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Chapter 8: Introduction to

Hypothesis Testing

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Hypothesis Testing

The general goal of a hypothesis test is to

rule out chance (sampling error) as a

plausible eplanation for the results from a

research study!

Hypothesis testing is a techni"ue to help

determine #hether a specific treatment

has an effect on the indi$iduals in apopulation!

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Hypothesis Testing

The hypothesis test is used to e$aluate the

results from a research study in #hich

1! & sample is selected from thepopulation!

2! The treatment is administered to the

sample!

%! &fter treatment' the indi$iduals in the

sample are measured!

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Hypothesis Testing (cont!)

If the indi$iduals in the sample are

noticeably different from the indi$iduals in

the original population' #e ha$e e$idence

that the treatment has an effect!

Ho#e$er' it is also possible that the

difference bet#een the sample and the

population is simply sampling error

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Hypothesis Testing (cont!)

The purpose of the hypothesis test is to decidebet#een t#o eplanations:

1! The difference bet#een the sample and

the population can be eplained by samplingerror (there does not appear to be atreatment effect)

2! The difference bet#een the sample and

the population is too large to beeplained by sampling error (there does

appear to be a treatment effect)!

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The +ull Hypothesis' the &lpha ,e$el' the

Critical -egion' and the Test .tatistic

The follo#ing four steps outline the

process of hypothesis testing and

introduce some of the ne# terminology:

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.tate the hypotheses and select an le$el! The null hypothesis' H0' al#aysstates that the treatment has no effect (no

change' no difference)! &ccording to thenull hypothesis' the population mean aftertreatment is the same is it #as beforetreatment! The levelestablishes acriterion' or 0cutoff0' for maing a decisionabout the null hypothesis! The alpha le$elalso determines the ris of a Type I error!

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,ocate the critical region! The criticalregionconsists of outcomes that are $eryunliely to occur if the null hypothesis is

true! That is' the critical region is definedby sample means that are almostimpossible to obtain if the treatment has noeffect! The phrase 3almost impossible4means that these samples ha$e aprobability (p) that is less than the alphale$el!

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.tep %

Compute the test statistic! The test

statistic(in this chapter a 6score) forms

a ratio comparing the obtained difference

bet#een the sample mean and the

hypothesi6ed population mean $ersus the

amount of difference #e #ould epect

#ithout any treatment effect (the standarderror)!

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& large $alue for the test statistic sho#s that theobtained mean difference is more than #ould beepected if there is no treatment effect! If it islarge enough to be in the critical region' #e

conclude that the difference is significantorthat the treatment has a significant effect! In thiscase #e re7ect the null hypothesis! If the meandifference is relati$ely small' then the test

statistic #ill ha$e a lo# $alue! In this case' #econclude that the e$idence from the sample isnot sufficient' and the decision is fail to re7ect thenull hypothesis!

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rrors in Hypothesis Tests

9ust because the sample mean (follo#ingtreatment) is different from the originalpopulation mean does not necessarily

indicate that the treatment has caused achange!

ou should recall that there usually is

some discrepancy bet#een a samplemean and the population mean simply asa result of sampling error!

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rrors in Hypothesis Tests (cont!)

;ecause the hypothesis test relies on

sample data' and because sample data

are not completely reliable' there is al#ays

the ris that misleading data #ill cause thehypothesis test to reach a #rong

conclusion!

T#o types of error are possible!

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Type I rrors

& Type I erroroccurs #hen the sample data appear tosho# a treatment effect #hen' in fact' there is none!

In this case the researcher #ill re7ect the null hypothesisand falsely conclude that the treatment has an effect!

Type I errors are caused by unusual' unrepresentati$esamples! 9ust by chance the researcher selects anetreme sample #ith the result that the sample falls inthe critical region e$en though the treatment has no

effect! The hypothesis test is structured so that Type I errorsare $ery unliely< specifically' the probability of a Type Ierror is e"ual to the alpha le$el!

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Type II rrors

& Type II erroroccurs #hen the sample doesnot appear to ha$e been affected by thetreatment #hen' in fact' the treatment does ha$ean effect!

In this case' the researcher #ill fail to re7ect thenull hypothesis and falsely conclude that thetreatment does not ha$e an effect!

Type II errors are commonly the result of a $erysmall treatment effect! &lthough the treatmentdoes ha$e an effect' it is not large enough tosho# up in the research study!

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=irectional Tests

>hen a research study predicts a specificdirection for the treatment effect (increaseor decrease)' it is possible to incorporate

the directional prediction into thehypothesis test!

The result is called a directional testor a

one-tailed test! & directional test includesthe directional prediction in the statementof the hypotheses and in the location ofthe critical region!

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=irectional Tests (cont!)

?or eample' if the original population has amean of @ 8/ and the treatment is predicted toincrease the scores' then the null hypothesis#ould state that after treatment:

H/: A 8/ (there is no increase)

In this case' the entire critical region #ould belocated in the righthand tail of the distribution

because large $alues for B #ould demonstratethat there is an increase and #ould tend to re7ectthe null hypothesis!

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Beasuring ffect .i6e

& hypothesis test e$aluates the statistical

significanceof the results from a research study!

That is' the test determines #hether or not it is

liely that the obtained sample mean occurred#ithout any contribution from a treatment effect!

The hypothesis test is influenced not only by the

si6e of the treatment effect but also by the si6e of

the sample!

Thus' e$en a $ery small effect can be significant

if it is obser$ed in a $ery large sample!

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Beasuring ffect .i6e

;ecause a significant effect does not necessarily

mean a large effect' it is recommended that the

hypothesis test be accompanied by a measure

of the effect size! >e use Cohen@s d as a standardi6ed measure

of effect si6e!

Buch lie a 6score' Cohen=s dmeasures the

si6e of the mean difference in terms of the

standard de$iation!

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o#er of a Hypothesis Test

The powerof a hypothesis test is defined

is the probability that the test #ill re7ect the

null hypothesis #hen the treatment does

ha$e an effect!

The po#er of a test depends on a $ariety

of factors including the si6e of the

treatment effect and the si6e of thesample!

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