Applications of Hypothesis Testing for Environmental Science

282 Pages·2020·7.896 MB·English

by  Abbas F. M. Alkarkhi

Applications of Hypothesis Testing
for Environmental Science
Applications of
Hypothesis Testing for
Environmental Science
Abbas F. M. Alkarkhi
University of Kuala Lumpur (Unikl)–Malaysia
Elsevier
Radarweg29,POBox211,1000AEAmsterdam,Netherlands
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom
50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
Copyright©2021ElsevierInc.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,
electronicormechanical,includingphotocopying,recording,oranyinformationstorageand
retrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowtoseek
permission,furtherinformationaboutthePublisher’spermissionspoliciesandour
arrangementswithorganizationssuchastheCopyrightClearanceCenterandtheCopyright
LicensingAgency,canbefoundatourwebsite:www.elsevier.com/permissions.
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightby
thePublisher(otherthanasmaybenotedherein).
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchand
experiencebroadenourunderstanding,changesinresearchmethods,professionalpractices,
ormedicaltreatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgein
evaluatingandusinganyinformation,methods,compounds,orexperimentsdescribed
herein.Inusingsuchinformationormethodstheyshouldbemindfuloftheirownsafety
andthesafetyofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,or
editors,assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatter
ofproductsliability,negligenceorotherwise,orfromanyuseoroperationofanymethods,
products,instructions,orideascontainedinthematerialherein.
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
AcatalogrecordforthisbookisavailablefromtheLibraryofCongress
ISBN:978-0-12-824301-5
ForInformationonallElsevierpublications
visitourwebsiteathttps://www.elsevier.com/books-and-journals
Publisher:JosephHayton
AcquisitionsEditor:MarisaLaFleur
EditorialProjectManager:GraceLander
ProductionProjectManager:SruthiSatheesh
CoverDesigner:ChristianBilbow
TypesetbyMPSLimited,Chennai,India
Dedication
To the memory of my parents (deceased)
To my beloved children Atheer, Hibah, and Farah
To my beloved wife
Abbas F. M. Alkarkhi
Contents
Preface xi
1 Introductiontostatisticalhypothesistesting 1
Learningoutcomes 1
1.1 Introduction 1
1.2 Whatishypothesistesting? 2
1.3 Thegeneralprocedureforperformingstatistical
hypothesistesting 6
1.4 Proceduresforperforminghypothesistesting 10
1.5 Typesoferrors 11
Furtherreading 12
2 Z-testforone-samplemean 13
Learningoutcomes 13
2.1 Introduction 13
2.2 Whatisnormaldistribution? 14
2.3 Whatisstandardnormaldistribution? 15
2.4 Findingtheareaunderthenormalcurve 16
2.5 Hypothesistestingforonesamplemean(Z-test) 25
Furtherreading 32
3 t-testforone-samplemean 33
Learningoutcomes 33
3.1 Introduction 33
3.2 Whatistdistribution? 34
3.3 Findingthetcriticalvalues 35
3.4 Hypothesistestingforaone-samplemean(t-test) 39
Furtherreading 46
4 Z-testforonesampleproportion 47
Learningoutcomes 47
4.1 Introduction 47
4.2 WhatisBernoullidistribution? 48
4.3 WhatisBinomialdistribution? 48
4.4 Hypothesistestingforonesampleproportion(Z-test) 52
Furtherreading 61
viii Contents
5 Chi-squaretestforonesamplevariance 63
Learningoutcomes 63
5.1 Introduction 63
5.2 Whatischi-squaredistribution? 64
5.3 Findingthechi-squarevalues(areaunderthechi-squarecurve) 65
5.4 Hypothesistestingforone-samplevarianceorstandarddeviation 71
Furtherreading 78
6 Theobservedsignificancelevel(P-value)procedure 79
Learningoutcomes 79
6.1 Introduction 79
6.2 Whatistheobservedsignificancelevel? 80
6.3 ComputingtheP-valueforaZ-test 80
6.4 Testingonesamplemeanwhenthevariance
isknown:P-value 85
6.5 ComputingtheP-valueforat-test 90
6.6 Testingonesamplemeanwhenthevariance
isunknown:P-value 95
6.7 Testingonesampleproportion:P-value 101
6.8 ComputetheP-valueforachi-squaretest 107
6.9 Testingone-samplepopulationvarianceor
standarddeviation:P-value 112
Furtherreading 118
7 Intervalestimationforonepopulation 121
Learningoutcomes 121
7.1 Introduction 121
7.2 Whatisintervalestimation? 122
7.3 Confidence intervalforonepopulationmean 123
7.4 Confidence intervalforonepopulationproportion 128
7.5 Confidence intervalforonepopulationvariance 131
Furtherreading 133
8 Theintervalestimationprocedure:hypothesistesting
foronepopulation 135
Learningoutcomes 135
8.1 Introduction 135
8.2 Thestepsfortheconfidenceintervalprocedure 136
8.3 Confidence intervalfortestingonemeanvalue:Z-test 136
8.4 Confidence intervalfortestingonemeanvalue:t-test 140
8.5 Confidence intervalfortestingoneproportionvalue 145
8.6 Confidence intervalfortestingonestandarddeviationvalue 149
Furtherreading 154
Contents ix
9 Hypothesistestingforthedifferencebetweentwopopulations 155
Learningoutcomes 155
9.1 Introduction 155
9.2 Thegeneralprocedurefortestingtwosamples 156
9.3 Testingthedifference betweentwomeanswhenthe
samplesizeislarge 161
9.4 Testingthedifference betweentwomeanswhenthe
samplesizeissmall 169
9.5 Testingtwodependentsamples 179
9.6 Testingthedifference betweentwoproportions 189
9.7 Testingtheratiooftwovariances 199
Furtherreading 208
10 Intervalestimationforthedifferencebetweentwopopulations 209
Learningoutcomes 209
10.1 Introduction 209
10.2 Thestepsfortheconfidenceintervalprocedurefor
thedifferencebetweentwopopulations 210
10.3 Confidence intervalforthedifference betweentwomeans
whenthesamplesizeislarge 210
10.4 Confidence intervalforthedifference betweentwomeans
whenthesamplesizeissmall 214
10.5 Confidence intervalfordependentsamples 218
10.6 Confidence intervalforthedifference betweentwoproportions 221
10.7 Confidence intervalfortheratiooftwovariances 225
Furtherreading 229
11 Theintervalestimationprocedure:hypothesistestingfortwo
populations 231
Learningoutcomes 231
11.1 Introduction 231
11.2 Thestepsfortheconfidenceintervalprocedureforthe
difference betweentwopopulations 232
11.3 Confidence intervalfortestingthedifference between
twomeanswhensamplesizeislarge 232
11.4 Confidence intervalfortestingthedifference between
twomeanswhenthesamplesizeissmall 238
11.5 Confidence intervalfortestingtwodependentsamples 246
11.6 Confidence intervalfortestingthedifference between
twoproportions 251
11.7 Confidence intervalfortestingtheratiooftwovariances 257
Furtherreading 263
Appendix 265
Index 275
Preface
In this book applications of hypothesis testing for environmental science are
describedinaneasyandenjoyablestyletohelpandassistgraduateandpostgraduate
studentsandresearchersworkinginthefieldsofenvironmentalscience,environmen-
talengineering,andrelatedfields.Thelackofknowledgeofmanyresearchersabout
the concept and application of hypothesis testing in the environmental field, includ-
ing how to state the hypothesis, how to choose the most suitable test, and how to
interpret and draw conclusions motivated me to write this book and introduce
hypothesistestingforoneandtwoparameterscoveringthreeproceduresforhypoth-
esis testing (critical value, P-value, and confidence interval) and to connect the the-
ory to the environmental field to meet the needs of researchers and enable them to
carryoutvarioustestingproceduresanddrawconclusions.
The concept of hypothesis testing, the type of test, interpretation of the results,
anddrawingconclusionsarecarriedoutinastep-by-step,easy,andclear mannerto
enable researchers to understand it clearly and use it in their research. The book
focuses on the application of hypothesis testing for one and two population para-
meters including Z-test, t-test, chi-square test, and F-test, which are frequently used
by researchers in the environmental field. Most of the tutorials in the book use real
dataobtainedthroughover18yearsofresearchinthisarea.
Finally,Iwishtothankmybelovedfamilyfortheircontinuoussupport.Ialsowould
liketoexpressmygratitudetoUniversityofKualaLumpur(Unikl)foritssupport.
AbbasF. M.Alkarkhi
June2020
1
Introduction to statistical
hypothesis testing
Abstract
Chapter 1 is the introductory chapter that presents the philosophy of hypothesis testing
and the concept of null and alternative hypotheses, rejection region, nonrejection region,
significance level, and critical values. Furthermore, the general procedure for performing
hypothesis testing isgiven along with thetwo typesof errors that could occur whileper-
forming the procedure. Types of hypothesis testing including one-tailed (right-tailed and
left-tailed)andtwo-tailedtestsaredeliveredinasimpleandeasytounderstandway.This
chapter includes examples obtained from the field of environmental science to show the
applicationofhypothesistestinginthefieldofenvironmentalscienceandengineering.
Keywords: Null and alternative hypotheses; critical value; rejection and nonrejection
regions;significancelevel;typesIandIIerrors
Learning outcomes
Aftercompletingthischapter,readerswillbeableto:
Understandthephilosophybehindhypothesistestingidea;
(cid:1)
Knowthegeneralprocedureforperforminghypothesistesting;
(cid:1)
Knowhowtodefinethetwohypotheses(nullandalternative);
(cid:1)
Describethestepsonhowtosetupthenullandalternativehypothesesintermsofmathe-
(cid:1)
maticalforms;
Understandthedifferencebetweenrejectionandnonrejectionregions;
(cid:1)
Knowhowtoobtaincriticalvaluesfromaprobabilitydistributiontable;
(cid:1)
Knowthelevelofsignificanceandhowtouseit;
(cid:1)
Knowhowtousethesampledataincalculatingtheteststatistic;
(cid:1)
Describethetwotypesorerrors(typeIandtypeIIerrors);
(cid:1)
Understandthedifferencebetweentwo-tailedandone-tailedtests;
(cid:1)
Understand the procedure for applying hypothesis testing in the field of environmental
(cid:1)
scienceorengineering;
Explaintheoutputsandmatchtheresultstoenvironmentalissues;
(cid:1)
Knowhowtoextracthelpfulinferencesregardingtheproblemunderstudy.
(cid:1)
1.1 Introduction
Researchers usually carry out various experiments using statistical techniques
for collecting, analyzing, summarizing, and draw conclusions to gather
ApplicationsofHypothesisTestingforEnvironmentalScience.DOI:https://doi.org/10.1016/B978-0-12-824301-5.00006-X
©2021ElsevierInc.Allrightsreserved.
2 ApplicationsofHypothesisTestingforEnvironmentalScience
information regarding the behavior of populations under investigation and help
in making judgments regarding the purpose of study. Statistical hypothesis test-
ing is used to make judgments regarding any claim or issue using the sample
data to obtain the correct decision that guides researchers to accomplish the
research and achieve the objectives. Gathering information can be done using a
representative sample chosen from the same population under investigation.
Hypothesis testing is an important method to examine various claims or state-
ments regarding different issues to support or deny the claims in order to plan
for the next step.
Hypothesis testing concerning population proportions, population means, and
population variances is of the subject of this text. A Z-test is used to carry out
hypothesis testing regarding population proportions. Furthermore, a Z-test can be
used for population means when the sample size is large, and a t-test is used when
the sample size is small. A chi-square test and F-test are used to test a hypothesis
concerningoneandtwovariances.
As an example, consider a researcher who wants to test a claim regarding the
concentration of total suspended solids (TSSs) in the surface water of a river, the
claim says that the mean concentration of TSSs is 465 (mg/L). A sample should be
selected from the river and tested for the concentration of TSSs and then the results
summarized and analyzed to finally decide whether to support the claim or to
rejectit.
1.2 What is hypothesis testing?
Hypothesis testing is a statistical technique used to make judgments regarding
claims or statements related to populations. The opinion and definition of hypothe-
sis testing with the general procedure of conductingthe analysis are clarified before
beingappliedtoenvironmentaldata.
Any claim or statement made on population parameters (mean, standard devia-
tion, and proportion) is called a hypothesis. A sample should be chosen from the
population of interest to understand the behavior of the issue under investigation
andmakeajudgmentbasedonthecollectedsampleinformation.
We can identify two types of hypotheses: the null hypothesis and the alternative
hypothesis. The two hypotheses are mutually exclusive (they cannot occur
together).
The null hypothesis is a statistical hypothesis that says no real difference
exists between a population parameter (such as proportion, average, and stan-
dard deviation) and some claimed value (given value). The symbol for null
hypothesis is H .
0
The alternative hypothesis is a statistical hypothesis that represents the opposite
of the null hypothesis. An alternative hypothesis says that a real difference exists
between a population parameter and a given value. The symbol for alternative
hypothesisisH .
1