiicchhaaeell HHaaeessee
for use with
IB Diploma
Programme
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MATHEMATICSFORTHEINTERNATIONALSTUDENT
MathematicsHL(Option):StatisticsandProbability
CatherineQuinn B.Sc.(Hons),Grad.Dip.Ed.,Ph.D.
PeterBlythe B.Sc.
RobertHaese B.Sc.
MichaelHaese B.Sc.(Hons.),Ph.D.
HaeseMathematics
152RichmondRoad,Marleston, SA5033,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haesemathematics.com.au
NationalLibraryofAustraliaCardNumber&ISBN 978-1-921972-31-7
©Haese&HarrisPublications2013
PublishedbyHaeseMathematics.
152RichmondRoad,Marleston, SA5033,AUSTRALIA
FirstEdition 2013
ArtworkbyBrianHouston.
CoverdesignbyPiotrPoturaj.
TypesetinAustraliabyDeanneGallasch.TypesetinTimesRoman10\Qw_.
PrintedinMalaysiathroughBookpacProductionServices,Singapore.
The textbook and its accompanying CD have been developed independently of the International
Baccalaureate Organization (IBO). The textbook and CD are in no way connected with, or endorsed by,
theIBO.
This book is copyright. Except as permitted by the CopyrightAct (any fair dealing for the purposes of
private study, research, criticism or review), no part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese
Mathematics.
Copying foreducational purposes:Where copies of part or the whole of the book are made under Part
VB of the CopyrightAct, the law requires that the educational institution or the body that administers it
has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the
CopyrightAgencyLimited.
Acknowledgements: While every attempt has been made to trace and acknowledge copyright, the authors
and publishers apologise for any accidental infringement where copyright has proved untraceable. They
wouldbepleasedtocometoasuitableagreementwiththerightfulowner.
Disclaimer:Alltheinternetaddresses(URLs)giveninthisbookwerevalidatthetimeofprinting.While
the authors and publisher regret any inconvenience that changes of address may cause readers, no
responsibilityforanysuchchangescanbeacceptedbyeithertheauthorsorthepublisher.
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FOREWORD
MathematicsHL(Option):StatisticsandProbabilityhasbeenwrittenasacompanionbooktothe
MathematicsHL(Core)textbook.Together,theyaimtoprovidestudentsandteacherswith
appropriatecoverageofthetwo-yearMathematicsHLCourse,tobefirstexaminedin2014.
Thisbookcoversallsub-topicssetoutinMathematicsHLOptionTopic7andFurtherMathematics
HLTopic3,StatisticsandProbability.
Theaimofthistopicistointroducestudentstothebasicconceptsandtechniquesofstatisticsand
probabilityandtheirapplications.
Detailedexplanationsandkeyfactsarehighlightedthroughoutthetext.Eachsub-topiccontains
numerousWorkedExamples,highlightingeachstepnecessarytoreachtheanswerforthatexample.
TheoryofKnowledgeisacorerequirementintheInternationalBaccalaureateDiplomaProgramme,
wherebystudentsareencouragedtothinkcriticallyandchallengetheassumptionsofknowledge.
DiscussiontopicsforTheoryofKnowledgehavebeenincludedonpages157to159.Theseaimto
helpstudentsdiscoverandexpresstheirviewsonknowledgeissues.
TheaccompanyingstudentCDincludesaPDFofthefulltextandaccesstospeciallydesigned
graphingsoftware.
GraphicscalculatorinstructionsforCasiofx-9860GPlus,Casiofx-CG20,TI-84PlusandTI-nspire
areavailablefromiconslocatedthroughoutthebook.
Fullyworkedsolutionsareprovidedatthebackofthetext,howeverstudentsareencouragedto
attempteachquestionbeforereferringtothesolution.
Itisnotourintentiontodefinethecourse.Teachersareencouragedtouseotherresources.Wehave
developedthisbookindependentlyoftheInternationalBaccalaureateOrganization(IBO)in
consultationwithexperiencedteachersofIBMathematics.TheTextisnotendorsedbytheIBO.
Inthischangingworldofmathematicseducation,webelievethatthecontextualapproachshownin
thisbook,withassociateduseoftechnology,willenhancethestudentsunderstanding,knowledge
andappreciationofmathematicsanditsuniversalapplications.
Wewelcomeyourfeedback.
Email: [email protected] CTQ PJB
Web: www.haesemathematics.com.au RCH PMH
ACKNOWLEDGEMENTS
The authors and publishers would like to thank all those teachers who offered advice and
encouragementonthisbook.
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USING THE INTERACTIVE STUDENT CD
TheinteractiveCDisidealforindependentstudy.
Students can revisit concepts taught in class and undertake their own revision
andpractice.TheCDalsohasthetextofthebook,allowingstudentstoleave
thetextbookatschoolandkeeptheCDathome.
By clicking on the relevant icon, a range of interactive features can be
accessed:
INTERACTIVE
(cid:2) Graphics calculator instructions for the Casio fx-9860G Plus, Casio fx- LINK
CG20,TI-84PlusandtheTI-nspire
(cid:2) Interactivelinkstographingsoftware
GRAPHICS
CALCULATOR
INSTRUCTIONS
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TABLEOFCONTENTS 5
TABLE OF CONTENTS
SYMBOLSANDNOTATIONUSEDINTHISBOOK 6
A Expectationalgebra 9
B Discreterandomvariables 26
C Continuousrandomvariables 42
D Probabilitygeneratingfunctions 52
E DistributionsofthesamplemeanandtheCentralLimitTheorem 66
F Pointestimation(unbiasedestimatorsandestimates) 82
G Confidenceintervalsformeans 90
H Significanceandhypothesistesting 100
I BivariateStatistics 124
ReviewsetA 146
ReviewsetB 148
ReviewsetC 151
ReviewsetD 153
THEORYOFKNOWLEDGE(TheCentralLimitTheorem) 157
THEORYOFKNOWLEDGE(PopulationParameters) 158
WORKEDSOLUTIONS 160
INDEX 207
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6
SYMBOLS AND NOTATION USED IN THIS BOOK
¼ is approximatelyequal to
> is greater than
> is greater than or equal to
< is less than
6 is less than or equal to
f......g the set of all elements......
2 is an elementof
2= is not an element of
N the set of all natural numbers f0, 1, 2, 3, ....g
Z the set of integers f0, §1, §2, §3, ....g
Q the set of rational numbers
R the set of real numbers
Z+ the set of positive integers f1, 2, 3, ....g
µ is a subset of
½ is a proper subset of
) implies that
)Á does not imply that
f : A!B f is a functionunder which each element of set A has an image in set B
f : x7!y f is a functionunder which x is mapped to y
f(x) the image of x under the function f
f ±g or f(g(x)) the compositefunctionof f and g
jxj the modulus or absolutevalue of x
[a, b] the closed interval a6x6b
]a, b[ the open interval a<x<b
un the nth term of a sequence or series with first term u1
fung the sequence with nth term un, if first term is u1
Sn the sum of the first n terms of a sequence
S1 the sum to infinity of a convergent series
Pn
ui u1+u2+u3+::::+un
i=1
Qn
ui u1£u2£u3£::::£un
i=1
lim f(x) the limit of f(x) as x tends to a
x!a
lim f(x) the limit of f(x) as x tends to a from the positive side of a
x!a+
lim f(x) the limit of f(x) as x tends to a from the negative side of a
x!a¡
maxfa, bg the maximumvalue of a or b
P1
cnxn the power series whose terms have form cnxn
n=0
dy
the derivativeof y with respect to x
dx
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7
f0(x) the derivativeof f(x) with respect to x
d2y
the second derivativeof y with respect to x
dx2
f00(x) the second derivativeof f(x) with respect to x
dny
the nth derivativeof y with respect to x
dxn
f(n)(x) the nth derivativeof f(x) with respect to x
R
y dx the indefiniteintegral of y with respect to x
R
b
y dx the definite integral of y with respect to x between the
a
limits x=a and x=b
ex exponentialfunctionof x
lnx the natural logarithmof x
sin, cos, tan the circular functions
csc, sec, cot the reciprocalcircularfunctions
arcsin, arccos, arctan the inverse circular functions
¡ ¢
n n!
r r!(n¡r)!
P(A) probabilityof event A
P(A0) probabilityof the event “not A”
P(AjB) probabilityof the event A given B
x1, x2, .... observations
P(x) probabilitydistributionfunction Px = P(X =x) of the discrete random
variableX
f(x) probabilitydensity function of the continuousrandom variable X
F(x) cumulativedistributionfunctionof the continuousrandom variable X
E(X) the expected value of the random variable X
Var(X) the variance of the random variableX
¹ populationmean Pn
(xi¡¹)2
¾2 populationvariance,the value ¾2 = i=1 , for a populationof size n
n
¾ populationstandarddeviation
x sample mean Pn
(xi¡x)2
s2 sample variance, the value s2 = i=1 , from a sample of size n
n n n
sn standarddeviationof the sample of size n
s2 unbiasedestimateof the populationvariance,the value
n¡1
Pn
(xi¡x)2
s2 = n s2 = i=1 , from a sample of size n
n¡1 n¡1 n n¡1
Pn
X the estimator of ¹, that is the function X = 1 Xi, where Xi, i=1, ...., n
n
i=1
are identicallydistributedindependentrandom variables each with mean ¹
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8
Pn
(Xi¡X)2
S2 the biased estimator of ¾2, that is the function S2 = i=1
n n n
where Xi, i = 1, ...., n are identically distributed independent random variables
each with variance ¾2
n
S2 the unbiasedestimatorof ¾2, that is the function S2 = S2
n¡1 n¡1 n¡1 n
DU(n) discreteuniform distributionwith parametern
B(1, p) Bernoullidistributionwith parameterp
B(n, p) binomialdistributionwith parametersn and p
Geo(p) geometricdistributionwith parameterp
NB(r, p) negativebinomialdistributionwith parametersr and p
Po(m) Poisson distributionwith mean m
X » DU(n) the random variable X has a discreteuniform distributionwith parametern
X » B(1, p) the random variable X has a Bernoullidistributionwith parameterp
X » B(n, p) the random variable X has a binomialdistributionwith parametersn and p
X » Geo(p) the random variable X has a geometricdistributionwith parameterp
X » NB(r, p) the random variable X has a negativebinomialdistributionwith parametersr and p
X » Po(m) the random variable X has a Poisson distributionwith mean m
U(a, b) continuousuniformdistributionwith parametersa and b
Exp(¸) exponentialdistributionwith mean 1
¸
N(¹, ¾2) normal distributionwith mean ¹ and variance ¾2
º number of degrees of freedom
t(º) Student’s t-distributionwith º degrees of freedom
X » U(a, b) the random variable X has a continuousuniformdistributionwith parametersa and b
X » Exp(¸) the random variable X has an exponentialdistributionwith mean 1
¸
X » N(¹, ¾2) the random variable X has a normal distributionwith mean ¹ and variance ¾2
T » t(º) the random variable T has the Student’s t-distributionwith º degrees of freedom
G(t) the probabilitygeneratingfunction E(tX) for a discrete random variableX which
takes values in N
p dependingon the context, a parameterof a distribution,a populationproportion,
or a p-value in a hypothesistest
pb a sample proportion
H0 null hypothesis
H1 alternativehypothesis
® significancelevel or probabilityof a Type I error
¯ probabilityof a Type II error
1¡¯ power of a hypothesistest
Cov(X, Y) covarianceof random variablesX and Y
½ product momentcorrelationcoefficient betweentwo random variables
R the sample product momentcorrelationcoefficient; an estimatorof ½
r the observed value of R for a given sample of bivariatedata; an estimateof ½
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STATISTICSANDPROBABILITY 9
A
EXPECTATION ALGEBRA
A random variable can take any one of a set of values from a given domain, according to given
probabilities. The domain may be discrete or continuous.
DISCRETE RANDOM VARIABLES
If X is a discrete random variable, then:
1 X has possible values x1, x2, x3, ..... To determinethe value of X we usually count.
P
2 X takes value xi with probabilitypi, where 06pi 61, i=1, 2, 3, ...., and pi =1.
3 X has a probabilitydistributionfunction(or probabilitymass function)
P(x), where P(xi)= P(X =xi)=pi, i=1, 2, 3, ..... P
reads “thesumforall
4 X has a cumulativediPstributionfunction(CDF) F(x), where xi6x valuesofxiless
F(x)= P(X 6x)= P(X =xi). thanorequaltox”.
x 6x
i
F(x) is the probabilitythat X takes a value less than or equal to x.
Examplesof discreteprobabilitydistributionsand random variables coveredin
the Core course include Bernoulli,Discrete Uniform,Binomial,and Poisson.
CONTINUOUS RANDOM VARIABLES
If X is a continuousrandom variable, then:
1 Thepossiblevaluesof X may be all x2R, or all realx in somedomain[a, b]. To determinethe
value of X we usually measure.
2 X has a continuousprobabilitydensity function(PDF) f(x), where:
y ² f(x)>0 for all x in the domain of f.
Z
y=f(x) 1
Area=1 ² f(x)dx=1 if the domain of f is R, or
¡1
Z
b
f(x)dx=1 if the domain of f is [a, b].
a
x
3 Suppose f and the PDF for X, have domain [a, b]. X has a cumulative distribution function
Z
x
(CDF) F(x), where F(x)= P(X 6x)= f(t)dt for x2[a, b].
a
y
² F(x) is the probabilitythat X takes a value less than
y=f(t)
or equalZto x. Area=F(x)
b
² F(b)= f(t)dt=1
a
t
a x b
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10 STATISTICSANDPROBABILITY
² The probabilitythat X takes a value in the interval
y
[c, d]µ[a, b] is given by
Z
d Area=P(c6X6d)
P(c6X 6d)= f(t)dt
c
=F(d)¡F(c)
t
a c d b
4 Since X has infinitely many possible values, the probability that X takes a single value X = x
is 0. However, since X is a continuousrandomvariable,for x2Z and x¡0:56X <x+0:5,
the value of X will be roundedto the integer x.
Thus, for x2Z, we define P(X =x)=P(x¡0:56X <x+0:5)
Z
x+0:5
= f(t)dt
x¡0:5
Youshouldrecognise
=F(x+0:5)¡F(x¡0:5)
theNormaldistribution
fromtheCorecourse.
Example1
Givenarandomvariable X »N(7:2,28), find P(X =10).
P(X =10)=P(9:56X <10:5)
¼0:0655
For a continuousrandom variable X, P(c6X <d)
=P(c6X 6d)
=P(c<X 6d)
=P(c<X <d)
sincethecorrespondingintegralsalldefinethesameareaunderthecurve y =f(t) between t=c
and t=d.
ExamplesofcontinuousprobabilitydistributionsandrandomvariablescoveredintheCorecourseinclude
ContinuousUniform, Exponential,and the Normal distribution.
EXPECTATION
The mean or expected value or expectationE(X) of a random variableX is defined as follows:
² If X is a discrete random variable with set of possible vaPlues x1, x2, .... and probabilitymass
function P(X =xi)=pi, i=1, 2, ...., E(X)=¹= xiP(X =xi)
Pi
= xipi, i=1, 2, ....
i
² If X is a continuousrandom variable with probabilitydensity function f(x) with domain [a, b],
Z
b
E(X)=¹= xf(x)dx.
a
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