INTRODUCTION TO INTRODUCTION TO TESTING OF HYPOTHESISTESTING OF HYPOTHESIS

SHWETA MOGRESHWETA MOGRE

Hypothesis TestingHypothesis Testing DefinitionDefinition:: HypothesisHypothesis

testingtesting isis aa procedureprocedureforfor makingmaking inferencesinferencesaboutabout aa populationpopulation..

Population

Parameter

Population

Sample

Statistic

Inference

PopulationPopulation

PopulationPopulation meansmeans thethe entireentire spectrumspectrum ofof aasystemsystem ofof interestinterest..

SampleSampleSampleSample SampleSample isis thatthat partpart ofof PopulationPopulation whichwhich wewe

selectselect forfor thethe purposepurpose ofof investigationinvestigation..

What Is Hypothesis ?What Is Hypothesis ?

AA hypothesishypothesis isis ananassumptionassumption aboutabout thethepopulationpopulation parameterparameter..

AA parameterparameter isis aa

What is a Hypothesis?

AA parameterparameter isis aacharacteristiccharacteristic ofof thethepopulation,population, likelike itsitsmeanmean oror variancevariance..

TheThe parameterparameter mustmust bebeidentifiedidentified beforebeforeanalysisanalysis..

Hypothesis is a Statement .

1984-1994 T/Maker Co.

What is Statement ?What is Statement ?

StatementStatement An statement is a declarative sentence which is An statement is a declarative sentence which is

true or false, but not both ,or in other words an true or false, but not both ,or in other words an statement is a declarative sentence which has a statement is a declarative sentence which has a definite truth values.definite truth values.definite truth values.definite truth values.

A hypothesis (plural "hypotheses") is a A hypothesis (plural "hypotheses") is a statement which may or may not be true. statement which may or may not be true.

Characteristics of HypothesisCharacteristics of Hypothesis HypothesisHypothesis shouldshould bebe clearclear andand preciseprecise.. HypothesisHypothesis shouldshould bebe statedstated asas farfar asas possiblepossible

inin mostmost simplesimple termsterms soso thatthat thethe samesame isis easilyeasilyunderstandableunderstandable byby allall concernedconcerned..understandableunderstandable byby allall concernedconcerned..

HypothesisHypothesis shouldshould statestate relationshiprelationship betweenbetweenvariables,variables, ifif itit happenshappens toto bebe aa relationalrelationalhypothesishypothesis..

HypothesisHypothesis mustmust actuallyactually explainexplain whatwhat ititclaimsclaims toto explainexplain..

Types of HypothesisTypes of Hypothesis

Null hypothesisNull hypothesis Alternative hypothesisAlternative hypothesis

Null HypothesisNull Hypothesis

It is a hypothesis which we want to test.It is a hypothesis which we want to test. It is denoted by HIt is denoted by H00 It is denoted by HIt is denoted by H00 HH00: Write statement here.: Write statement here.

Null HypothesisNull Hypothesis

NullNull hypothesishypothesis isis thethe hypothesishypothesis ofof nonodifference,difference, whichwhich showsshows thethe nullnull oror neutralneutralattitudeattitude ofof thethe statisticianstatistician..

Null HypothesisNull Hypothesis

ForFor example,example, inin aa clinicalclinical trialtrial ofof aa newnewdrug,drug, thethe nullnull hypothesishypothesis mightmight bebe thatthat thethenewnew drugdrug isis nono better,better, onon average,average, thanthan thethecurrentcurrent drugdrug.. WeWe wouldwould writewritecurrentcurrent drugdrug.. WeWe wouldwould writewrite

HH00:: ThereThere isis nono differencedifference betweenbetween thethe twotwodrugsdrugs onon averageaverage..

Alternative HypothesisAlternative Hypothesis

The hypothesis which differs from given null The hypothesis which differs from given null hypothesis.hypothesis.

It is denoted by HIt is denoted by H11 It is denoted by HIt is denoted by H11 HH11: Write statement here.: Write statement here.

Alternative HypothesisAlternative Hypothesis

TheThe alternativealternative hypothesishypothesis HH11,, isis aa statementstatement ofofwhatwhat aa statisticalstatistical hypothesishypothesis testtest isis setset upup totoestablishestablish..

ForFor example,example, inin aa clinicalclinical trialtrial ofof aa newnew drug,drug, ForFor example,example, inin aa clinicalclinical trialtrial ofof aa newnew drug,drug,thethe alternativealternative hypothesishypothesis mightmight bebe thatthat thethenewnew drugdrug hashas aa differentdifferent effect,effect, onon average,average,comparedcompared toto thatthat ofof thethe currentcurrent drugdrug..

Alternative HypothesisAlternative Hypothesis IfIf wewe wantwant toto testtest thethe hypothesishypothesis thatthat thethe populationpopulation

hashas aa specifiedspecified meanmean saysay 00 thenthen thethe nullnull hypothesishypothesiswouldwould bebe

HH00:: == 00andand thethe alternativealternative hypothesishypothesis couldcould bebe

1.1. HH :: 1.1. HH11:: 00

2.2. HH11:: > 00

Alternative HypothesisAlternative Hypothesis

The alternative hypothesis in 1 is known as a The alternative hypothesis in 1 is known as a two tailed alternative in 2 & 3 are known as two tailed alternative in 2 & 3 are known as two tailed alternative in 2 & 3 are known as two tailed alternative in 2 & 3 are known as left tailed & right tailed alternatives.left tailed & right tailed alternatives.

Critical RegionCritical Region TheThe testingtesting ofof hypothesishypothesis isis donedone onon thethe basisbasis

ofof thethe divisiondivision ofof samplesample spacespace intointo twotwomutuallymutually exclusiveexclusive regionsregions..

OneOne isis regionregion forfor acceptanceacceptance ofof HH andand anotheranother OneOne isis regionregion forfor acceptanceacceptance ofof HH00 andand anotheranotherisis regionregion forfor rejectionrejection ofof HH00..

Critical RegionCritical Region

Rejection Region (Critical Region))

Acceptance Region

There are two decisions that wemake; reject or fail to reject.Each could possibly be a wrong

Error

Each could possibly be a wrongdecision; therefore, there aretwo types of errors.

There are two types of ErrorsThere are two types of ErrorsType of I ErrorType of I Error Type of II ErrorType of II Error

Types of Error Types of Error

Decision based Decision based On sampleOn sample

HHo o is Trueis True HHo o is Falseis False

Reject HReject H00 Wrong Decision Wrong Decision (Type I error)(Type I error)

Right Decision Right Decision (No error)(No error)

Accept HAccept H00 Right Decision Right Decision (No error)(No error)

Wrong Decision Wrong Decision (Type II error)(Type II error)

Type I errorType I error

WhenWhen youyou rejectreject thethe nullnull hypothesishypothesis thatthat isisreallyreally truetrue isis knowknow asas TypeType II errorerror..

TheThe probabilityprobability ofof committingcommitting TypeType II errorerror isis TheThe probabilityprobability ofof committingcommitting TypeType II errorerror isiscalledcalled thethe sizesize ofof typetype II errorerror andand itit isis denoteddenotedbyby ..

Type II ErrorType II Error

The decision of accepting the null hypothesis The decision of accepting the null hypothesis H0 when it is false is called Type II error.H0 when it is false is called Type II error.

The probability of committing type II error is The probability of committing type II error is The probability of committing type II error is The probability of committing type II error is called the size of the type II error and it is called the size of the type II error and it is denoted by denoted by ..

Reduce probability of one errorand the other one goes up.

& Have an Inverse Relationship

Continue.Continue. Both Both & & can not be reduced at once so we can not be reduced at once so we

try to reduce try to reduce for a prefixed for a prefixed ..

The The seriousnessseriousness of the error types is of the error types is The The seriousnessseriousness of the error types is of the error types is determined by the specific situationsdetermined by the specific situations..

Consider a murder trial:Consider a murder trial:What are the hypotheses?What are the hypotheses?

Type I error Type I error ConsequenceConsequence::

Decide the defendant is guilty when really innocent

H0: defendant is innocent

Ha: defendant is guilty

ConsequenceConsequence::

Type II error Type II error ConsequenceConsequence::

An innocent person goes to prison

Decide defendant is not guilty when really guilty

A guilty person goes free

Continue.Continue. ItIt maymay bebe notednoted thatthat inin legallegal systemsystem thethe

commissioncommission ofof thethe errorerror designateddesignated asas TypeType IIerrorerror hashas beenbeen consideredconsidered toto bebe farfar moremoreseriousserious thanthan aa typetype IIII error,error, thusthus wewe saysay thatthat ititseriousserious thanthan aa typetype IIII error,error, thusthus wewe saysay thatthat ititisis aa moremore grievousgrievous mistakemistake toto convictconvict ananinnocentinnocent manman thanthan toto letlet aa guiltyguilty manman gogo freefree..

Lays Chip Company decides to accept atruckload of potatoes based upon resultsfrom a sample of potatoes from the truckload.

What are the hypotheses?

Type I error? Decide the potatoes are bad when they really are good

H0: potatoes good

Ha: potatoes bad

Type II error?

From the suppliers viewpoint,which is more serious? A type I error

From the chip companys viewpoint,which is more serious?

Decide the potatoes are bad when they really are good

Decide the potatoes are good when they really are bad

A type II error

Sometimes, the seriousness

depends upon the persons point-of-

view

The Level of SignificanceThe Level of Significance

The maximum value of the probability of type The maximum value of the probability of type I error which we would be willing to risk is I error which we would be willing to risk is I error which we would be willing to risk is I error which we would be willing to risk is called level of significance.called level of significance.

Continue.Continue. MostMost widelywidely usedused valuevalue ofof areare 00..0101 ((11%%)) &&

00..0505 ((55%%))..

55%% levellevel ofof significancesignificance meansmeans thatthat outout ofof 100100 55%% levellevel ofof significancesignificance meansmeans thatthat outout ofof 100100onon anan averageaverage therethere areare 55 chanceschances ofof rejectingrejectingaa correctcorrect HH00..

Power of The TestPower of The Test

ProbabilityProbability ofof rejectionrejection ofof aa wrongwrong nullnullhypothesishypothesis isis calledcalled powerpower ofof testtest..

SymbolicallySymbolically itit isis denoteddenoted byby SymbolicallySymbolically itit isis denoteddenoted byby11-- ..

Normal DistributionNormal Distribution

Level of Significance , , , , and the Rejection Region

Rejection Regions

Left Tail Hypothesis

/2

RegionsRight Tail Hypothesis

Two Tail Hypothesis

Statistical TestsStatistical Tests AA statisticalstatistical testtest isis aa twotwo actionaction decisiondecision

problem,problem, whenwhen aa decisiondecision ofof rejectionrejection ororacceptanceacceptance ofof nullnull hypothesishypothesis isis toto bebe takentakenonon thethe basisbasis ofof thethe informationinformation suppliedsupplied bybyonon thethe basisbasis ofof thethe informationinformation suppliedsupplied bybythethe samplesample observationobservation..

TheseThese testtest maymay bebe ofof twotwo typestypes1.1. OneOne tailedtailed testtest2.2. TwoTwo tailedtailed testtest

Procedure for Testing of Procedure for Testing of SignificanceSignificance

FormulationFormulation ofof nullnull hypothesishypothesis && alternativealternativehypothesishypothesis..

SelectionSelection ofof testtest statisticstatistic.. SelectionSelection ofof thethe appropriateappropriate levellevel ofof significancesignificance.. SelectionSelection ofof thethe appropriateappropriate levellevel ofof significancesignificance.. FormulationFormulation ofof criticalcritical regionregion.. ConductingConducting studystudy CalculateCalculate testtest statisticstatistic valuevalue.. MakingMaking decisiondecision..

Thank YouThank You