Chap 10

Two-sampleTests

USINGSTATISTICS @,BLK Foods 10.1 COMPARING THE MEANSOF TWO INDEPENDENT POPULATIONS ZTestfor theDifferenceBetween TwoMeans Pooled-Variance I Testfor the DifferenceBetween Two Means ConfidenceInterval Estimatefor the Difference BetweenTwo Means Separate-Variance t Testfor the Difference BetweenTwo Means 10.2 COMPARINGTHE MEANS OF TWO RELATEDPOPULATIONS Pairedt Test ConfidenceInterval Estimatefor the Mean Difference 10.3 COMPARINGTWO POPULATION PROPORTIONS ZTest for the DifferenceBetweenTwo Proportions ConfidenceInterval Estimatefor the Difference BetweenTwo Proportions

EXCEL COMPANION TO CHAPTER10 E10.1 Using theZTest for the Difference BetweenTwo Means(Unsummarized Data) E10.2 Using theZTest for the Difference BetweenTwo Means(SummarizedData) E10.3 Usingthe Pooled-Variance /Test (UnsummarizedData) E10.4 Usingthe Pooled-Variance /Test (SummarizedData) 810.5 Usingthe Separate-Variance I Test for the DifferenceBetweenTwo Means (UnsummarizedData) E 10.6 Usingthe Paired/ Testfor the Difference BetweenTWoMeans(UnsummarizedData) E10.7 Using theZTest for the Difference BetweenTwo Proportions(Summarized Data) E10.8 UsingtheFTest for the Difference BetweenTwoVariances(UnsummarizedData) E10.9 UsingtheFTest for the Difference BetweenTwo Variances(Summarized D ata)

10.4 F TESTFOR THE DIFFERENCE BETWEEN TWO VARIANCES Findins Lower-TailCritical Values

In this chapteq you learn how to use hypothesistesting for comparing the difference between: I

The means of two independentpopulations

t

The means of two related populations

I

Two proportions

I

The variancesoftwo independentpopulations

370

Tests CHAPTERTENTwo-Sample

Using Statistics@ BLK Foods Doesthe type of displayusedin a supermarketaffectthe salesof the regionalsalesmanagerfor BLK Foods,you want to comparethe sales ume of BLK cola whenthe productis placedin its the normalshelf to the salesvolume whenthe productis featuredin a specialend-aisledi To testthe effectiveness of the end-aisledisplays,you select20 stores the BLK supermarketchain that all experiencesimilar storewidesales umes.You thenrandomlyassignI 0 of the 20 storesto group I and l0 to 2.The managers of the l0 storesin group I placethe BLK colain the shelflocation,alongsidethe othercolaproducts.The l0 storesin group the specialend-aislepromotionaldisplay.At the end of oneweek,the BLK cola are recorded. How can you determine whether sales of BLK

usingthe end-aisledisplaysarethe sameasthosewhenthe colais the normalshelflocation?How canyou decideif the variabilityin BLK salesfrom storeto storeis the samefor the two types of displays?How could you use answersto thesequestionsto improvesalesof BLK colas?

Lfypothesis testingprovidesa confirmatoryapproachto dataanalysis.In Chapter9, procedures that relateto a si I llearned a varietyof commonlyusedhypothesis-testing sampleof dataselectedfrom a singlepopulation.In this chapter,you will learn how to hypothesis testing to procedures that compare statistics from two samples of data taken 'Are two populations. One such extension would be asking, the mean weekly salesof BLK

whenusingan end-aisledisplayequalto themeanweeklysalesof BLK colawhenplacedin normalshelflocation?"

1O.1 COMPARINGTHE MEANS OF TWO INDEPENDENT POPULATIONS Z Test for the Difference Between Two Means Supposethat you take a randomsampleof n, from one populationand a randomsample from a second population. The data collected in each sample are from a numerical vari

the first population,the meanis represented by the symbolp,, and the standarddevi represented by the symbolor. In the secondpopulation,the meanis represented by the p2, and the standarddeviation is representedby the symbol or. The test statrstic used to determinethe differencebetweenthe popu\ation meansis on the differencebetweenthe samplemeans(X t - X z).If you assumethatthesamples are domly and independentlyselectedfrom populationsthat are normally distributed,this follows the standardizednormal distribution.If the populationsare not normally di the Ztest is still appropriateif the samplesizesare large enough(typically n, andnr2 theCentralLimitTheorem in Section7.4).Equation(10.1)definesthe Ztest for the betweentwo means.

ZTESTFORTHE DIFFERENCE BETWEEN TWO MEANS -_-( X r - X ) - ( p r - p z )

l a ?T -o 4

r/-

\nt

n2

(10.1)

populations 3l I 10.l: Comparingthe Meansof Two Indepcndent

where X, : mean of the sample taken from population I : F1 meon of population I of : variance of population I

I

nr: size of the sample taken from population 1 X2:

meanof the sample taken from population 2

Ir2: mean of population2

) I )

o22: varianceof population 2 n2: size of the sample taken from population 2 The test statistic Z follows a standardizednormal distribution.

Pooled-Variancet Test for the Difference Between Two Means In most cases,the variancesof the two populationsare not known. The only information you usually have are the samplemeansand the samplevariances.If you assumethat the samplesare randomly and independentlyselectedfrom populationsthat are normally distributedand that the population variancesare equal (that is, oi - oi), you can use a pooled-variance t test to deterrnine whetherthere is a significant differencebefweenthe meansof the two populations.If the populations are not normally distributed,the pooled-variance/ test is still appropriateif the samplesizes are large enough(typically n, and nr> 30; seethe Central Limit Theorem in Section7.4). To test the null hypothesisof no difference in the means of two independentpopulations: HoiVt- F2orp,-Fz:0 againstthe alternativethat the rleans are not the same: lWhenthe two samplesizes areequal (that is, n, = n2), theeguation for the pooled variancecan be simpltfied

- 2 S ?+ s 4

La"p

2

H r : p t , + F 2 o r p ,- V 7 + 0 you usethe pooled-variance l-test statisticshown in Equation( 10.2).The pooled-variance / test gets its^namefrom the {act that the test statisticpools or combines the two sample variances Sf and S2rto compute Srj, the best estimate of the variance common to both populations under the assumptionthat the two populationvariancesare equal.l

POOLED-VARIANCE t TESTFORTHEDIFFERENCE BETWEEN TWO MEANS ., -_- ( X r - X ) - ( t r r

-trz)

(10.2)

l) L( t . i r ; l - + n z )| !'\nt

where

.s2- ( n r - \ S ? + @ z - D S j (rr -l)+(n2-l)

*qp2 : pooled variance

N1 : mean of the sample taken from population I s,2: variance of the sample taken from population I

.-l

n l : size of the sample taken from population I

X 2: meanof the sampletakenfrom population2 S22: varianceof the sampletakenfrom population2 nr: sizeof the sampletakenfrom population2 The test statistic / followsa 1distributionwith nt * ttt - 2 degreesof freedom.

372

CHAPTERTENTwo-samoleTests

Thepooled-variance /-teststatisticfollowsa I distributionwith n , * nt - 2 degreesof For a givenlevelof significance,crt,, in a twotail test,you rejectthenull hypothesisif the / teststatisticis greaterthanthe upper-tailcritical valuefrom the / distributionor ifthe teststatisticis lessthanthe lower-tailcriticalvaluefrom the / distribution.Fizure 10.I regionsof rejection.In a one-tailtestin which therejectionregionis in the lowertail, you null hypothesis if thecomputedteststatisticis lessthanthelower-tailcriticalvaluefromthet bution.[n a one-tailtestin whichtherejectionregionis in theuppertail.you rejectthenull esisif thecomputedteststatisticis greaterthanthe upper-tailcriticalvaluefrom the/

10.1 FIGURE Regionsof rejection and nonrejectionfor the oooled-variance t test for the difference betweenthe means (two-tailtest)

To demonstrate the useof thepooled-variance I test,returnto the Using Statistics on page370.Youwantto determinewhetherthe meanweeklysalesof BLK colaarethe whenusinga normalshelflocationandwhenusingan end-aisledisplay.Therearetwo tionsof interest.The first populationis the setof all possibleweeklysalesof BLK colaifall BLK supermarkets usedthe normalshelflocation.The secondpopulationis the setof all sibleweeklysalesof BLK cola if all the BLK supermarkets usedthe end-aisledisplays. first samplecontainsthe weeklysalesof BLK cola from the 10 storesselectedto usethe mal shelflocation,and the secondsamplecontainsthe weeklysalesof BLK cola fromthe storesselectedto usethe end-aisledisplay.Table 10.1containsthe cola sales(in number cases)for the two samples(seethe!![ffr file). TABLE 10.1 ComparingBLKCola WeeklySalesfrom Two DifferentDisplay (in Number Locations of Cases)

Display Location Normal

22 40

34 64

52 84

62 56

30 59

52 83

7l 66

76 90

54 77

The null andalternativehypotheses are Ho:h:p2orp, -Vz:0 Hr: p,,+ F2or p, - V2+0 Assumingthat the samplesare from underlyingnormal populationshaving equal ances,you can usethe pooled-variance I test.The r test statisticfollows a / distribution l0 + l0 - 2 : 18 desreesof freedom.Usinethe cr: 0.05levelof sisnificance. vou divide rejectionregioninto the two tails for this two-tail test(thatis, two equalpartsof 0.025 TableE.3 showsthat the critical valuesfor this two-tailtestare +2.1009and -2.1009. shownin Figure10.2,the decisionrule is RejectHoif t > rrs +2.1009 orift < -lra: -2.1009; otherwise,do not rejectHn.

10.1: Comparing the Means of Two IndependentPopulations

10.2 test of isfor the betweenthe at the 0.05 level ficancewith of freedom

t

-2.100e

Region r nofo fI Reiection :tion I I Critical Value

Regionof Nonrejection

I Regionof lRejection Critical\r-f;, Value

is0.0070. andthep-value forthistestis-3.0446, t statistla lromFiguel0 3,thecomputa0 10.3 :ft Excelttest for the two locations

5ttJ 72 3flr,6rt8 r5t3333 t0 t0 Verlancc 25frxt66 llypolhcdredtecn Dlfierenco 0 CT t8 t Stat 30{{6 Pft<'{ orrroll 0J035 Crldcal one{all 1.73t1 F{f<-{ tro{6ll 0"0d?0

SecrionE10.3 to create

?.rmt

UsingEquation(10.2)on page371 andthe descriptivestatisticsprovidedin Figure10.3,

- (-x t - x ) - ( t q - u z )

l ^ - , (r

l)

1 "\'t

n z)

tJ:l -+-l

where

z-DSl s3=( n(rn- r\ -S1?) ++(@ n2-l) 9(3s0.6778) + 9(l 57.3333) = 254.0056 9 +9 Therefore,

(s0.3-72.0\-0.0

zs+.ooso[! + l) ( r 0 r 0)

=#

= 4.0446

{50.801

Yourejectthenull hypothesisbecauset : -3 .0446< /r 8 -2.1009.Thep-value(ascomputed from MicrosoftExcel)is 0.0070.In otherwords,the probabilitythat t> 3.0446or t < -3.0446 is equalto 0.0070.Thisp-value indicatesthat if the populationmeansare equal,the probability of observinga differencethis largeor largerin the two samplemeansis only 0.0070.Because

374

CHAPTER TEN Two-Samole Tests

thep-valueis lessthana : 0.05,thereis sufficientevidenceto rejectthe null hypothesis. can concludethat the meansalesaredifferentfor the normalshelf locationand theend-a location.Basedon theseresults,the salesare lower for the normal location(thanfor end-aislelocation). Example10.I providesanotherapplicationof thepooled-variance t-test.

E X A M P L E1 0 . 1

TESTING FORTHEDIFFERENCE IN THEMEANDELTVERY TIMES

A local pizzarestaurantand a local branch ofa national chain are located acrossthe streetfi' a college campus.The local pizza restaurantadvertisesthat it delivers to the dormitoriesfal than the national chain. In order to determine whether this advertisementis valid, you andso friends have decided to order 10 pizzas from the localpizza restaurantand l0 pizzasfromt national chain, all at different times. The delivery times, in minutes (see the [[!!@l!fiI are shown in Table 10.2.

TABLE 10.2 DeliveryTimesfor Local PizzaRestaurant and NationalPizzaChain

Local

Chain

16.8 11.'l 15.6 16.7 l 1. 5

22.0 t5.2 18.7 15.6 20.8

Chain

Local

18.1 14.1 21.8 13.9 20.8

19.5 17.0 19.5 16.5 24.0

At the 0.05 level of significance, is there evidencethat the mean delivery time for theloc ptzza restaurantis less than the mean delivery time for the national pizza chain?

SOLUTION Becauseyou want to know whether the mean is lower for the local pizza resta rant than for the national pizza chain, you have a one-tail test with the following null and alte native hypotheses:

Ho:vt ) p, (The mean delivery time for the local pizza restaurantis equal to or greater than the mean delivery time for the national pizza chain.)

Hi vr < F2 (The mean delivery time for the local pizza restaurantis lessthan the mean delivery time for the national pizza chain.) Figure 10.4 displays Microsoft Excel results of the pooled I test for these data.

FIGURE10.4 MicrosoftExcelresults of the pooled t test for the pizzadelivery time data

See Section E10.3 to create this.

Local 4 :Mean 5 ,Variance 6 Obseryallons 7 ,PooledVarlance I HypotheslzedMean Olfierence s jdf 10 ,t Siat

l1 ;Pff.-0 ono-tail 1_2. t Crltlcalone-tall 13 rP[f<-$ two-tall 14 rt Crlticaltwo-tail

Cilein

16.7 18.S 9.58228.2151 10 10 8.8987 0 18 -1.6341 0.0598 1.7341 0.1196 ?.1009

10.I : Comparingthe Meansof Two IndependentPopulations 37 5

(10.2)on page371, UsingEquation

(Xt-X)-(trr-pz)

rl Frr. nz) " l n r ! . from faster some m the file),

where

e2 _

(nr-l)Si+(rz-l)Sz2 (rr-l)+(n2-l) 9 ( 9 . s 8 2+2 e) ( 8 . 2 l s l=) 9 +9

8.8987

Therefore.

( t 6 . 7- r 8 . 8 8 ) - 0 . 0 - 2 . 1 8

sqszfl+I I (10 r0)

ffi

= -1.6341

local

staurlter-

Youdo not rejectthe null hypothesis because t: -1.6341> /r8 -1.7341.Thep-value(as computedfrom MicrosoftExcel)is 0.0598.This p-valueindicatesthat the probabilitythat t < -1 .6341is equalto 0.0598.In otherwords,if the populationmeansareequal,the probability thatthe samplemeandeliverytime for the localpizzarestaurant is at least2.18minutesfasterthanthe nationalchainis 0.0598.Becausethep-valueis greaterthan cr : 0.05, thereis insufficientevidenceto rejectthe null hypothesis. Basedon theseresults,therers insufficientevidencefor the localpizzarestaurant to makethe advertisingclaim that it has a fasterdelivervtime.

In testing for the difference between the means, you assumethat the populations are normally distribute4 with equal variances.For situations in which the two populations have equal variances,the pooled-variance/ test is robust (or not sensitive) to moderate departuresfrom the assumption of normality, provided that the sample sizes are large. ln such situations, you can use the pooled-varianceI test without serious effects on its power. However, if you cannot assumethat the data in each group are from normally distributed populations, you have two choices.You can use a nonparametricprocedure,such as the Wilcoxon rank sum test (covered in Section 12.5),that doesnot dependon the assumptionof normality for the two populations, or you can use a normalizing transformation (seereference5) on each of the outcomesand then use the pooled-varianceI test. To check the assumptionof normality in each of the two groups, observethe box-andwhisker plot of the salesfor the two display locationsin Figure 10.5.There appearsto be only moderate departurefrom normality, so the assumptionof normality neededfor the / test is not seriously violated.

37 6

Tests TEN Two-sample CHAPTER Box-and-WhiskerPlot for Sales Location

10.5 FIGURE MicrosoftExcelbox-andwhiskerplot for the sales for two aislelocations

See Section E3.4 to create this.

Confidence Interval Estimate of the Difference Between Two Insteadof, or in addition to, testing for the difference in the meansof two independent tions, you can use Equation(10.3)to developa confidenceintervalestimateof the differencei the means.

OF THEDIFFERENCE ESTIMATE INTERVAL CONFIDENCE BETWEEN TWO MEANS

f'l tX,- x rl+,,,*,,-r./s;[,f z-.1 r'Ir, . nz

(10.3)

)

or

lAt-

A2l-trt+rt-2

trr-r) \*o\,,

n z)

1 where /,, +," -2 is the critical value of the I distribution with n, * nz - 2 degreesof freedom for an area of alZ in the upper tail.

Using 95% confidence,the samplestatisticsreportedin Figure10.3on page373and ( 10.3), Equation

andr,, = 2.1009: nr = 50.3,nr = 10,Xz = 72.n2 = 10.S; = 254.0056, - 7 2 ) ! ( 2 . t 0 0 e ) 2s4.oos6(! (50.3 * Il (10 t0)

- 2t.7X (2.r00ex7.| 275) - 2 1 . 7t 1 4 . 9 7 - 3 6 . 6 7S p r - S t 2{ - 6 . 7 3

10.1:Comparing theMeansof TwoIndependent Populations377 Therefore,you are 95% confident that the differencein meansalesbetweenthe normal shelf locationandthe end-aislelocationis between-36.67 casesof cola and-6.73 casesof cola.In otherwords,the end-aislelocation sells,on average,6.73 to 36.67casesmore than the normal aisle location. From a hypothesistesting perspective,becausethe interval does not include zero,you rejectthe null hypothesisof no differencebetweenthe meansof the rwo populations.

Separate-Variancet Test for the Difference Between Two Means In testingfor the differencebetweenthe meansof two independentpopulationswhen the population variancesareassumedto be equal,the samplevariancesarepooledtogetherinto a common estimat", 53. However,if you cannot make this assumption,then the pooled-variance I test is inappropriate.In this case,it is more appropriateto use the separate-varianceI test developedby Satterthwaite(seereference4). In the Satterthwaiteapproximationprocedure, you includethe two separatesamplevariancesin the computationof the /-teststatistic-hence, / test.The computationsfor the separate-variance / test arecomplithe nameseparate-variance presents the outputfrom the sepacatedbut canbe carriedout by Microsoft Excel. Figure 10.6 rate-varianceI test from Microsoft Excel for the cola data.

10.6 Excelresults separate-variance forthe display data

'e*

Anmlng Two-Scmplo

Oiccrntlom llypottrdzed lcan Dlfirrencr

E10.5to create

df Stt P(f<-! onc{all Crltlcal onc-tcll P(f<-Q mo{all

Crhlcalrwo.rall

Vcrhnccr

350.6ilt0, tf/3EB 10 10 0 15 3-0tr6 OIIF9

r:i{rei o"mn, 2.1$a

In Figure 10.6,the test statisticis t: -3.0446 and thep-valueis 0.0077< 0.05.Thus, the resultsfor the separate-variance t test are almost exactlythe sameas thoseof the pooledvarianceI test. The assumptionof equality of population varianceshad no real effect on the t testsconflict results.Sometimes,however,the resultsfrom the pooled-and separate-variance becausethe assumptionof equalvariancesis violated.Therefore,it is importantthat you evaluate the assumptionsand use thoseresultsas a guide in appropriatelyselectinga test procedure.In Section10.4.the F test is usedto determinewhetherthereis evidenceof a difference in the two populationvariances.The resultsof that test can help you determinewhich of the I tests-pooled-varianceor separate-variance-ismore appropriate.

the Basics 10.1 Givena sampleof nr:40 from a population with known standarddeviation o, : 20 and an independentsampleof nr:50 from another ion with known standarddeviation o, : 10"what rcvalueof the Z test statisticfor testingHo: trr : p2if =72 andXr: ee't

10.2 What is your decisionin Probleml0.l if you are testingHo: t\: F2 againstthe two-tail alternativeHl Fr + ;j, using the level of significancecr:0.01? 10.3 What is thep-valuein Probleml0.l if you are testingHo: l\: lr2 againstthe two-tail alternative11,: tt1* 1tr?

378

CHAPTERTENTwo-SamoleTests

10.4 Assumethat you havea sampleof r, : 8, with the samplemeanX1: 42 anda samplestandard deviationof ,S,: 4. and vou have an independentsampleoflr: l5 from anotherpopulationwith a samplemeanof X 2 : 34 and the samplestandarddeviation S, : 5. /-teststatistic a. What is the value of the pooled-variance for testingHo: 1tr: 1t"r? b. In finding the critical value ofthe test statistic/, how manydegreesof freedomarethere? c. Usingthe levelof significancecx,:0.01,whatis the critHo: trt,t trt, ical valuefor a one-tailtestof the hypothesis againstthe alternativeHr: 1tr> 1tr? d. What is your statisticaldecision? 10.5 What assumptions aboutthe two populationsare necessary in Problem10.4? 10.6 Referringto Problem10.4,constructa 95o/oconfidence interval estimate of the population difference betweenF1andpr.

Applying the Concepts 10.7 Theoperations managerat a light bulb factory wantsto determinewhetherthereis any differencein the mean life expectancyof bulbs manufacturedon two differenttypesof machines.The populationstandarddeviationof machineI is 110hoursandof machineII is 125hours.A randomsampleof 25 light bulbs from machineI indicatesa samplemeanof 375 hours,and a similarsampleof 25 from machineII indicatesa sample meanof 362 hours. a. Using the 0.05 level of significance,is there any evidenceof a differencein the meanlife of bulbsproduced by the two typesof machines? b. Computethep-valuein (a) andinterpretits meaning. 10.8 The purchasingdirector for an industrial parts factory is investigatingthe possibility of purchasinga new type of milling machine.She determines that the new machinewill be boughtif thereis evidencethat the partsproducedhavea highermeanbreaking strengththan thosefrom the old machine.The population standarddeviationofthe breakingstrengthfor the old machineis l0 kilogramsandfor the newmachineis 9 kilograms.A sampleof 100partstakenfrom the old machine indicatesa samplemeanof 65 kilograms,and a similar sampleof 100 from the new machineindicatesa sample meanof 72 kilograms. a. Usingthe0.01levelof significance,is thereevidencethat the purchasingdirectorshouldbuy the new machine? b. Computethep-valuein (a) andinterpretits meaning. 10.9 Millions of dollarsarespenteachyearon diet foods. Trendssuchas the low-fat diet or the low-carbAtkins diet

have led to a host of new products.A study by Dr. Sternof the PhiladelphiaVeteransAdministration comparedweight lossbetweenobesepatientsona diet and obesepatientson a low-carb diet (Extracted R. Bazell. "Studv CastsDoubt on Advantasesof Diet," msnbc.com,May 17,2004).Let p, meannumberof poundsobesepatientson a low-fat lose in six monthsand p, representthe mean pounds obese patients on a low-carb diet lose i

months. a. State the null and alternative hypothesesifyou

testwhetherthe meanweisht lossbetweenthetwo is equal. b. In the contextof this study,what is the meaning TypeI error? c. In the contextof this study,what is the meaning Type II error? d. Suppose thata sampleof 100obesepatientsona diet losta meanof 7.6poundsin six months,witha dard deviationof 3.2 pounds,while a sampleof obesepatientson a low-carbdiet lost a meanof pounds in six months,with a standarddeviationof pounds.Assuming that the population variances is there equalandusinga 0.05levelof significance, dence of a difference in the mean weight loss of patients between the low-fat and low-carb diets?

10.10 Whendo childrenin the United Statesdevelop erencesfor brand-nameproducts?In a studyreportedin Journal of ConsumerPsychology(Extractedfrom G. AchenreinerandD. R. John,"The Meaningof Brand to Children:A DevelopmentalInvestigation,"Journal ConsumerPsychology,2003, 13(3), pp. 205-219), ketersshowedchildren identicalpicturesof athletic One picture was labeled Nike, and one was labeled K-

The childrenwereaskedto evaluatethe shoesbasedon qualiry price,prestige,favorableness, appearance, and erencefor owning.A scorefrom 2 (highestproduct tion possible)to1(lowest productevaluationpossible) recordedfor each child. The following table reports resultsof the study: Age by Brand

Age8 Nike K-Mart Age12 Nike K-Mart Age15 Nike K-Mart

Sample Size

Sample Mean

SampleStandard Deviation

27 22

0.89 0.86

0.98

39 4l

0.88 0.09

1.01 1.08

35 33

0.41 -0.29

0.81 0.92

r.07

'il l0.l:

a. Conducta pooled-variance/ test for the difference betweentwo meansfor eachof the three age groups. Usea levelof significanceof 0.05. b. Whatassumptionsareneededto conductthe testsin (a)? c. Write a brief summaryof your findings. 10.11 According to a survey conducted in October2001,consumerswere trying to reduce their credit card debt (Extractedfrom M. Price, "CreditDebtsGet Cut Down to Size,"Newsday,November 25,2001,p. F3). Basedon a sampleof 1,000consumers in October2001 and in October2000, the meancredit carddebtwas $2,41I in October2001 as comparedto $2,814in October2000.Supposethat the standarddeviationwas$847.43 in October2001and $976.93in October 2000. a. Assumingthat the populationvariancesfrom both years areequal,is there evidencethat the mean credit card debtwaslower in October2001 than in October2000? (Usethe cx: 0.05levelof significance.) b. Determinethep-value in (a) and interpretits meaning. c. Assumingthat the populationvariancesfrom both years are equal,constructand interpret a 95o/oconfidence intervalestimateof the differencebetweenthe populationmeansin October2001andOctober2000. 10,12 The Computer Anxiety Rating Scale (CARS) measuresan individual'slevel of computer anxiety,on a scalefrom 20 (no anxiety) to (highest level of anxiety). Researchersat Miami 100 ty administeredCARS to l'12 businessstudents. of the objectives of the study was to determrne thereis a differencein the level of computeranxrexperiencedby female and male businessstudents. foundthe following: Males

40.26 13.35 100

Females 36.85 9.42 72

' ExtractedfromT Broome and D. Havelka, "Determinants ComputerAnxiety in BusinessStudents,"The Review of Business

Systems, Spring2002,6(2),pp. 9-16

At the 0.05level ofsignificance,is thereevidenceofa differencein the meancomputeranxietyexperiencedby femaleandmalebusinessstudents? Determinethep-value and interpretits meaning. Whatassumptionsdo you have to make about the two populations in orderto justify the useofthe r test? 13 A companymakingplasticoptical componentswas ing inconsistencies in an optical measurementcalled Twodifferenttypes of pins usedin the mold produced followingresults:

Comparing the Means of Two IndependentPopulations

Taper Locks

1.262 0.297

x ,s n20

319

Locking Pins

0.561 0.307 20

Extracted in YourProcess? WhoYa Source: fromJ.Duncan,"Ghosts May2005,pp.52-57. Goingto Call?" QualityProgress, a. Assumingthat the populationvariancesare equaland the populationsare normally distributed,at the 0.05 levelofsignificance,is thereevidenceofa differencein the meansbetweentaperlocks and locking pins? b. Repeat(a), assumingthat the population variancesare not equal. c. Comparethe resultsof (a) and(b). 10.14 A bankwith a branchlocatedin a commercialdistrict of a city has developedan improvedprocessfor serving customersduring the noon-to-l p.m. lunch period.The waiting time (operationallydefined as the time elapsed from when the customerentersthe line until he or she reachesthe teller window) of all customersduringthis hour is recordedover a period of one week.A randomsample of 15 customersis selected(and stored in the file El[trED, and the results(in minutes)are as follows: 4.2r 5.55 3.02 5.13 4.77 2.34 3.54 3.20 4.s0 6.10 0.38 5.12 6.46 6.19 3.79 Supposethat anotherbranch,locatedin a residentialarea, is also concernedwith the noon-to-l p.m. lunchperiod.A randomsampleof l5 customersis selected(and storedin the file EE&[D, and the resultsare as follows: 9.66 5.90 8.02 s.79 8.13 3.82 8.01 8.35 10.49 6.68 5.64 4.08 6.r7 9.91 s.47 a. Assumingthat the populationvariancesfrom both banks are equal,is there evidenceof a differencein the mean (Usea: 0.05.) waitingtime betweenthe two branches? b. Determinethep-value in (a) and interpretits meaning. is necessary in (a)? c. What otherassumption d. Assuming that the population variancesfrom both branchesare equal,constructand interpreta95o/oconfidenceinterval estimateof the differencebetweenthe populationmeansin the two branches. 10.15 RepeatProblem10.14(a), assumingthatthepopulation variances in the two branches are not equal. Comparethe resultswith thoseof Probleml0.la (a). 10.16 A problem with a telephoneline that preventsa customerfrom receivingor making calls is disconcertingto both the customerand the telephonecompany.The datain the file EfiftllE represent samples of 20 problems reportedto two different offices of a telephonecompany and the time to clear theseproblems(in minutes)from the customers'lines:

380

CHAPTERTEN Two-SamoleTests

Central Office I Time to Clear Problems(Minutes) 1 . 4 8 1 . 1 50 . 7 8 2 . 8 5 0 . 5 2 1 . 6 04 . 1 5 3 . 9 7 1 . 4 8 3 . 1 0 1.02 0.53 0.93 1.60 0.80 1.05 6.32 3.93 s.45 0.97 Central Office ll Time to Clear Problems(Minutes) 7.55 3.7s 0.10 1.r0 0.60 0.s2 3.30 2.10 0.58 4.02 3.75 0.65 1.92 0.60 1.53 4.23 0.08 1.48 r.65 0.72 a. Assumingthat the populationvariancesfrom both offices are equal,is there evidenceof a differencein the meanwaiting times betweenthe two offices?(Use o(: 0.05.) b. Determinethep-valuein (a) andinterpretits meaning. c. What otherassumption is necessary in (a)? d. Assuming that the population variancesfrom both offices are equal,constructand interpreta 95Yoconfidenceinterval estimateof the differencebetweenthe populationmeansin the two offices. 10.17 RepeatProblem10.16(a), assumingthatthe population variancesin the two offices are not equal.Compare theresultswith thoseof Problem10.16(a). 10.18 In intaglioprinting,a designor figure is carved beneaththe surfaceofhard metalor stone.Supposethat an experimentis designedto comparedifferencesin mean surfacehardnessof steelplatesused in intaglio printing (measuredin indentationnumbers),basedon two different surfaceconditions-untreatedand treatedby lightly polishingwith emerypaper.In the experiment,40 steelplates are randomlyassigned-20 that are untreated,and20 that are treated.The data are shownhere and storedin the file

@EE' Untreated r64.368 1s9.018 153.871 165.096 157.184 154.496 160.920 164.917 169.09r 175.276

177.t35 163.903 167.802 160.818 t67.433 163.s38 164.525 t'71.230 174.964 1 6 6 . 3I 1

Treated

r58.239 r38.216 168.006 t49.654 145.456 16 8 . 7 18 t54.321 162.763 161.020 167.706

150.226 155.620 151.233 158.653 tst.204 150.869 t61.657 r57.016 156.670 t47.920

a. Assumingthat the populationvariancesfrom both conditions are equal,is thereevidenceofa differencein the mean surfacehardnessbetweenuntreatedand treated steelplates?(Usecr: 0.05.) b. Determinethep-valuein (a) andinterpretits meaning. c. What otherassumptionis necessaryin (a)? d. Assuming that the population variancesfrom untreatedandtreatedsteelplatesare equal,constructand interpreta 95%oconfidenceinterval estimateof the

differencebetweenthe population meansin the conditions. 10.19 RepeatProblem10.18(a),assuming thatthe tion variances from untreated and treated steel olatesare

equal.Comparetheresultswith thoseof Problem10.18 'lO.2O The director of trainins for an electronic ment manufactureris interestedin determinins different training methodshavean effect on the ity of assembly-line employees.Sherandomlyassigns 42 recentlyhired employeesinto two groupsof 21. Thefint group receivesa computer-assisted, individual-based training program,and the other receivesa team-basedtraining program.Upon completionof the training, the employees areevaluatedon the time (in seconds)it takesto assemble a part.The resultsare in the datafilel[@f!!. a. Assumingthat the variancesin the populationsof haining methodsare equal,is thereevidenceof a difference between the mean assemblytimes (in seconds)of employeestrained in a computer-assisted individualbasedprogramand thosetrainedin a team-based program?(Usea 0.05levelof significance.) b. What otherassumption is necessary in (a)? c. Repeat(a), assumingthat the populationvariances are not equal. d. Comparethe resultsof (a) and(c). e. Assumingequalvariances,constructandinterpreta 95% confidenceinterval estimateof the differencebetween the populationmeansof the two training methods. 10.21 Nondestructive evaluationis a methodthat is used to describethe propertiesof componentsor materials without causing any permanentphysical changeto the units.It includesthe determination of propertiesof materials andthe classificationof flawsby size,shape,type,and location.This methodis most effectivefor detectingsurfaceflaws and characterizing surfacepropertiesof electrically conductivematerials.Recently,datawere collected that classifiedeachcomponentas having a flaw or not basedon manual inspectionand operatorjudgment and also reportedthe sizeof the crackin the material.Do the componentsclassifiedas unflawedhave a smallermean crack size than componentsclassifiedas flawed?The resultsin termsof cracksize(in inches)arein the datafile @@Q (extractedfrom B. D. Olin and W. Q. Meeker, 'Applicationsof StatisticalMethods to Nondestructive Evaluation,"Technometrics, 38, 1996,p. 10 1.) a. Assuming that the populationvariancesare equal,is thereevidencethat the meancrack sizeis smallerfor the unflawed specimensthan for the flawed specimens? (Usea: 0.05.) b. Repeat(a), assumingthat the populationvariancesare not equal. c. Comparethe resultsof (a) and(b).

10.2: ComparingtheMeansof TwoRelatedPopulations 381

1O.2 COMPARING THE MEANSOF TWO RELATED POPULATIONS proceduresexaminedin Section10.1enableyou to make comparisons The hypothesis-testing and examinedifferencesin the meansof two independentpopulations.In this section,you will learn about a procedurefor analyzingthe differencebetweenthe meansof two populations when you collect sampledata from populationsthat are related-that is, when resultsof the first populationarenol independentofthe resultsofthe secondpopulation. There are two casesthat involve relateddata betweenpopulations.In the first case,you take repeated measurementsfrom the sameset of items or individuals.In the secondcase, items or individualsare matched accordingto somecharacteristic.In either case,the variable of interestbecomesthe dffirence betweenthe valuesratherthan the valuesthemselves. The first casefor analyzingrelatedsamplesinvolvestaking repeatedmeasurements on the sameitemsor individuals.Underthe theorythat the sameitemsor individualswill behavealike if treatedalike, the objectiveis to showthat any differencesbetweentwo measurements of the sameitems or individuals are due to different treatmentconditions.For example,when performing a taste-testingexperiment,you can use eachpersonin the sampleas his or her own control so that you can haverepeatedmeasurements onthe sameindividual. The secondapproachfor analyzingrelatedsamplesinvolvesmatchingitemsor individuals accordingto somecharacteristicof interest.For example,in testmarketinga productundertwo different advertisingcampaigns,a sampleof test marketscan be matchedon the basisof the test marketpopulationsize and/ordemographicvariables.By controlling thesevariables,you arebetterableto measurethe effectsof the two different advertisingcampaigns. Regardlessof whetheryou havematchedsamplesor repeatedmeasurements, the objective is to study the differencebetweentwo measurements by reducingthe effect of the variability that is due to the items or individuals themselves.Table 10.3 showsthe differencesin the individualvaluesfor two relatedpopulations.To readthis table,let X1, Xt2, . . . , Xrrrepresentthe n valuesfrom a sample.And let X2t, X22,. . . , X2, representeither the corresponding n matchedvalues from a secondsampleor the correspondingn repeatedmeasurements from the initial sample.Then,Dp D2, . . . , Dnwill representthe correspondingset of n differencescoressuchthat D t : X t t - X 2 t , D 2 : X 1 2 - X 2 2 , .. . , a n dD r : X r n - X r n

Group

10.3 ininqthe BetweenTwo Populations

Value I 2

Difference

xtt xtz

":,

i,. samplesize is large, Centnl LimitTheorem

page268)ensures you tt the sam pling di stributi o n follows a normal

xzz

Dr: Xtt - Xrt Dz: xr..z-Xn

i,,

Dr: xtt- xn

Xz,

Dn: Xrn- Xrn

Xzt

To test for the meandifferencebetweentwo relatedpopulations,you treat the difference scores,eachD,, as valuesfrom a singlesample.If you know the populationstandarddeviation of the differencescores,you usetheZtest definedin Equation(10.q.2This Ztest for themean differenceusing samplesfrom two relatedpopulationsis equivalentto the one-sampleZtest for the meanof the differencescores[seeEquation(9.1)on page334].

382

CHAPTERTEN Two-Samole Tesrs

Z TESTFORTHEMEAN DIFFERENCE Z*

D-po

(10.4)

6D -r

,\ln

where n

S n-I.

.L/

D:i=r

n meandifference Fp: hypothesized or:

populationstandarddeviationofthe differencescores

r : samplesize The teststatisticZ followsa standardized normaldistribution.

Paired t Test In mostcases,the populationstandarddeviationis unknown.The only informationyou havearethe samplemeanandthe samplestandarddeviation. lf you assumethat the difference scores are randomly and independentlyselectedfiom population that is normally distributed you can use the paired I test for the mean differenr

in relatedpopulationsto determinewhetherthereis a significantpopulationmeandiffe Like the one-sampleI test developedin Section 9.4 [see Equation (9.2) on page 347], thet statistic developedhere follows the r distribution, with n - I degreesof freedom.Although

mustassume thatthepopulationis normallydistribute{as longasthe samplesizeis not smallandthe populationis not highly skewedyou canusethe pairedr test. To test the null hypothesisthat there is no differencein the meansof two related Hoi Fo- 0 (where Fn: \

-,ttz)

againstthe alternativethat the means are not the same: Hr: P'o+0 you computethe I test statisticusing Equation( 10.5).

PAIRED t TESTFORTHEMEAN DIFFERENCE D -po sp '""r1; where \ - ! -

/-/"i

D:

;-l

, n sa.^

:., ).\u; -Df

so:

,-l

The teststatisticl followsa / distributionwith n - l desreesof freedom.

(r0.s)

10.2:Comparing theMeansof TwoRelatedPopulations 383 For a two-tail test with a given level of significance, o, you reject the null hypothesisif the computed I test statisticis greater than the upper-tail critical value tr_, from the I distribution or if the computed test statistic is less than the lower-tail critical value -t, , from the / distribut i o n .T h e d e c i s i o nr u l e i s R e j e c tH o i f t > t n _ l or if / < -/, ,: otherwise, do not reject 11n. The following example illustrates the use of the / test for the mean difference. The Automobile Assocation of America (AAA) conducted a mileage test to compare the gasoline mileage from real-life driving done by AAA members and results of city-highway driving done according to current (as of 2005) government standards(extracted from J. Healey, "Fuel Economy Calculationsto Be Altered," USA Today,January 11,2006, p. 1B). What is the best way to design an experiment to compare the gasoline mileage from reallife driving done by AAA members and results of city-highway driving done according to current (as of 2005) government standards?One approachis to take two independentsamplesand then use the hypothesistestsdiscussedin Section I 0. | . In this approach,you would use one set of automobiles to test the real-life driving done by AAA members.Then you would use a second set of different automobiles to test the results of city-highway driving done according to current (as of2005) government standards. However,becausethe first set of automobilesto test the real-life driving done by AAA membersmay get lower or higher gasoline mileage than the secondset of automobiles,this is not a good approach.A better approach is to use a repeated-measurements experiment. In this experiment, you use one set of automobiles. For each automobile, you conduct a test of real-life driving done by an AAA member and a test of city-highway driving done according to current (as of 2005) government standards.Measuring the two gasol i n e m i l e a g e s f o r t h e s a m e a u t o m o b i l e s s e r v e st o r e d u c e t h e v a r i a b i l i t y i n t h e g a s o l i n e mileagescomparedwith what would occur if you used two independentsetsof automobiles. This approach focuses on the differences between the real-life driving done by an AAA member and the city highway driving done according to current (as of 2005) government standards.

llv na lce ce. est /ou 3ry NS:

Table10.4displaysresults(storedin the file EEE@EEffr) from a sampleof

n : 9 automobilesfrom such an exoeriment.

T A B L E1 0 . 4 Repeated Measurements of Gasoline Mileagefor Drivingby Real-Life MA Membersand Driving City-Highway DoneAccordingto (asof 2005) Current Government Standards

Model

Members

2005FordF-150 2005ChevroletSilverado 2002HondaAccordLX 2002HondaCivic 2004HondaCivic Hybrid 2002FordExplorer 2005ToyotaCamry 2003ToyotaCorolla 2005ToyotaPrius

14.3 15.0 27.8 27.9 48.8 16.8 23.7 32.8 37.3

Current 16.8 17.8 26.2 tJ.z

47.6 18.3 28.5 33.1 44.0

You want to determine whether there is any difference in the mean gasoline mileage between the real-life driving done by an AAA member and the city-highway driving done according to current (as of2005) government standards.In other words, is there evidencethat

384

Two-SampleTests CHAPTERTEN the mean gasolinemileage is different betweenthe two types of driving? Thus,the alternativehypothesesare Ho: Fn: 0 (Thereis no differencein meangasolinemileagebetweenthe real-lifedr Uy an ene memberand the city-highway driving doneaccordingto currentfasof ernmentstandards.) Ht: Fn* 0 (Thereis a differencein meangasolinemileagebetweenthe real-lifedr Uy anlen memberand the city-highway driving doneaccordingto current[asof ernmentstandards.) Choosingthe level of significanceof cr : 0.05 and assumingthat the differences you usethepairedI test[Equation(10.5)].For a sampleof n=9 mally distributed, :8 degreesof freedom.UsingTableE.3,the decisionrule is therearen | RejectHoif t> tr:2.3060; or if r < -te: -2.3060; otherwise,do not rejectIlo. \sr{\ren--\t\\srursts\st\S$lt\\),$-t.rss$s,rsssN\\RsxsN\Rbls

Yo

'

Ll

D=i=t

ar

l

_-Lt't=_2.3444 n9

and n

\rn, - D)' SD=

,-1

= 2.893575

From Equation (10.5) on page 382,

D -v, ,SD

_-2.3444-0 =_2.4301 2.893575

\n

llo (t.. Figure Becauset: -2.4307 is lessthan-2.3060,you rejectthe null hypothesis, Thereis evidenceof a differencein meangasolinemileagebetweenthe real-life driving by an AAA memberand the city-highway driving doneaccordingto current(asof 2005) ernmentstandards.Real life driving resultsin a lower meangasolinemileage. FIGURE10.7 Two-tailpairedttest at the 0.05level with of significance B degreesof freedom

Regionof I Rejection I I

Q / Regionof Nonrejection

+2.306f f8 I Regionof lRejection

10.2:Comparing theMeans ofTwoRelated Populations385 You can computethis test statisticalong with the p-value by using Microsoft Excel (see Figure10.8).Because thep-value:0.0412 < o:0.05, you rejectHo.The p-valueindicatesthat if the two typesof driving havethe samemeangasolinemileage,the probabilitythat onetype of driving would havea meanthat was 2.3444miles per gallon lessthan the othertype is 0.0412. Becausethis probabilityis lessthancr:0.05, you concludethatthe alternativehypothesisis true.

10.8 A MicrosoftExcel ultsof paired t test thecarmileagedata

Condadon ilocn Dlferencr

I 0T613. 0

e:

Jrrot. onc{all Crldcal onc.tcll

two{rll

0@-

1.8S||Il

0.0t1?1

Crldcal two{dl

B MicrosoftExcel

Box-and-Whisker Plot for casolaneMileageDifierences

iskerplot thecarmileagedata

From Figure 10.8,PanelB, observethat the box-and-whiskerplot showsapproximate symmetry.Thus,the datado not greatly contradictthe underlyingassumptionof normality. If an exploratorydata analysisrevealsthat the assumptionof underlyingnormality in the population is severelyviolated"then the / test is inappropriate.Ifthis occurs,you can either use a nonparametricprocedurethat doesnot makethe stringentassumptionof underlyingnormality (seeReferencesI and 2) or makea datatransformation(seereference5) and then recheckthe assumptionsto determinewhetheryou shouldusethe I test.

X A M P L E1 0 . 2

PAIREDI-TE5T OF PIZZA DELIVERYTIMES Recall from Example l0.l on page 374 that a local pizzarestaurantlocatedacrossthe street from a collegecampusadvertisesthat it deliversto the dormitoriesfasterthan the local branch of a nationalpizza chain. In order to determinewhetherthis advertisementis valid, you and somefriends havedecidedto order l0 pizzasfrom the local pizza restaurantand I 0 pizzasfrom the nationalchain.In fact, eachtime you orderedapizza from the local pizza restaurant,your friendsorderedapizza from the nationalpizzachain.Thus,you havematcheddata.For eachof

386

CHAPTERTEN Two-Samole Tests the ten times pizzas were ordered"you have one measurementfrom the local pizza and one from the national chain. At the 0.05 level of significance, is the mean delivery time the local pizza restaurantless than the mean delivery time for the national pizza chain? SOLUTION Use the paired r test to analyzethe data in Table 10.5(seethe file [@@@.

TABLE 10.5 DelivervTimesfor Local rrzzaKestaurant ano NationalPizzaChain

Time I

2 J

4 5 6 7 8 9 10

Local

Chain

Difference

16.8 tt.7 15.6 16.7 17.5 18.1 t4.l 2t.8 13.9 20.8

22.0 15.2 18.7 15.6 20.8 19.5 17.0 19.5 16.5 24.0

-5.2 -3.5 -3.I l.l

-3.3 -1.4 -2.9 z.J

-2.6 -3.2 -21.8

Figure 10.9 illustrates Microsoft Excel paired I test results for the pizza delivery data.

FIGURE10.9 MicrosoftExceloaired t test resultsfor the pizzadeliverydata

t-l l-ll

l-l SeeSectionE10.6to create this.

Local

4 Mean -5 Variance 6 Observallors 7 rPeartonCorrelatlon I Hypotheslzed MeanDlfference gidf 10 t Stat 1.'ltPff."q one-tail 12.t Crltlcalone-tall 13 Pfr.=q rro-rall 14 rt Crlrlcalrwo-rail

Cltp'in

16.7 18.S 9.58?2 8.2151 10 10 0.714'l 0 I 3.0448 0.0070 1.8331 0.0139 2.2622

The null and alternativehypothesesare Ho: Vn > 0 (Mean delivery time for the local pizzarestaurant is greater than or equal tothe mean delivery time for the national pizza chain.) Hi Fp < 0 (Mean delivery time for the local pizzarestaurantis less than the mean delivery time for the national pizza chain.) Choosing the level of significance o(: 0.05 and assumingthat the differencesare normally distribute4 you use the paired / test [Equation(10.5) on page 382]. For a sampleof r : 10 delivery times, there are n - | -- 9 degreesof freedom. Using Table E.3, the decision rule is R e j e c tH o i f t < t n : - 1 . 8 3 3 1 ; otherwise, do not reject 11n. For n : l0 differences(seeTable 10.5).the sample mean difference is

tr, -

D_,=,

-7rR = -"" =_2.13 n10

10.2: Cornparingthe Meansof Two Relatecl Populations 387

.rant : for

and the sample standard deviation of the difference is

s1.'= (10.5) FromEquation onpage382, D -Vn

s,

t;

-) tR-0 2.2641

tTt

Becauset - -3.0448 is less than - I .833I, you reject the null hypothesisHn (the 7r-valueis 0 . 0 0 7 0 < 0 . 0 5 ) .T h e r e i s e v i d e n c et h a t t h e m e a n d e l i v e r y t i m e i s l o w e r f o r t h e l o c a l p i z z a restaurantthan for the national pizza chain. This conclusionis differentfrom the one you reachedwhen you usedthe pooled-variance r test for these data. By pairing the delivery tirnes, you are able to focus on the differences betweenthe two pizza delivery servicesand not the variability createdby ordering pizzas at differenttimes of day.The pairedI test is a more powerful statisticalprocedurethat is betterable to detect the difference between the two pizza delivery services.

Confidence Interval Estimate for the Mean Difference Insteadof, or in additionto, testingfor the differencebetweenthe meansof two relatedpopulations, you can use Equation 10.6to constructa confidenceinterval estimateof the mean difference.

CONFIDENCE INTERVAL ESTIMATE FORTHEMEAN DIFFERENCE (10.6)

D - tr-t

s^

*U

P=u,< D + t . - , G "!n

Returnto the exarnplecomparinggasolinenileagegenerated by real-lifedrivingandby government (10.6).D - -2.3444.tD- 2.8936, standards. UsingEquation n:9, andt :2.306 : (for 95%confidence andn I 8 degrees of freedom), lisliv-

_ 2 . 3 4 4t4 ( 2 . 3 0 62) ' 8 9 : ] 6 r/q -2.3444+ 2.2242 -4.5686(U,S-0.1202 Thus,with 95% confidence, the meandifference in gasolinemileagebetween thereal-lifedriving doneby an AAA memberandthe city highwaydrivingdoneaccording to current(asof 2005)government standards is between-4.5686and -0.1202milesper gallon.Because the intervalestirnate containsonlyvalueslessthanzero,you canconclude thatthereis a difference in the populationmeans.The meanmilesper gallonfor the real-lifedrivingdoneby an AAA memberis lessthanthe meanrnilesper gallonfor thecity-highwaydrivingdoneaccording to current(asof2005)government standards.

388

CHAPTERTEN

Tlvo-SamDleTestg

(codedto maintainconfi

Learning the Basics

fil. @!@ss[

EnnFq rc.22 An experimentaldesignfor a paired/ test lAsslsr I has,as a matchedsample,20 pairsof identical twins. How many degreesof freedomaretherein this I test?

an analyzerduring the production processand from analyticallab (extractedfrom M. Leitnaker," MeasurementProcesses:In-line Versus Analvti Measuremen ts," QuaI i Q Engin eering, | 3, 2000-2001, 293-298). a. At the 0.05 level of significance,is there evidence differencein the mean measurements in-line and an analyticallab? b. What assumptionis necessaryto perform this test? c. Use a graphicalmethodto evaluatethe validityof assumption in (a). d. Constructand interpret a 95o/oconfidenceinterval mateof the differencein the meanmeasurements inli and from an analyticallab.

q 10.23 An experimentrequiresa measurement EE of a stimulusto lAsslsTI beforeand afterthe presentation eachof 15 subjects.In the analysisof the data collectedfrom this experiment,how many degreesof freedom aretherein the test?

Applying the Concepts 10.24 The Septemberissuesof monthly magazines typically carry the most advertising pagesfor any issueduring the year.The followgive the number ing data (stored in the file lilEltEIE) of advertisingpagesin September2004 and September 2005: Magazine Martha StewartLiving GoodHousekeeping Parenting Glamour(specialissue) PopularMechanics Ebony Cosmopolitan (specialissue) Ladies'HomeJournal Parents Vogue Harper s Bazaar Elle Esquire Real Simple Men s Health GQ InStyle Details

2004 52.14 tt5.t2 r23.84 184.78 67.44 r22.32 227.35 125.21 t39.14 650.63 261.09 342.27 165.58 163.10 r39.76 292.8s 382.96 206.97

2005 75.25 t49.41 1s8.37 236.00 85.02 141.77 248.60 136.99 r49.68

690.ss 274.06 346.94 167.53 163.80 t40.16 288.27 376.00 202.13

a. At the 0.05 level of significance,is thereevidenceof a differencein the meannumberof advertisingpagesin September 2004 andSeptember 2005? b. What assumptionis necessaryto perform this test? c. Determinethep-valuein (a) andinterpretits meaning. d. Constructand interpreta 95o/oconfidenceinterval estimate of the differencein the meannumberof advertising pagesin September 2004and September 2005. 10.25 In industrial settings,alternativemethodsoften exist for measuringvariablesof interest.The data in the

reDresentmeasurementsin-line that were collected

10.26 Can studentssavemoney by buying their books at Amazon.com?To investigatethis possibility, randomsampleof 14textbooksusedduringthe 2006 mer sessionat Miami Universitvwas selected.Theori for thesetextbooksat both a local bookstoreand Amazon.comwererecorded.The oricesfor the tex includingall relevanttaxesand shipping,are given (and are storedin the fileS!g!!pQ): Textbook Concepts in Federal TAxafion Intermediate A ccounting The Middle East and CentralAsia Wests Business Law Leadership: Theory and Practice Making Choicesfor Multicultural Education Direct Instruction Reading Essentials of Economics Marriage and Family America and lts People Oceanography Calculus : E arly T?anscendental Single Variable Access to Health Womenand GIobalizati on

Book Store A

r38.2r rst.92

143.95. rs2.7a

s2.06 1s9.31 49.59

53.00 143.95 48.95

71.74 98.12 102.12 106.92 100.44 105.l8

56.95 97.35 99.64 100.98 9s.20 128.95

I I 5.00 93.47 29.54

133.50 88.60 18.48

a. At the 0.01 level of significance,is thereevidenceof differencebetweenthe meanprice of textbooksat local bookstorcand Amazon.com? b. What assumptionis necessaryto perform this test? c. Constructa 99o/oconfidenceinterval estimateof the meandifferencein price. Interpretthe interval. d. Comparethe resultsof (a) and (c).

10.2:Comparing theMeans ofTwoRelated Populations389 immediatelyprior to the stemcell transplantand at the time of the completeresponse:

A newspaperarticlediscussedthe openingof a Whole Marketin the Time-Warnerbuilding in NewYork City.

followingdata(storedin the fileEEEEEE@

co--

Patient

thepricesof somekitchen staplesat the new Whole Marketand at the Fairwaysupermarketlocatedabout fromtheTime-Warnerbuilding:

1 2 3 4 s 6 7

Whole Foods Fairway

milk eggs orangejuice (64 oz.) of Bostonlettuce round, I lb. Beetuna.6 oz. can Smithapples(l lb.) DeCeccolinguini steak,I lb, chicken,per pound

2.r9 2.39 2.00 1.98 4.99 1.79 1.69 1.99 7.99 2,t9

1.35 r.69 2.49 1.29 3.69 1.33 r.49 1.59 5.99 1.49

: Extractedfrom W Grimes, "A Pleasure Palace Without the

" TheNewYorkTimes,February18, 2004,pp. Fl , F5

tAtthe 0.01 level of significance,is there evidencethat 'themeanprice is higher at Whole FoodsMarket than at theFairwaysupermarket? Interpretthe meaningof thep-value in (a). Whatassumptionis necessaryto performthe testin (a)? Multiple myeloma,or blood plasmacancer,is charby increasedblood vesselformulation(angiogenin the bone marrow that is a prognostic factor in sur. Onetreatmentapproachusedfor multiple myelomais cell transplantationwith the patient'sown stem cells. followingdata(storedin the file fi!$@l@ represent bonemarrowmicrovesseldensityfor patientswho had a responseto the stemcell transplant,as measured blood and urine tests. The measurementswere taken

Before

After

158 189 202 353 416 426 44t

284 2r4 lOt 227 290 176 290

Source: Extractedfrom S. V Rajlamaa R. Fonseca, T E. llitzig, M. A. Gertz, and P R. Greipp, "Bone MarowAngiogenesis in Patients Achieving Complete ResponseAfter Stem Cell Transplantationfor Multiple Myeloma," Leukemi4 1999, 13,pp. 469472.

a. At the 0.05levelof significance,is thereevidencethatthe meanbone marrow microvesseldensityis higherbefore the stemcell transplantthan after the stemcell transplant? b. Interpretthe meaningof thep-value in (a). c. Constructand interpreta 95o/oconfidenceinterval estimateof the meandifferencein bonemarrowmicrovessel densitybeforeand after the stemcell transplant. d. What assumptionis necessaryto perform the testin (a)? 10.29 Over the past year, the vice presidentfor human resourcesat a latge medical centerhas run a seriesof threemonth workshopsaimedat increasingworker motivationand performance.To checkthe effectivenessofthe workshops,she selecteda randomsampleof 35 employeesfrom thepersonnel files and recordedtheir most recentannualperformanceratings,alongwith their ratingsprior to attendingthe workshops. The dataarestoredin the file[@@. TheMicrosoft Excel resultsin PanelsAandB provideboth descriptiveandinferential information so that you can analyzethe resultsand examine the assumptionsof the hypothesistestused: Stateyour findings and conclusionsin a report to the vice presidentfor humanresources.

Dlflslrlnce

sJtrfi1

rslle

'5, -t0, Dovladon 115,?32 Varlancr 1T2:t0,,g 1.1038 Skawrer 0.1103, Rango fl-l

ttnlmrm {3xlmum Sum Counf

llsq{0

34i

u,:

.:t&f, 3il

n' aa:

for Xcatr

7t5(B E0"gt2537.16f7 It 35

Olrorallom Pearon Conelatlon o.r3{2 Hypoitrcdrrd liin Dlfrerencr . . . 0 df

sr||

k-{ oncsft

Cdtlcal one{all

i3{

-2599n

o"arr

t5s obios ,rffi.

390

CHAPTERTEN Two-SampleTests

10.30 The datain the file@EErepresent the compressivestrength,in thousandsof poundsper squareinch (psi),of 40 samplesof concretetakentwo and sevendays afterpouring.

a. At the 0.01 level of significance, is there evidence the mean strength is lower at two days than at seven

b. What assumption is necessary to performthistest? c. Findthep-valuein (a) andinterpretits meaning.

Source: Extracted from O. Caruillo-Gamboa and R. F Gunst, "Measurement-Error- ModeI Col linearities," Technometrics,-34, 1992,pp. 454-464.

10.3

COMPARINGTWO POPULATIONPROPORTIONS Often, you need to make comparisons and analyze differences between two population proportions. You can perform a test for the difference between two proportions selectedfrom independent samplesby using two different methods.This section presentsa procedurewhosetest statistic,Z, is approximatedby a standardizednormal distribution. In Section 12.1,a procedure whose test statistic, 12, is approximatedby a chi-squaredistribution is developed.As you will see,the results from these two testsare equivalent.

Z Testfor the DifferenceBetweenTwo Proportions In evaluatingdifferences betweentwo populationproportions,you canusea Z testfor thedif. ferencebetweentwo proportions.The teststatisticZ is basedon the differencebetween two (pr- pz). This teststatistic,givenin Equation(10.7),approximatelyfolsampleproportions lows a standardized normaldistributionfor largeenoughsamplesizes.

ZTESTFORTHE DIFFERENCE BETWEEN TWO PROPORTIONS (10.7)

- ( r- + - 1 1 )

P)l \rt

n z)

with Y.

Xr+X1 l)r=-

\*n2

n,

' ' l

P2=-

x) n2

wnere p1 : proportionof successes in sampleI X, : numberof successes in sample1 n,: samplesizeof sample1 fil : proportionof successes in population1 in sample2 P2-- proportionof successes X, : numberof successes in sample2 nr: samplesizeof sample2 fi2: proportionofsuccesses in population2 p : pooledestimateof the populationproportionof successes The test statisticZ approximatelyfollows a standardizednormal distribution.

10.3:Comparing TwoPopulation Proportions 391 Under the null hypothesis,you assumethat the two population proportions are equal (n, : n).Because the pooled estimatefor the populationproportion is basedon the null hypothesis,you combine,or pool, the two sampleproportionsto computean overall estimate of the commonpopulationproportion.This estimateis equal to the numberof successesin the two samplescombined (Xt + X) divided by the total samplesize from the two samplegroups(nr+ n2). As shownin the following table,you can usethis Ztestfor the differencebetweenpopulation proportionsto determinewhetherthereis a differencein the proportionof successes in the two groups(two-tail test) or whetherone group has a higher proportion of successes than the other group (one-tailtest): TWo-TailTest

Ho:nr: n, H r :n r * n ,

One-Tail Test

One-Tail Test

Ho:nr) n, Hr: nr
Ho:nr< n, Hr: nr> n,

where in populationI fil : proportionofsuccesses in population2 7[2: proportionofsuccesses To test the null hypothesisthat thereis no differencebetweenthe proportionsof two independentpopulations:

Ho:nr: n, proportions thatthetwopopulation againstthealternative arenotthesame: Hl T\+T.2 usethe test statisticZ, givenby Equation(10.7). For a given level of significanceo, rejectthe null hypothesisif the computedZ test statisticis greaterthan the upper-tailcritical value from the standardizednormal distribution or if the computedtest statisticis lessthan the lower-tail critical value from the standardizednormal distribution. To illustratethe useof the Z test for the equality of two proportions,supposethat you are the managerof T.C. ResortProperties,a collectionof five upscaleresorthotelslocatedon two resort islands.On one of the islands,T.C. ResortPropertieshas two hotels,the Beachcomber andthe Windsurfer.In tabulatingthe responses to the singlequestion,'Are you likely to choose this hotel again?"163of 227 guestsat the Beachcomberrespondedyes,and 154of 262 guests at the Windsurferrespondedyes.At the 0.05 level of significance,is thereevidenceof a significant differencein guestsatisfaction(as measuredby the likelihood to return to the hotel) betweenthe two hotels? The null and alternativehypothesesare H o : n , : r 2 o r n .- n r : 0 Hr: n, + Tc2or fil - n2+ 0 Usingthe 0.05levelof significance,the criticalvaluesne -1.96 and+1.96(seeFigure10.10), and the decisionrule is RejectHoif Z<-1.96 or if Z> +1.96; otherwise,do not reject.Flo.

392

Tests CHAPTERTENTwo-SamPle

FI M fo di

10.10 FIGURE Reqions of reiection a n i n o n r e j e c i i o nw h e n testing a hyPothesisfor the difference between two proportions at the 0.05 level of significance

tV

h P

Regionof Rejection

Regionof Rejection Critical Value

Regionof Nonrejection

Criticat Value

UsingEquation(10.7)on Pagel q o

p(r -

-.( r . ; ) u'li

where

x1 Pt

ny

163 -0.7181 227

rnL . = I z - 1 5 4 = 0 . 5 8 7 8 262 fi2

and P-

Xr+X, _163+154 _311 =0.64g3 489 227 + 262 nr + n2

so that

- (0) ( 0 . 7 1 8-1 0.s878)

_o64sr(*.#)

0.6483(l

0 . 13 0 3

l(urrsxo.oo8r) 0 . 13 0 3 - o ' 1 3 0 3= + 3 . 0 0 8 8 0.0432 z: +3'0088> +1'96' Using the 0.05 level of significance,rejectthe null hypothesisbecause from iable E.2 or from the MicrosoftExcelresultsof Figure Thel-value is 0.0026(cai-culated that aZtest statisticis 10.11).This meansttratlf the null hypothesisis true, the probability statisticis greaterthan Z test a that probability -3.0088 the is 0.0013,an4 similarly' lessthan + 0'0013:0'0026' 0'0013 is thep-value test, +3.0088is 0.0013.Thus,for this two-tail to concludethat evidence is There : Because0.0026< cr 0.05,you rejectthe null hypothesis. greaterprop0r' a guest satisfaction; the two hotelsare significantlydifferentwith respectto Windsurfer' thanto the tion of guestsarewilling to returnto the Beachcomber

l

S tt

10.3: ComparingTwo PopulationProportions

10.11 FIGURE Microsoft Excelresults fortheZ test for the difference between for the twoproportions hotelguestsatisfaction problem

393

1

t ? 4

b

iq 9 l0 11 17

ts .87,'88 -Bt0l811 -814 .815 -(87]810),'(88+8rll -{8rG .81)/soRT(817'(t - 814'(1/88 + 1rBl1}}

14 t5

i6 17 !8 1e .-20

SeeSectionE10.7 to create this.

-r{ORl{Slf{v{85,?} -xoRrstilv(l - 85.") =2' (1 - iloRHsDlsT{ABs(818D} -lF(823 < 85, 'RoJeclthe oull hypothods", "Do nol r.Jocl tho o{ll byporhGls)

,2 73 24

E X A M P L E1 0 . 3

TESTINGFOR THE DIFFERENCE IN TWO PROPORTIONS Money worriesin the United Statesstartat an earlyage.In a survey,660 children(330 boys and 330 girls) ages6 to 14 wereaskedthe question,"Do you worry abouthavingenough money?"Of the boyssurveyed20l (60.9%)saidyes,and 178(539%) of the girls surveyed saidyes(extractedfrom D. HaralsonandK. Simmons,"Snapshots," USAToday,May24,2004, p. lB).At the 0.05 levelof significance, is the proportionof boyswho worry abouthaving enoughmoneygreaterthanthe proportionof girls? SOLUTION Becauseyou wantto know whetherthereis evidencethat the proportionof boys who worry abouthavingenoughmoneyis greaterthantheproportionof girls,you havea onetail test.The null andalternativehypotheses are Hr: n, < n, (The proportionof boyswho worry abouthavingenoughmoneyis lessthan or equalto the proportionof girls.) H ,: n, > n, (Theproportionof boyswho worry abouthavingenoughmoneyis greaterthanthe proportionof girls.) Using the 0.05 level of significance,for the one-tailtest in the uppertail, the criticalvalueis +1.645.Thedecisionrule is R e j e cH t o i fZ > + 1 . 6 4 5 ; otherwise,do not rejectF1o. UsingEquation(10.7)on page390, Z

96. ure :is 1an

26. hat ror-

where

Pr = X t

n1

-

2ol = 0.609 330

rz

x2 - 1 7 8= 0.539 n2

330

394

Tests CHAPTERTEN Two-Samole and

Xr+X,

n=

2 0 1+ 1 7 8

379

330+ 330

660

=.........'.................'-=_-n<1L"'

nl+n2

(0.60e-0.s3e)-(0)

Z_

330/

\"0

\ 0.07

^l1o.z++s11o.00606 ) 0.07

ffi =

tr:t

0.07 = +l'818 or*

Usingthe 0.05levelof significance, rejectthe null hypothesis because Z : +1.818> + 1.645, Thep-valueis 0.0344(calculatedfrom TableE.2).Therefore,if the null hypothesis is true,the probabilitythata Z teststatisticis greaterthan+l .818is 0.0344(whichis lessthano : 0.05). Youconcludethatthereis evidencethattheproportionofboys who worry abouthavingenough moneyis greaterthantheproportionof girls.

Lei IA\

I. I

t b.( +

Confidence Interval Estimate for the Difference Between Two Proportions Insteadof, or in addition to, testing for the difference between the proportions of two independent populations, you can construct a confidence interval estimate of the difference between the two proportions,using Equation( 10.8).

l!!!t tIA\

-

7.t

I [.r

INTERVAL FORTHEDIFFERENCE CONFIDENCE ESTIMATE BETWEEN TWO PROPORTIONS (p1- p2)+Z

(n-pz)-Z

Pr(l-p),p20-pz) nr n2

(10.8)

- nr) p2(l - p2) p,1(1 T--1!i1 n1

S(pr-p)+Z

A5

-ii2

n2

I

T-

n2

To construct a 95o/oconfidence interval estimate of the population difference betweenthe percentagesof guests who would return to the Beachcomberand who would return to the Windsurfer, you use the results on page 392 or from Figure 10.11 on page 393:

X, Dr= 1= n1

_1 6 3= 0 . 7 1 8 1 zzl

x,

mt asl mt L.

P{r- p) , p20- p2) n1

tfEl

154

=0.5878 Dt=i=__-_ n2 262

10.3: ComparingTwoPopulationProportions 395

UsingEquation(10.8),

( 0 . 7 1 8- 10 . 5 8 7 8t )( 1 . 9 6 ) " / L \l r (1.96x0.0426) 0.1303

227

262

0.1303 r 0.0835 0.0468( (nr - TE)< 0.2138 Thus,you have95o/oconfidencethat the differencebetweenthe populationproportionof guestswhowouldreturnagainto theBeachcomber andtheWindsurferis between 0.0468and the difference 0.2138.In percentages, is between 4.68%and,2l.35Yo. is Guestsatisfaction higherat theBeachcomber thanat theWindsurfer.

the Basics 10.31Assume thatnr: 100,Xr: 50,nr: 100, andXr:30. At the 0.05 level of significance,is thereevidenceof a significant differencebetweenthe two population proportions? Constructa95Yoconfidenceinterval estimateof the differencebetweenthe two populationproportions.

10.32Assume thatnr: 100,Xr: 45,nr: 50, andXr:25. At the 0.01 level of significance,is there evidenceof a significant differencebetweenthe two population proportions? Constructa99o/oconfidenceinterval estimateof the differencebetweenthe two populationproportions.

the Concepts 10.33 A sampleof 500 shoppers wasselectedin a largemefopolitan areato determinevariousinforion concerningconsumerbehavior.Among the questions was, "Do you enjoy shoppingfor clothing?" Of 240 yes. yes.Of 260 females,224 answered 136answered Is there evidenceof a significant differencebetween malesandfemalesin the proportionwho enjoy shopping for clothingat the 0.01levelof significance? Findthep-value in (a) and interpretits meaning. Constructand interpreta 99o/oconfidenceinterval estimateof the differencebetweenthe proportion of males andfemaleswho enjoy shoppingfor clothing. What are your answersto (a) through (c) if 206 males enjoyedshoppingfor clothing? An article referencing a survey conducted by -LOL Learning Servicesclaims that parentsare more

confident than teachersthat their schoolswill meet the standardssetby the No Child Left BehindAct. The survey askedparents(and teachers),"How confident are you that your child's school(the schoolwhereyou work) will meet the standardsby the deadline."The responsesto that question are given in the following table: Parents

Veryconfident Not veryconfident Totals

401 684 1,085

Teachers

162 648 810

Source: Adaptedfrom B. Feller "kachersMoreLikelySkeptics of No Child,"TheCincinnati Enquirer, April20,2006,p. A4. a. Set up the null and alternativehypothesesneededto try to prove that the population proportion ofparents that are very confidentthat their child's schoolwill meetthe standardsby the deadlineis greaterthan the population proportion ofteachers that are very confident that the school where they work will meet standardsby the deadline. b. Conductthe hypothesistest defined in (a), using a 0.05 levelofsignificance. c. Doesthe resultof your test in (b) makeit appropriatefor the article to claim that parentsare more confidentthan teachers? 10.35 The resultsof a study conductedas part of a yield-improvementeffort at a semiconductor manufacturingfacility provideddefectdatafor a sampleof 450 wafers.The following contingencytable presentsa sunmary of the responses to two questions:"Was a particle found on the die that produced the wafer?" and "Is the wafer good or bad?"

396

Two-SampleTests cHAPTERTEN

Quality of Wafer

PARTICLES Yes No Totals

Good

Bad

Totals

t4 320 334

36 80 116

50 400 450

Source: Extractedfrom S.W Hall, "Analysis of Defectivity of SemiconductorWafersby Contingency Table,"ProceedingsInstitute ofEnvironmental Sciences,Vol.I, 1994,pp. 177-183.

a. At the 0.05 level of significance,is thereevidenceof a significantdifferencebetweenthe proportionof good andbad wafersthat haveparticles? b. Determinethep-value in (a) and interpretits meaning. c. Constructand interpreta 95o/oconfidenceinterval estimate of the differencebetweenthe population proportion of good andbad wafersthat containparticles. d. What conclusionscan you reachfrom this analysis? 10.36 Accordingto an Ipsospoll, theperception of unfairnessin the U.S. tax codeis spreadfairly evenly acrossincome groups,age groups,and educationlevels.In an April 2006 surveyof 1,005adults, Ipsosreportedthat almost600/oof allpeoplesaidthe codeis unfair, while slightly more that 60% of thosemaking more than $50,000viewed the code as unfair (Extractedfrom "People Cry Unfairness," The Cincinnati Enquirer, Aptll 16,2006,p.A8). Supposethat the following contingency tablerepresentsthe specificbreakdownofresponses: Income Level U.S.Tax Code

LessThan $50,000

More Than $50.000

225 280 505

180 320 500

Fair Unfair Total

Total

405 600 I,005

a. At the 0.05 level of significance,is thereevidenceof a differencein the proportionof adultswho think the U.S. tax codeis unfair betweenthe two incomegroups? b. Find thep-value in (a) and interpretits meaning. A sur10,37 Is goodgasmileagea priority for carshoppers? Insuranceaskedthis question vey conductedby Progressive to both men andwomenshoppingfor new cars.The datawere reportedaspercentages, andno samplesizesweregiven: Gender

GASMILEAGE A PRIORITY?

Men

Women

Yes No

76% 24%

84% t6%

Source: Extractedfrom " Snapshots,"usatoday.com,June 2 I, 2004.

a. Assumethat 50 men and 50 women were r the survey.At the 0.05 level of significance, evidenceof a differencein the population tion of males and females who made gas ml priority? b. Assumethat 500 men and 500 womenwere inc

the survey.At the 0.05level of significance,is there denceof a differencebetweenmalesand femalesin proportionwho madegasmileagea priority? c. Discuss the effect of sample size on the Z test for the ferencebetween two proportions.

10.38 An experiment was conductedto study choicesmadein mutual fund selection.Unde and MBA studentswere presentedwith different 500 index funds that were identical except for studentsand 100 MBA Suppose100 undergraduate dentswere selected.Partialresultsare shownin the lowins table:

STUDENT GROUP FUND Highest-cost fund fund Not-highest-cost

Undergraduate 27 73

|

Source: Extractedfrom J. Choi, D. Laibson, and B. Madrian, "Why Does the Law of One Practice Fail? An Experiment on Mutual Funds," www.som.yale. edu/faculty/jj c83/fees,pdf.

a. At the 0.05 level of significance,is thereevidence and MBA students differencebetweenundergraduate the proportionwho selectedthe highest-costfund? b. Find thep-valuein (a) and interpretits meaning. 10.39 Wherepeopleturn for newsis different for age groups (Extractedfrom P. Johnson,"Young Turn to the Web for News." USATodav.March 23.2006. 9D). Supposethat a study conductedon this issue who werebetweenthe agesof basedon 200 respondents and 50 and 200 respondentswho wereaboveage50.Of 200 respondentswho were betweenthe agesof 36 and 82 got their news primarily from newspapers.Of the respondentswho were aboveage 50, 104 got their primarily from newspapers. a. Is thereevidenceofa sisnificantdifferencein the portion who get their news primarily from ne betweenthoserespondents36 to 50 yearsold and above50 yearsold? (Usecr: 0.05.) b. Determinethep-value in (a) and interpretits meaning. confidenceinterval c. Constructand interpret,a95Yo mate of the differencebetweenthe population tion of respondentswho get their news primarily newspapers betweenthoserespondents 36 to 50 old and thoseabove50 yearsold.

10.4: F Tcst for the DifferenceBetweenTwo Variances

397

1O.4 F TESTFORTHE DIFFERENCEBETWEENTWO VARIANCES Often, you need to determine whether two independentpopulations have the same variability. This determinationis made by testing variances.One important reasonto test for the difference between the variancesof two populations is to determine whether to use the pooled-variance/ test (equal variance case)or the separate-variance/ test (unequal variance case). The test for the difference between the variancesof two independentpopulations is based on the ratio of the two sample variances.If you assumethat each population is normally distributed then the ratio Sf /S2' follows the F distribution (seeTable E.5). The critical values of the F distribution in Table E.5 depend on two sets of degreesof freedom. The degreesof freedom in the numerator of the ratio are for the first sample, and the degreesof freedom in the denominator are for the secondsample.Equation ( 10.9) defines the F test statistic for testing the equality of two variances.

F TESTSTATISTIC FORTESTINGTHE EOUALITYOF TWO VARIANCES TheF-teststatistic is equalto thevariance of sample1 dividedby thevariance of sample 2. '- J f

-)

r^") i

(10.e)

where

si : rt

variance of sample I

: variance of sample 2

n l : size of sample taken from population I n2 : size of sample taken from population 2 n t - 1 : degreesof freedom from sample 1 (that is, the numerator degreesof freedom)

n z - 1 : degreesof freedom from sample 2 (that is, the denominator degreesof freedom)

TheteststatisticF followsan F distributionwith n, - 1 andn, - | degrees of freedom. For a givenlevelofsignificance, o, to testthenull hypothesis ofequalityofvariances:

nn:ol : ol againstthe alternativehypothesisthat the two population variancesare not equal:

)

H,'

" t

l'

ol +o)

you reject the null hypothesisif the computed tr test statistic is greater than the upper-tail critical value, Fy, from the Fdistribution with n, - I degreesof freedom in the numerator and n, - I degreesof freedom in the denominator,or if the computed Ftest statisticis less than the lower-tail critical value, F., from the F distribution with r, - I and n, - 1 degreesof freedom in the numerator and denominator, respectively.Thus, the decision rule is

n

R e j e c tH o l f F > F u orifF
S

This decisionrule and rejectionregionsare displayedin Figure 10.12.

398

c H A P T E R T E NTwo-SampleTcsts -t

FIGURE10.12

l

R e g i o n so f r e j e c t i o n a n d n o n r e j e c t i o nf o r the two-tail F test

I

o

i

I.a-

FL

R e g i o no f Rejection

rLJ

R e g i o no f Nonrejection

1 1

r

R e g i o no f Rejection

To illustrate how to use the F test to determine whether the two variancesare equal,return to the Using Statisticsscenarioconcerningthe salesof BLK cola in two differentaislelocations. To determinewhether to use the pooled-varianceI test or the separate-variance I testin Section 10.I , you can test the equalityof the two populationvariances.The null and alternative hypothesesare I t

tII

l).

) ol ) ol

o) 6;

Becausethis is a two-tail test,the rejectionregion is split into the lower and uppertailsof thef distribution.Using the level of significancecr : 0.05, each rejectionregion contains0.025of the distribution. Becausethere are samplesof l0 stores for each of the two display locations,thereare l0 - I :9 degreesof freedom for group I and also for group 2. FL,, the upper-tailcritical value of the Fdistribution, is found directly frorr-rTable E.5, a portion of which is presentedin Table 10.6.Becausethere are 9 degreesoffreedom in the numeratorand 9 degreesoffreedon in the denominator,you find the upper-tailcritical value,.Fr, by looking in the column labeled 9 and the row labeled9. Thus, the upper-tailcritical value of this Fdistribution is 4.03.

Numeratord/,

Denominator

T A B L E1 0 . 6 F i n d i n g F , , ,t h e U p p e r t a i lL n t | c a tv a t u eo T F with9and9Degrees o f F r e e d o mf o r U p p e r Tail Area of 0.025 f -

df1 I

2 3

7 8

641.80 799.s0 864.20 38.51 39.00 39.17 t7.44 16.04 1s.44

8.07 7. 5 7

6.54 6.06

5.89 < A''

948.20 39.36 14.62

956.70 39.37 14.54

4.99 4.53

4.90 4.43

30 39 47

Sottrce: Extnttted /iont TctbleE.5.

Finding Lower-TailCritical Values You computeFr_,a lower-tailcriticalvalue on the trdistribution with n, - I degreesof freedom in the numeratorand r, - I degreesof lreedornin the denominatoqby taking the reciprocal of F u*, an upper-tailcritiial value on the F distributionwith degreesof freedom"switched"(that is, nr- I degreesof freedornin the numeratorand n, - I degreesof freedom in the denominator). This relationshipis shown in Equation( 10.10).

10.4: F Test for the Difference BetweenTwo Variances

399

VALUESFROMTHE F DISTRIBUTION CRITICAL FINDINGLOWER-TAIL -

f r = LE'

.t I

(10.10)

tu*

wheref'* is from an F distributionwith n, ! degreesof freedomin the numeratorand n, I degreesof freedomin the denominator'

In the cola salesexample,the degreesof freedomare9 and9 for both the numeratorsamjust takethe ple anddenominatorru-pi., sothereis no "switching"of degreesof freedom;you reciprocal.Therefore,to computethe lower-tail0.025criticalvalue,you needto find theupperfreetail b.ozs criticalvalueof F with 9 degreesof freedomin the numeratorand9 degreesof page this 398, on 10.6 Table in shown dom in the denominatorand take its reciprocal.As upper-tailvalueis 4.03.UsingEquation(10.10),

lrN iL-

;in ive

_l f r = fU*

-

1 = 0.248 4.03

As depictedin Figure10.13,the decisionrule is RejectHoif F> Fu:4.03 orifFlF,:0.248; otherwise,do not rejectHn.

10.13 ionsof rejectionand for two-tail reiection forequalityof at the variances levelof significance 9 and9 degrees

Regionol Rejection

colasalesdata(seeTablel0.l on page372), UsingEquation(10.9)on page397 and,the the F teststatisticis F={ S;

= 3 5 0 ' 6 7 7 8= 2.228e G;d FL:0.248 < F:2.2289 . Fu:4.03, you do not reject110.Thep-valueis 0.2482for Because in a two-tailiest (twice thep-value for ihe one-tailtest shownin the MicrosoftExcel results in difference is no significant there that > you conclude Figure 10.14).Because0.2482 0.05, locations' the variabilityof the salesof cola for the two display

400

CHAPTERTEN Two-Samole Tests

FTGURE 10.14 Microsoft ExcelFtest results for the BLKcola salesdata

o||G-tsll Crltlcal ono{.ll

See Section E10.8 to create this.

4t0.6778 1t.333 10 10 99 2.2,,'.9 0.1211 3.1789

In testinsfor a differencebetweentwo variances usinstheF testdescribedin this you assumethat eachof the two populationsis normally distributed.The Ftest is very

plotsor normalprobabilityplotssuggest to thenormalityassumption. lf box-and-whisker mild from normality you shouldnot usethe,F a departure for eitherof thetwo populations, (seereferencesI and2). If this happens,a nonparametric approachis moreappropriate In testing for the equality of variances.as part of assessingthe validity of the

variancer test procedure,the F test is a two-tail test. However,when you are interestedin variabilityitself,the F testis oftena one-tailtest.Thus.in testingtheequalityof two vari you canuseeithera two-tailor one-tailtest,dependingon whetheryou aretestingwhether two populationvariancesaredffirent or whetheronevarianceis greaterthan theothervari Figure10.15illustratesthethreepossiblesituations.

t

,A

\

/\ /\ oFr

F

PanelA Two-tailtest

PanelB One-tailtest

PanelC One-tail test

Ho:cl, = 6f

Ho:ol>ol H.,;ol < of

uo: ol < o22

H r :o l * o l

H.,:ol> oj

*

Regionof Rejection Regionof Nonrejection

FIGURE10.15 Determining the rejectionregionfor testingthe equalityof two populationvariances Often,the samplesizesin the two groupsdiffer.Example10.4demonstrates how to finda lower-tailcriticalvaluefrom the F distributionin this situation.

E X A M P L E1 0 . 4

FINDING THE LOWER-TAILCRITICALVALUE FROM THE F DISTRIBUTION IN A TWO-TAILTESTOF A HYPOTHESIS You selecta sampleof n, : 8 from a normally distributedpopulation.The variancefor this sampleSf is 56.0.You selecta sampleof nr-- l0 from a secondnormal$ distributedpopulation (independent of the first population).The variancefor this sampleSj is 24.0.Usingthe level of significancecr:0.05, test the null hypothesisof no differencein the two population variancesagainstthe two-tailalternativethatthereis evidenceof a significantdifferencein the populationvariances. SOLUTION The null andalternativehypotheses are

ol o7 6? otr

10.4:F Testfor theDifferenceBetweenTwoVariances 401 The F test statisticis given by Equation ( 10.9) on page 397:

D - "- 1- t

s? S;

You use Table E.5 to find the upper and lower critical valuesof the F distribution.With nt- l:7 degrees of freedomin thenumerator,nr-l:9 degrees of freedomin thedenominator, and o, : 0.05 split equally into the lower- and upper-tailrejectionregionsof 0.025 each,the uppercriticalvalue,Fu, is 4.20 (seeTable10.7). To find the lower critical va|ue,Fp with 7 degreesof freedomin the numeratorand 9 degreesof freedomin the denominator,you takethe reciprocalof Fr* with degreesof freedom switchedto 9 in the numeratorandT in the denominator. Thus,from Equation(10.10)on page 399 andTable10.7,

F, 7e

| - I =0.207 F,= " Fu* 4.82

T A B L E1 0 . 7 FindingFu. and Fr, withTand9Deorees Usingthe ofFreedom, Level of Significance u = 0.05

Denominator

Numerator df,

df1 I I

2 J

647.80 799.50 864.20 38.51 39.00 39.17 17.44 16.04 15.44

7.57

.20 .36 .62

956.70 9( .30 39.37 3 .39 14.54 I .47

4.43 4.10

4.36 4.03

Source; Extrectedfrom Table E.5.

The decisionrule is RejectHoif F > Fu: 4.20 orifF.FL:0.207; otherwise,do not rejectHo. FromEquation(10.9)on page397,theF teststatisticis

-si S;

=16'q-211 24.0 Because FL:0.207 < F:2.33 . Fu:4.20,you do not rejectHo.Usinga 0.05levelof significance,you concludethatthereis no evidenceofa significantdifferencebetweenthe variances populations. in thesetwo independent

402

Tests CHAPTERTEN Two-Sample

Learningthe Basics 10.40 DetermineFu and Fr, the upper- and lower-tail critical valuesof F. in eachof the following two-tail tests: a . c x ' : 0 . 1 0r , : 1 6 , n r : 2 1 b. a:0.05,r,:16,nr:21 c . c [: 0 . 0 2n, r : 1 6 ,n r : 2 l d. cr: 0.01,n, : 16,nr-- 2l 10.41 DetermineFr,the upper-tailcriticalvalueof d in eachof the followingone-tailtests: a . 0 : 0 . 0 5n, r : 1 6 ,n r : 2 l b . a : 0 . 0 2 5n, r : 1 6 ,n r : 2 1 c . c r: 0 . 0 1n, , : 1 6 ,n r : 2 1 d . c r: 0 . 0 0 5n, , : 1 6 ,n r : 2 1 ',O.42 DetermineFr,rhe lower-tailcritical valueof F, in eachof the followingone-tailtests: a . c t , : 0 . 0 5n, , : 1 6 ,n r : 2 1 b . c r: 0 . 0 2 5n, r : 1 6 ,n r : 2 1 c . c r: 0 . 0 1n, r : 1 6 ,n r : 2 1 d . c r: 0 . 0 0 5n, r : 1 6 ,n r : 2 l 10.43 The following information is available for two samplesdrawn from independentnormally distributedpopulations:

nt = 25

S? = Bl.l

n z= 2 5

Si = 16l.g

What is the valueof the F test statisticif you aretestingthe null hypothesis//6: o? = a3t 10.44 In Problem10.43,how many degreesof @| lAsslsTI freedomare therein the numeratorand denominator of the F test? 10.45 In Problems10.43and 10.44,whatare the critical values for F, and F, if the level of significance,cr,is 0.05andthe alternativehypothesisis .F11: of * oi? '10.46 In Problems10.43through10.45,what is your statisticaldecision? 10.47 The following information is availablefor two samplesselectedfrom independentbut very right-skewed populations: nr = 16

S? = 4l.t

nz = 13

Sl = 36.q

Shouldyou use the F test to test the null hypothesisof equalityof variances(Hs: ol = o3)? Discuss.

10.48 In Problem 10.47,assumethat two samples normally distributedpopu selectedfrom independent a. At the 0.05 level of significance,is there evidence differencebetweenof and ai? b. Supposethat you want to perform a one-tailtest.At 0.05 level of significance,what is the upper-tailcri value of the F test statisticto determinewhethert evidencethat of > o]Z Whut is your statisticaldecisi c. Supposethat you want to perform a one-tailtest.At 0.05 level of significance,what is the lower-tailc valueof the .F test statisticto determinewhetherthere evidencethat of < c3?What is your statisticaldecisi

Applying the Concepts

S? = 210.2

nz = l0

Sl. = 3e.S

a. At the 0.05 level of significance,is there evidence supportthe professor'sclaim? b. Interpretthep-value. c. What assumptiondo you needto make in (a) about two populations in order to justify your use of .Ftest? l-lsELFl 10.50 The Computer Anxiety Rating Scale (CARS) measuresan individual'slevel of comG puter anxiety,on a scalefrom 20 (no anxiety)to 100 (highest level of anxiety). Researchersat Miami UniversityadministeredCARS to 172 businessstudents. One of the objectives of the study was to determine whetherthereis a differencebetweenthe level of computer anxiety experiencedby femalestudentsand male students. They found the following: Males

x

^t n

40.26 13.35 100

d

1 t ir \) c I

10.49 A professorin the accounting of a businessschool claims that thereis more variability in the final exam scoresof dentstakingthe introductoryaccountingcourseasa mentthanfor studentstakingthe courseaspart of a maj accounting.Randomsamplesof 13 non-accounting (group l) and l0 accountingmajors(group2) aretaken the professor'sclassroster in his large lecture,andthe lowingresultsarecomputedbasedon the final exam nt = 13

b c

Females

36.85 9.42 72

Source: ExtractedfromT. Broome and D. Havelka, "Determinants of ComputerAnxiety in BusinessStudents,"The Review of Business Information Systems,Spring 2002, 6(2), pp. 9-16.

a

s I

10.4:F Testfor the DifferenceBetweenTwoVariances 403

the 0.05level of significance,is there evidenceof a ce in the variability of the computeranxiety

Central Office ll Time to Clear Problems(Minutes) 7. 5 5 3 . 7 5 0 . 1 0 l . l 0 0 . 6 0 0 . 5 2 3 . 3 0 2 . t 0 0 . 5 8 4 . 0 2 3.75 0.65 t.92 0.60 1.53 4.23 0.08 1.48 1.65 0.72

assumptiondo you need to make about the two ionsin orderto justiff the useof the f'test? Basedon (a) and (b), which I test defined in Section I shouldyou useto testwhetherthereis a significant ifferencein mean computeranxiety for female and students? A bankwith a branchlocatedin a commercialdisof a city hasdevelopedan improvedprocessfor servduringthe noon-to-1p.m. lunchperiod.The customers ing time (defined as the time elapsedfrom when the entersthe line until he or shereachesthe teller of ) all customersduring this hour is recordedover of oneweek.A randomsampleof 15 customersis (andstoredin the file !@), andthe results(in ) areasfollows: 4.21 5.55 3.02 5.13 4.77 2.34 3.s4 3.20 4.50 6.10 0.38 5.12 6.46 6.t9 3.79 that anotherbranch,locatedin a residentialarea, with the noon-to-l p.m. lunchperiod.A concerned sampleof l5 customersis selected(and storedin file @@), and the results (in minutes) are as

9.66 5.90 8.02 5.79 8.73 3.82 8.01 8.35 10,496.68 5.64 4.08 6.17 9.91 5.47 Isthereevidenceof a differencein the variability of the (Usea:0.05.) waitingtime betweenthe two branches? Determine thep-value in (a) and interpretits meaning. Whatassumptionis necessaryin (a)? Is the assumption mlid for thesedata? Basedon the resultsof (a), is it appropriateto use the pooled-variance I test to comparethe meansof the two branches? 10.52 A problem with a telephoneline that prevents IJJ a customerfrom receiving or making is disconcertingto both the customerand the telecompany.The following data (stored in the file representsamplesof 20 problemsreportedto two t offices of a telephonecompanyand the time to theseproblems(in minutes)from the customers'lines: Office I Time to Clear Problems(Minutes) 1 . 7 50 . 7 8 2 . 8 5 0 . 5 2 1 . 6 0 4 . 1 5 3 . 9 7 1 . 4 8 3 . 1 0 0.53 0.93 1.60 0.80 1.05 6.32 3.93 s.45 0.97

a. Is there evidenceof a differencein the variability of the waiting times betweenthe two offices? (Use cr: 0.05.) b. Determinethep-value in (a) and interpretits meaning. c. What assumptionis necessary in (a)?Is the assumption valid for thesedata? d. Based on the results of (a), which / test defined in Section10.1 shouldyou use to comparethe meansof the two offices? 10.53 The director of training for a companythat manufactureselectronicequipmentis interestedin determining whether different training methodshave an effect on the productivity of assembly-lineemployees.She randomly assigns42 recentlyhired employeesto two groups of 21. The first group receivesa computer-assisted, individualbasedtraining program, and the other receivesa teambasedtraining program.Upon completionof the training, the employeesare evaluatedon the time (in seconds)it takesto assemblea part. The resultsare in the data file

@. a. Usinga 0.05levelof significance,is thereevidenceof a differencebetweenthe variancesin assemblytimes (in seconds)of employeestrainedin a computer-assisted, individual-basedprogram and those trained in a teambasedprogram? b. On the basisof the resultsin (a), which r test defined in Section10.1 shouldyou use to comparethe meansof the two groups?Discuss. 10.54 Is therea differencein the variationof the yield of different types of investmentbetweenbanks?The following data, from the fil" El@![f[, representthe nationwide highest yields for money market accountsand oneyearCDs as ofJanuary24,2006: Money MarketAccounts I

One-YearCD

4.ss 4.s0 4.40 4.38 4.38 | 4.94 4.90 4.85 4.85 4.85 Source: Extractedfrom Bankrate.com,January 24, 2006.

At the 0.05levelof significance,is thereevidenceof a differencein the varianceof the yield betweenmoneymarket accountsand one-yearCDs?Assumethat the population yields arenormally distributed.

404

CHAPTER TEN Two-Samolc Tests

TA Su

In this chapter,you were introducedto a variety of twosar.nple tests.For situationsin which the samplesare independent,you learnedstatisticaltest proceduresfor analyzi n g p o s s i b l ed i f f e r e n c e sb e t w e e nm e a n s ,v a r i a n c e s .a n d proportions.ln addition, you learneda test proceclurethat is frequentlyused when analyzingdifferencesbetweenthe meansof two relatedsamples.Rcmernberthat you needto

select thc test that is most appropriate for a givensetof conditions and to critically investigatethe validity of the a s s u m p t i o n su n d e r l y i n g e a c h o f t h e h y p o t h e s i s - t e s t i n g procedures. The roadmap in Figurc 10.l6 illustrates the stens neededin determiningwhich two-sar.nple test of hypothesis to use:Thc fbllowing arethe questionsyou needto consider,

Two-Sample Tests

tn \

I

Z' Type of Data

Categorical

Ztest for the d ifference between two proportions

Yes

CentraI Tendency

No

Separate-Variance t Test

o?=otrz

Numerical

Independent Samples?

No

Variability

Focus

FTest

Yes

roa r l =ol

Paired t Test

Pooled-Variance f Test

FIGURE'10.'16Roadmapfor selectinga two-sampre test of hypothesis l. What typc of datado you lrave'/If you are clealingwith categorical variables, use the Z rest for the differencc betweentwo proportions.(This test assumesindeoen_ d e n ts a m p l e s . . l 2. lf you have a numerical variable, deten.ninewhether you have independentsar.nples or related samples.lf you haverelatedsamples,use the pairedI test. 3. Ifyou haveindependentsamples,is your focus on variability or central tendency'?If the focus is variabilitv. u s et h e , t r t e s t .

4 . If your focus is central tendency,determine whether you can assumethat the variancesof the two groups a r e e q u a l . ( T h i s a s s u r n p t i o nc a n b e t e s t e du s i n gt h e F test.) If yoLrcan assumethat the two groups have equal varia n c e s ,u s e t h e p o o l e d - v a r i a n c eI t e s t . I f y o u c a n n o t assumethat the two groups have equal variances,use the separate-variance / test. Table 10.8providesa list of topics coveredin this chapter.

Kcy Equations 405

I cf :le 1g

Typesof Data

TABLE 10.8 Summary of Topics i nC h a p t e 1r 0

Type of Analysis

Numerical

Categorical

Comparingtwo populations

Z and t tests for the difference in the means of two independentpopulations ( S e c t i o n1 0 .I )

Z test for the difference betweentwo proportions ( S e c t i o n1 0 . 3 )

ps ;is

P a i r e dI t e s t( S e c t i o n1 0 . 2 ) F test for differencesbetween two variances(Section 10.4)

ZTestfor the Difference Between Two Means

Z-

-trz)

(Xt-X)-(pr

f ,

)

loi

r rl ! /7t

ConfidenceInterval Estimatefor the Mean Difference

(r0.1)

(10.6)

o;

n2

D-t,,-r#=u,
Pooled-Variance I Test for the Difference BetweenTwo Means

- X t ) - ( $ r- p z )

(r0.2)

Z Test for the Difference BetweenTwo Proportions

pz\- (nr-/t.)

, _(pt-

( r 0.7)

ConfidenceInterval Estimateof the Difference in the Means of Two IndependentPopulations

ConfidenceInterval Estimatefor the Difference BetweenTwo Proporttont

(r0.3)

( . p t -p z ) + z ^ l p , o -p , l * p z o - p : t

(10.8)

lntn2 or (pt-p)-Z

FEB

S(X, - Xrl+t,,,.,,._,

her rps the lrinot

use

ZTestfor the Mean Difference

z =D

_ltn 9n

plt - p)

pt(l - pt)

n1

n2

a(h-pz)+Z

t1l

s.'

f = -;

s:'

Pairedt Test for the Mean Difference

s, ^Fn

p2(l- p2l t7-t

F Test Statistic for Testing the Equality of Two Variances

( r0.4)

J;

D -Vn

pt(l - pr)

< (nr- nz)

(10.9)

Finding Lower-TailCritical Values

(r0.s)

liom the F Distribution

-1 =

rt

r[]*

(10.10)

406

Tests CHAPTERTEN Two-Sample t( f(

F distribution 397 .Fteststatisticfor testingthe equality of two variances 397 matched 381 pairedI test for the meandifference in relatedpopulations 382

pooled-variance / test 371 repeatedmeasurements 381 robust 375 separate-variance / test 311

CheckingYour Understanding 10.55 What are someof the criteriausedin the selection procedure? of a particularhypothesis-testing 10.56 Underwhat conditionsshouldyou usethe pooledvariance/ testto examinepossibledifferencesin the means populations? of two independent '10.57 Underwhat conditionsshouldyou usethe F testto examinepossibledifferencesin the variancesof two independentpopulations? 10.58 What is the distinctionbetweentwo indeoendent populationsandtwo relatedpopulations? 10.59 What is the distinctionbetweenrepeatedmeasurementsandmatched(or paired)items? 10.60 Underwhat conditionsshouldyou usethe pairedr testfor themeandifferencebetweentwo relatedpopulations? 10.61 Explainthe similaritiesand differencesbetween for the differencebetweenthe means the testof hypothesis populationsandthe confidenceinterval oftwo independent estimateof the differencebetweenthe means.

Applying the Concepts 10.62 A study comparedmusic compactdisc pricesfor retailersand traditional brick-and-mortar Internet-based retailers(Extractedfrom L. Zoonky and S. Gosain,'A for MusicCDsin Electronic LongitudinalPriceComparison and Brick-and-Mortar Markets: Pricing Strategiesin EmergentElectronic Commerce,"Journal of Business Strategies, Spring2002,| 9(1), pp.5 5-72). Beforecollecting the data,the researchers carefully defined severalresearch hypotheses, including: l. The price dispersionon the Internetis lowerthan the pricedispersionin the brick-and-mortar market. 2. Pricesin electronicmarketsare lower than pricesin physicalmarkets. a. Considerresearch hypothesisL Write thenull andalternative hypothesesin terms of populationparameters. used. Carefullydefinethe populationparameters in (a). b. Define a TypeI andTypeII errorfor the hypotheses

Z test for differencebetween two means 370 Ztestfor the differencebetween two proportions 390

ti I\ n S

p o a

c. What typeof statisticaltestshouldyou use? d. What assumptionsare neededto performthetest selected? hypothesis 2. e. Repeat(a) through(d) for research 10.63 The pet-drug market is growing very Beforenew pet drugs can be introducedinto the place, they must be approved by the U.S. Food and

Administration(FDA). In 1999,the Novartiscompany trying to get Anafranil, a drug to reducedog anxi approved.Accordingto an article (E. Tanouye,"The0w Bowwow: With Growing Market in Pet Drugs, RevampClinical Trials," The WqllStreetJournal,Api,l 1999).Novartishadto find a wav to translatea doe's ety symptomsinto numbersthat could be usedto prove the FDA that the drug had a statisticallysignificant e on the condition. a. What is meant by the phrasestatisticallysi effect? b. Consideran experimentin which dogssuffering anxietyare dividedinto two groups.One groupwill given Anafranil,and the otherwill be givena (thatis, a drugwithoutactiveingredients). Howcan translatea dog's anxiety symptomsinto numbers? other words, define a continuous variable,Xr,the

surementof effectivenessof the drug Anafranil,and the measurement of the effectiveness of theplacebo. c. Building on your answerto part (b), definethenull alternativehypothesesfor this study. 10.64 In responseto lawsuitsfiled againstthe industry,manycompanies,suchas Philip Morris,are ning televisionadvertisements that aresupposed to teenagers aboutthe dangersof smoking.Are these Are industryantismokingcampaignssuccessful? sponsoredantismokingcommercialsmore effective? article (G. Fairclough,"Philip Morris's Antismoki CampaignDraws Fire," The Wall StreetJournal, Apri.l 1999,p. B I ) discussed a studyin Californiathat commercials made by the state of California and

showed the cials producedby Philip Morris. Researchers group of Californi stateadsandthe Philip Morris adsto a

Chapter ReviewProblems 407 and measuredthe effectivenessof both. The concludedthat the stateadsweremore effecin relayingthe dangersof smoking than the Philip is ads.The article suggests,however,that the study is statisticallyreliable becausethe samplesize was too and becausethe study specifically selectedparticiwho areconsideredmore likelv to startsmokinethan How do you think the researchersmeasuredeffectiveDefinethe null and alternativehypothesesfor this study. Explainthe risks associatedwith Type I and Type II enorsin this study. Whattype of test is most appropriatein this situation? Whatdo you think is meant by the phrasestatistically reliable2 The FedEx St. JudeClassicprofessionalgolftourt is held eachyear in Memphis,Tennessee. FedEx this PGA tournament,and part of the proceedsgo theSt. JudeChildren'sResearchHospital. In 2003, the raised$679,115for the hospital.This type of sponsorship is known ascause-related marketing. sampleof spectatorsat the tournamentwere surveyed askedto respondto a seriesof statements on a 5-point (l = StronglyDisagree,2 : Disagree,3 : Neutral, Agree,and 5 : StronglyAgree).The following are ofthe questionsasked: Cause-related marketing createsa positive company image. I would be willing to pay more for a servicethat supportsa causeI careabout. Cause-related marketingshouldbe a standardpart of a company's activities. Basedon its supportof St. Jude,I will be more likely to useFedExservices. eachquestion,the researchers testedthe null hypothethatthe meanresponsefor malesand femalesis equal. alternativehypothesisis that the meanresponseis diffor malesand females.The followine table summatheresults:

a. b. c. d. e.

Interpretthe resultsofthe r test for question1. Interpretthe resultsofthe t test for question2. Interpretthe resultsofthe t testfor question3. Interpretthe resultsofthe I test for question4. Write a short summary about the differencesbetween malesand femalesconcerninstheir views towardcauserelatedsponsorship.

10.66 Two professorswantedto study how studentsfrom their two universitiescomparedin their capabilitiesof using Excel spreadsheets in undergraduateinformation systems courses(Extractedfrom H. Howe, and M. G. Simkin, "FactorsAffecting theAbility to DetectSpreadsheet Errors,'o Decision SciencesJournal of InnovativeEducation, January 2006, pp. l0l-122). A comparisonof the studentdemographicswasalsoperformed.Oneschoolis a stateuniversity in the WesternUnited States.and the otherschoolis a state universityin the EasternUnited States.The following table containsinformation regardingthe agesof the students:

School

Western Eastern

4.46 4.09 4.26 4.12

4.26 3.86 3.91 4.06

t

1.907 2.105 3.258 0.567

p-Value

0.0s7 0.035 0.001 0.571

: Extractedfrom R. L. Irwin, T Lachowetz, T. B. Cornwell, J. S. Cook, "Cause-RelatedSport Sponsorship:AnAssessment :SpectatorBeliefs,Attitudes, and Behavioral Intentions," Sport

Quarterly,2003,I 2(3),pp. I 3I-l 39.

23.28 21.16

Standard Deviation

6,29 1.32

The following table containsinformation regardingthe yearsofspreadsheet usageofthe students:

Western Eastern

Female Male (nr=137) (2, =305)

93 r35

Mean Age

a. Using a 0.01levelof significance,is thereevidenceof a differencebetweenthe variancesin age of studentsat the Westernschooland at the Easternschool? b. Discussthe practicalimplicationsof the test performed in (a). Address, specifically, the impact equal (or unequal)variancesin agehason teachingan undergraduateinformation systemscourse. c. To test for a differencein the meanageof students,is it most appropriateto usethe pooled-variance/ test or the separate-variance I test?

School SampleMean

Sample Size

Sample Size

Mean Years

Standard Deviation

93 135

2.6 4.0

2.4 2.1

d. Usinga 0.01levelof significance,is thereevidenceof a differencebetweenthe variancesin yearsof spreadsheet usage of studentsat the Western school and at the Easternschool? e. Basedon the resultsof (d), usethe most appropriatetest to determine,at the 0.01 level of significance,whether there is evidenceof a differencein the meanyearsof spreadsheet usageofstudentsat the Westernschooland at the Easternschool?

408

Tests CHAPTERTENTwo-Sample

10.67 The manager of computer operations of a large company wants to study computer usage of two departments within the company-the accounting department and the researchdepartment.A random sample of five jobs from the accounting department in the past week and six jobs from the research department in the past week are selected"and the processingtime (in seconds)for eachjob is recorded(and stored in the l@$!!

Processing Time (in Seconds)

Department Accounting Research

file):

9 4

38',7 13109

l2 96

Usea levelof significanceof 0.05. a. Is thereevidencethat the meanprocessingtime in the is greaterthan6 seconds? research department b. Is there evidenceof a differencebetweenthe variances in theprocessing time of the two departments? c. Is thereevidenceof a differencebetweenthe meanprocessingtime of the accountingdepartmentand that of the researchdepartment? d. Determinethep-valuesin (a) through(c) and interpret theirmeanings. e. Constructand interpreta 95o/oconfidenceintervalestimate of the differencein the meanprocessingtimes betweenthe accountingand researchdepartments. 10.68 A computerinformationsystemsprofessoris interestedin studyingtheamountof time it takesstudentsenrolled in the introductionto computerscourseto write and run a programin Visual Basic.The professorhiresyou to analyze the following results(in minutes)from a randomsampleof ninestudents(thedataarestoredin the![ft! file): 10 t3 9

15 12 13 11 13 12

a. At the 0.05 level of significance,is thereevidencethat thepopulationmeanamountis greaterthan 10minutes? What will you tell the professor? b. Supposethe computerprofessor,when checking her results,realizesthat the fourth studentneeded5l minutesratherthanthe recordedl5 minutesto write andrun the Visual Basicprogram.At the 0.05 level of significance,reanalyzethe questionposedin (a), using the reviseddata.Whatwill you tell theprofessornow? c. The professoris perplexedby theseparadoxicalresults andrequestsan explanationfrom you regardingthejustification for the differencein your findings in (a) and (b). Discuss. d. A few dayslater,the professorcalls to tell you that the dilemmais completelyresolved.Theoriginalnumber15 (the fourth data value) was correct, and thereforeyour findingsin (a) arebeingusedin the articlesheis writing for a computerjournal. Now shewantsto hire you to comparethe resultsfrom that group of introductionto computersstudentsagainstthosefrom a sampleof 1l

computermajorsin orderto determinewhetherthereh evidencethat computermajorscan write a VisualBasis programin lesstime than introductory students.Forthc and computermajors,the samplemeanis 8.5 minutes, the samplestandarddeviationis 2.0 minutes.At the0.05 level of significance,completelyanalyzethesedata, What will you tell theprofessor? e. A few dayslater,theprofessorcallsagainto tell youthat a reviewerof her article wantsher to includethep-value for the "correct" result in (a). In addition,the professor problem,whichthe inquiresaboutan unequal-variances reviewerwantsher to discussin her article. In yourown the words,discussthe conceptof p-value and describe problem.Determinethe p-valuein unequal-variances problem (a) and discusswhetherthe unequal-variances had any meaningin the professor'sstudy. 10.69 An article in USA Today(D. Sharp,"Cellphones Reveal Screaming Lack of Courtesy," USA Today, September2001,p. 4,{) reportedthat accordingto a poll, the meantalking time per month for cell phoneswas372 minutesfor men and 275 minutesfor women,whilethe meantalking time per month for traditional homephones was 334 minutes for men and 510 minutesfor women. thatthepoll wasbasedon a sampleof 100menand Suppose 100 women, and that the standarddeviationof the talking time per month for cell phoneswas 120 minutesfor men of and 100minutesfor women,while the standarddeviation the talking time per month for traditionalhomephoneswas 100 minutes for men and 150 minutes for women. Use a level of significance of 0.05. a. Is there evidence of a difference in the mean monthly talking time on cell phones for men and women? b. Is there evidence of a difference in the mean monthly talking time on traditional home phones for men and women? c. Construct and interpret a 95o/oconfidence interval estimate of the difference in the mean monthly talking time on cell phones for men and women. d. Construct and interpret a 95o/oconfidence interval estimate of the difference in the mean monthly talking time on traditional home phones for men and women. e. Is there evidence of a difference in the variance ofthe monthly talking time on cell phones for men and women? f. Is there evidence of a difference in the variance of the monthly talking time on traditional home phones for men and women? g. Based on the results of (a) through (f , what conclusions can you make concerning cell phone and traditional home phone usage between men and women? 'lO.7O A survey of 500 men and 500 women designedto study financial tensions between couples asked how likely they were to hide purchases, cash, or investments from their partners.The results were as follows:

Chapter ReviewProblems 409 Likely to Hide

Men 66 t26 62 79 96 74 76 53 Extractedfrom L. Wei, "Your Money Manager as Financial " The Wall StreetJournal, November 5-6, 2005, p. 84.

Foreachtypeof purchase,determinewhetherthereis a rencebetweenmen and women at the 0.05 level of ificance.

As moreAmericansuse cell phones,they question it is okayto talk on cell phones.Thefollowingis a table results,in percentages, for 2000 and2006(extractedfrom Koch,"BusinessPut a Lid on Chatterboxes i USATbday, 7, 2006,p. 3A). Supposethe surveywasbasedon respondents in 2000and 100respondents in 2006.

YEAR TOTALKONA CEILPHONE INA 2OOO 39 ll 76 60 52 31

2006 38 2 63 66 45 zl

For eachtype of location, determinewhetherthere is a fferencebetween2000 and 2006 in the proportion who it is okayto talk on a cell phone(usethe 0.05 level of ificance).

72 The lengthsof life (in hours) of a sampleof 40 light bulbs producedby manufacturerA and a e of 40 100-wattlight bulbs producedby manufacB are in the file![!$fr. Completelyanalyzethe difbetweenthe lengthsof life of the bulbs produced (usecr: 0.05). twomanufacturers 73 The datafile [!!@@!E containsthe ratings for decor,service,and the price per personfor a sample 50 restaurants locatedin an urban arca and,50restaulocatedin a suburbanarea.Completely analyzethe ncesbetweenurbanand suburbanrestaurantsfor the iablesfood rating,decorrating, servicerating, andprice person, usingc: 0.05. : Extractedfrom Zagat Survey 2002: NewYork City and Zagat Survey 200 l-2002: Long Island Restaurants.

10.74 Management of a hotel was concerned with increasingthe return rate for hotel guests.One aspectof first impressionsby guestsrelatesto the time it takesto deliverthe guest'sluggageto the room after check-into the hotel.A randomsampleof 20 deliverieson a particularday were selectedin Wing A of the hotel, and a randomsample of 20 deliverieswere selectedin Wing B. The resultsare storedin the file lEtl*lElltl. Analyzethe data and determine whetherthere is a differencein the mean delivery time in the two wings of the hotel (useo(: 0.05). '10.75 In manufacturingprocesses,the term work-inprocess (often abbreviatedWIP) is often used.In a book manufacturingplant, WIP representsthe time it takes for sheetsfrom a pressto be folded,gathered,sewn,tipped on end sheets,and bound.The following data(storedin the file @$) representsamplesof 20 books at eachof two productionplantsand the processingtime (operationally definedasthe time, in days,from whenthe bookscameoff the pressto when they werepackedin cartons): PlantA 5.O 5.29 16.2s t0.92 1r.46 21.628.45 8.58 5.4r 11.42 rr.62 7.29 7.s0 7.96 4.42 10.s07.s8 9.29 7.54 8.92 Plant B 9.54 11.4616.6212.6225.7515.4114.2913.1313.7110.04 5.7512.46 9.17r3.2r 6.00 2.3314.25 5.37 6.25 9.11 Completelyanalyzethe differencesbetweenthe processing times for the two plants,using o:0.05, and write a summary of your findings to be presentedto the vice president for operationsof the company. 10.76 Do marketingpromotions,such as bobble-head giveaways,increaseattendanceat Major LeagueBaseball games?Anarticlereportedon the effectiveness of marketing promotions(extractedfrom T. C. Boyd and T. C. Krehbiel, 'An Analysisof SpecificPromotionTypes Attendance on at Major LeagueBaseballGames,"Mid-AmericanJournal of Business, 2006, 21, pp. 21,-32). The data file [!!@ includesthe following variablesfor the KansasCity Royals during|he2002baseballseason: Game-Home games,in the order in which they were played Attendance-Paid attendancefor the game Promotion-l : if a promotionwasheld;0: if no promotion was held a. At the 0.05 level of significance,is thereevidenceof a differencebetweenthe variancesin the attendanceat gameswith promotionsand gameswithout promotions? b. Basedon the resultof(a), conductthe appropriatetestof hypothesisto determinewhetherthereis a differencein the mean attendanceat gameswith promotionsand gameswithoutpromotions.(Usea: 0.05.) c. Write a brief summarvof vour results.

410

CHAPTER TEN Two-Samole Tests

10,77 The manufacturerof Bostonand Vermontasphalt shinglesknowsthat productweightis a major factorin the customer'sperceptionof quality.Moreover,the weight representsthe amountof raw materialsbeing used and is thereforevery importantto the companyfrom a cost standpoint. The last stageof the assembly-line packagesthe shinglesbeforethey areplacedon woodenpallets.Oncea pallet is full (a pallet for most brandsholds l6 squaresof shingles),it is weighed,and the measurement is recorded. The datafilep@[!contains the weight(in pounds)from a sampleof 368 palletsof Bostonshinglesand 330 pallets ofVermontshingles.Completelyanalyzethe differencesin theweightsof the BostonandVermontshingles,usingcr :

0.0s.

2005 return-Twelve-month return in 2005 Three-yearreturn-Annualized return, 2003-2005 Five-yearreturn-Annualized return, 20012005 Risk-Risk-of-loss factor of the mutual fund average,or high) 10.80 Completelyanalyzethe differencebetweenmutual funds without feesand mutual funds with feesin terms 2005 return. three-vear return. five-vear return. expenseratio. Write a report summarizingyour findings.

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10.81 Completely analyzethe differencebetweenmutml funds that have a growth objectiveand mutual fundsthat havea value objective in terms of 2005 return, three-yeu return, five-year return, and expenseratio. Write a report yourfindings. summarizing

10.78 The manufacturerof Bostonand Vermontasphalt shinglesprovidesits customers with a 2)-yearwarrantyon Student Survey Data Base most of its products.To determinewhethera shinglewill lastas long as the warrantyperiod,accelerated-life testing 10.82 Problem1.27on page 15 describesa surveyof 50 is conductedat the manufacturingplant.Accelerated-life undergraduatestudents(see the file GEEEElilil$ED. testingexposesthe shingleto the stresses it would be subFor thesedata, ject to in a lifetimeof normalusein a laboratorysettingvia a. at the 0.05 level of significance,is thereevidence ofa an experimentthat takesonly a few minutesto conduct.In differencebetweenmales and femalesin gradepoint this test,a shingleis repeatedlyscrapedwith a brushfor a average,expectedstartingsalary,salaryexpectedin five shortperiodof time, and the shinglegranulesremovedby years,age,andspendingon textbooksandsupplies? the brushingareweighed(in grams).Shinglesthat experib. at the 0.05 level of significance,is thereevidence ofa encelow amountsof granuleloss are expectedto last differencebetweenthose studentswho plan to goto longer in normal use than shinglesthat experiencehigh graduateschoolandthosewho do not plan to go to gradamountsof granuleloss.In this situation,a shingleshould uate school in gradepoint average.expectedstarting experience no more than 0.8 gramsof granulelossif it is salary,salary expectedin five years,age,and spending expected to lastthe lengthof the warrantyperiod.The data on textbooksandsupplies? file !@ containsa sampleof 170 measurements 10.83 Problem1.27on page15 describes a surveyof 50 madeon the company'sBostonshinglesand 140measure(see undergraduate the file@![[[!@!!f$. students mentsmadeon Vermontshingles.Completelyanalyzethe a. Selecta sampleof 50 undergraduate studentsat your differencesin the granulelossof the BostonandVermont schoolandconducta similarsurveyfor them. shingles, usingcr: 0.05. b. For the data collected in (a), repeat (a) and (b) of Repoft Writing Exercise Problem10.82. 10.79 Referring to the results of Problems 10.77 and c. Comparethe resultsof (b) to thoseof Problem10.82. 10.78concerningthe weightand granulelossof Boston 10.84 Problem1.28on page15 describes a surveyof50 andVermontshingles,write a reportthat summarizes your MBA students(seethe file f.EEllllllElll$. For thesedata, conclusions. at the 0.05 levelof significance,is thereevidenceof a difTeamProject ferencebetweenmalesand femalesin age,undergraduate The data file @tIEslE containsinformation regardgrade point average,graduate grade point average, ing ninevariablesfrom a sampleof 838 mutualfunds.The GMAT score,expectedsalaryupon graduation,salary variablesare expectedin five years,and spendingon textbooksand Category-Type of stockscomprisingthe mutual fund supplies? (smallcap,mid cap,or largecap) 10.85 Problem1.28on page15 describes a surveyof50 Objective-Objectiveof stockscomprisingthe mutual MBA students(seethe fileFEElliltEliE$. fund (growthor value) a. Selecta sampleof 50 graduatestudentsin your MBA Assets-In millionsof dollars programandconducta similar surveyfor thosestudents. Fees-Salescharges(no or yes) b. For the datacollectedin (a),repeatProblem10.84. Expenseratio-Ratio of expensesto net assets,in c. Comparethe resultsof (b) to thoseof Problem10.84. percentage

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A marketingdepartment team is charged with improving thetelemarketingprocess in order to increasethe number of home-deliverysubscriptions sold. After several brain$ormingsessions,it was clear that the longer a caller speaks to a respondent,the greater the chance that the callerwill sell a home-delivery subscription.Therefore,the teamdecidedto find ways to increase the length of the phonecalls. Initially,the team investigatedthe impact that the time ofa call mrght have on the length of the call. Under current arrangements, calls were made in the evening hours, between 5:00 p.m. and 9:00 p.m., Monday through Friday. Theteamwantedto comparethe length of calls made early in the evening(before 7:00 p.m.) with those made later in theevening(after 7:00 p.m.) to determine whether one of these time periods is more conducive to lengthier calls and,

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correspondingly,to increasedsubscription sales.The team selected a sample of 30 female callers who staff the telephone bank on Wednesday evenings and randomly assigned 15 of them to the "early" group and l 5 to the "later" group. The callers knew that the team was observing their efforts that evening but didn't know which calls were monitored. The callers had been trained to make their telephone presentationsin a structured manner.They were to read from a script, and their greeting was personal but informal ("Hi, this is Mary Jones from the Springville Herald. May I speakto Bill Richards?"). Measurementswere taken on the length of the call (defined as the difference, in seconds,betweenthe time the person answeredthe phone and the time he or she hung up). The results (stored in the file !fff[!) are presentedin T a b l eS H l 0 . l .

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EXERCISES S H I O . 1A n a l y z e t h e d a t a i n T a b l e S H l 0 . l a n d w r i t e a report to the rnarketing departmentteam that indicatesyour findings. Include an attachedappendix in which you discussthe reasonyou selecteda particular statisticaltest to compare the two independent groupsofcallers. SH10.2Suppose that instead of the research design describedhere, there were only 15 callers sampled,

and each caller was to be monitored twice in the evening-once in the early time period and once in the later time period. Suppose that in Table SH I 0. I, each pair of values representsa particular caller's two measurements.Reanalyze these data and write a report for presentationto the team that indicatesyour findings. SHl0.3 What other variablesshould be investisatednext?

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Web Case Applyltour knowledge about hypothesis testing in this Web Case,which continues the cereal-fill packaging dispute WebCuse fiom Chapters 7 and 9.

Even after the recent public experiment about cereal box weights, the Consumers Concerned About Cereal Cheaters(CCACC) remains convinced that Oxford Cereals

412

TEN Two-Sample Tests CHAPTER

has misled the public. The group has createdand posteda documentin which it claims that cerealboxesproducedat PlantNumber 2 in Springvilleweigh lessthan the claimed meanof 368 grams.Visit the CCACCMore Cheatingpage (or at www.prenhall.com/Springville/MoreCheating.htm page folder) from the text CD's Web Case file openthis Web andthenanswerthe following:

l. Conover,W. J.,PracticalNonparametricStatistics,3rd ed. (New York: Wiley, 2000). 2. Daniel, W., Applied NonparametricStatistics,2nd ed. (Boston:HoughtonMifilin, 1990). 3. MicrosoftExcel 2007 (Redmond,WA: Microsoft Corp., 2007). F. E., 'An ApproximateDistributionof 4. Satterthwaite, Estimatesof VarianceComponents l' BiometricsBulletin, 2(1946):I 10-114.

l . Do the CCACC'sresultsprove that there is a stati

differencein the meanweightsof cerealboxes at PlantNumbersI and2? ) Performthe appropriateanalysisto test the CC hypothesis.What conclusionscanyou reachbasedon data?

5 . Snedecor. G. W."andW. G. Cochran.Statistical (Ames. IA: Iowa StateUniversiWPress,1989). 8th ed. 6. Winer.B. J..D. R. Brown.andK. M. Michels. Principles in ExperimentalDesign,3rd ed. (New McGraw-Hill.1989).

Excel Companionto Chapter 10

413

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Youcan use Microsoft Excel to conduct all six two-sample hypothesis testsdiscussedin Chapter 10. Some testsrequire theuse ofToolPak procedure. Others require that you use PHStat2or one of the workbooks stored on the Student CD-ROM.Sometestshave Excel versionsin which you can useonly unsummarizeddata, and others have Excel versions in which you must use summarized data. (A few tests have Excelversionsfor either unsummarizedor summarized data.) Use Table El0.l below to help choosethe right test for yourdataand as a guide for the rest of this Excel Companion.

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62 6_,iNormal 7 ,Normal 8:Normal _9 :Normal l0,Normal 11rNormal

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Two-Sample Data Arrangements Unsummarizeddata can be entered in a stacked or unstackedarrangement.In a stacked arrangement,all the valuesfor a variable appear in a single column, next to a columnthat identifies the sampleor group to which individualvaluesbelong. In an unstackedarrangement,the values for the samplesor groups appear in separatecolumns. For example,Figure El0. I shows the stackedand unstacked versionsof the cola display location sales analysis data of T a b l el 0 . l o n p a g e 3 7 2 .

T A B L EE 1 0 . 1 Two-Sample Tests inMicrosoftExcel

Two-SampleTest

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UnsummarizedData SummarizedData (ExcelCompanionSection) (ExcelCompanionSection)

Z test for the difference ToolPakz-Test:Two Samnle betweentwo means for Means(E 10.I )

Z'lwo Means.xls or PHStat2 Z Testfor Differencesin TwoMeans(E10.2)

Pooled-variance/ test for the difference betweentwo means

ToolPakt-Test:Two-Sample Pooled-variance T.xlsor AssumingEqualVariances PHStat2t Testfor Differences (E10.3) BetweenTwo Means(E10.4)

Separate-variance I test for the difference betweentwo means

ToolPakt-Test:Two-Sample Not included AssumingUnequalVariances

Paired I test

ToolPakt-Test:PairedTwo Samplefor Means(E10.6)

Not included

Z test for the difference between two proportions

Not included

ZTwo Proportions.xls or PHStat2 Z Testfor DifferencesBetween Two Proportions(E 10.7)

,F test for the difference between two variances

ToolPakF-TestTwo-Sample F TwoVariances.xls or (E10.8) for Variances PHStat2F Testfor Differences (E10.9) BetweenTwo Variances t'E10.9 wo Variances

(Er0.s)

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4I4

EXCELCOMPANION to Chanter 10

Specific statisticalproceduresor Excel worksheetsfor analysesinvolving two or more groups require that the data be arranged either as stacked or unstackeddata. The twosample hypothesis tests for unsummarized data discussed in this E,xcel Companion require that your data be unstacked. If you need to change stacked data into its unstacked equivalent, you can sort the data by sample or group and then cut and paste the second sample's data to a new column. Likewise, to stack unstackeddata, you can copy the data from the secondsampledirectly below the first sample and then add a column that identifies the group. PHStat2 can automate these tasks by using either the Unstack Data or Stack Data procedures found in the PHStat2Data Preparation submenu.

E10.1 USINGTHE Z TESTFORTHE DIFFERENCE BETWEENTWO MEANS(UNSUMMARTZED DATA) For unsummarizeddata,you conduct a Ztest for the difference between two means by selecting the ToolPak z-Test: Two Sample for Means procedure. Open to the worksheetthat containsthe unsummarized data for the two samples.SelectTools ) Data Analysis, select z-Test: Two Sample for Means from the Data Analysis list, and click OK. In the procedure'sdialog box (shown below), enter the cell range of one sample as the Variable 1 Range and the cell rangeof the other sampleas the Variable 2 Range. Enter the Hypothesized Mean Difference, the population variance of the first sample as the Variable I Variance (known), and the population variance of the second sample as the Variable 2 Variance (known). SelectLabels and click OK. Resultsappearon a new worksheet.

E1O.2USINGTHEZ TESTFORTHE DIFFERENCE BETWEEN TWO M EANS(SUMMARTZED DATA) For sunrmarized data, you conduct a Z test for the diffuence between two means by either selecting the PHStat2 Z Test for Differences in Two Means procedureor by making entries in the !@fll@[! workbook.

Usinq Pl'{Stat2 Z Test for the Diffirence Betu/een Two Means Select PHStat ) Two-Sample Tests ) Z Testfor Differences in Two Means. In the Z Test for the Differences in Two Means dialog box (shown below),enter the Hypothesized Difference and, if necessary,change the Level of Significance. Enter the sample size, sample mean, and population standarddeviation for the population 1 sample and the population 2 sample.Click one of thete$ options.entera title as the Title, and click OK.

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Using Z Two Means.xls Open to theZTwo Meansworksheet of theEIEEEEEE (seeFigureE10.2)usesthe workbook. Thisworksheet NORMSINV?
from the Data Analysis list and click OK. In the procedure's dialog box (shown below), enterthe cell rangeof one sample as the Variable I Range and the cell rangeof the other sample as the Variable 2 Range.Enterthe HypothesizedMean Difference,click Labels,and click OK. Resultsappearon a new worksheet. Figure 10.3 on page 373 shows the results for the Table 10.I BLK cola salesdata.

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E10.3USINGTHE POOLED-VARIANCE t TEST(UNSUMMARIZEDDATA) For unsummarized data, you conduct a pooled-variance I testby selectingthe ToolPak t-Test:Two-SampleAssuming EqualVariancesprocedure. Open to the worksheetthat containsthe unsummarized datafor the two samples.SelectTools t Data Analysis, selectt-Test: Two-Sample Assuming Equal Variances

E10.4 USINGTHE POOLED.VARIANCE DATA) t TEST(SUMMARTZED F o r s u m r n a r i z e dd a t a , y o u c o n d u c t a p o o l e d - v a r i a n c e I test by either using the PHStat2 t Test for Differences in Two Means procedureor by making entries in the

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workbook.

416

EXCELCoMPANIoNto Chaoter lo

Usinq PHStat2 t Test for the Diffe-rencesin Two Means Select PHStat ) Two-Sample Tests ) t Test for Differences in Two Means. In the t Test for Differences in Two Means dialog box (shown at right), enter values for the Hypothesized Difference and the Level of Significance and then enter the Sample Size, Sample Mean, and Sample Standard Deviation for the population I sample and the population2 sample.Click one of the test options, entera title as the Title, and click OK. To include a confidence interval estimate of the difference between the two means (similar to one found in the PVt worksheet d e s c r i b e db e l o w ) , c l i c k C o n f i d e n c e I n t e r v a l E s t i m a t e before you click OK.

Using Pooled-VarianceT.xls Opento the PVt worksheet of thefi!![!!f,lftfifiEE w o r k b o o k . T h i s w o r k s h e e t( s e e F i g u r e E 1 0 . 3 ) u s e s t h e TINY(l-confidence level, degrees of freedom) function to determine the lower and upper critical values. To compute the p-values, this worksheet uses the TDIST(ABS(I), degreesoffreedom,fails) function, in which ABS(/) is the absolutevalue of the I test statistic, and tsils is either 1, for a one-tailtest, or 2. for a two-tail test.The worksheetuses the IF function in cells 83l and 836 to determinewhich one of two values computed in a calculations area (not shown in Fisure E 10.3)to use.

FIGURE 10.3 PVtworksheet

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810.7: Using the ZTest for the DifferenceBetweenTwo Proportions(SummarizedData)

The worksheetcontainsentries basedon the Table I 0.1 BLK cofasalesdata on page 312.To adaptthis worksheet to otherproblems, change, as necessary,the hypothesized level of significance,and sample statisticsof difference, t h et w o s a m p l e si n c e l l s 8 4 , 8 5 , B l : 8 9 , a n d B l l : B 1 3 . I f youdo not want to include a confidence interval estimate in your worksheet(see Figure E 10.4), select and delete the c e l l r a n gD e3:E16.

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E 1 0 . 6U S I N GT H EP A I R E D t T E S TF O R BETWEEN THEDIFFERENCE TWO MEANS(UNSUMMARIZED DATA) For unsummarized data,you conduct a paired l test for the difference between two means by selecting the ToolPak t-Test: PairedTwo Sample for Mean procedure. Open to the worksheetthat contains the unsummarized data for the two samples.SelectTools ) Data Analysis, select t-Test: Paired Two Sample for Means from the Data Analysis list and click OK. In the procedure'sdialog box (shown below), enter the cell range of one sample as the Variable I Range and the cell range of the other sample as the Variable 2 Range. Enter the Hypothesized Mean Difference, click Labels, and click OK. Results appearon a new worksheet.Figure 10.8on page385 shows the resultsfor the Table 10.4car milease data.

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intervalestimatearea FIGUREE10.4Confidence

E10.5USINGTHE SEPARATE. VARIANCEt TESTFORTHE BETWEENTWO DIFFERENCE (UNSUMMARIZED DATA) MEANS Forunsummarizeddata,you conduct a separate-variance/ testfor the difference between two means by selecting the ToolPakt-Test:Two-SampleAssuming Unequal Variances procedure. Opento the worksheetthat containsthe unsummarized datafor the two samples.SelectTools ) Data Analysis, t-Test:Two-SampleAssuming Unequal Variances select fromthe Data Analysis list, and click OK. In the procedialog box (shown below), enter the cell range of one dure's sampleas the Variable I Range and the cell range of the other sample as the Variable 2 Range. Enter the HypothesizedMean Difference, click Labels, and click 0K. Figure 10.6 on page377 shows the results of applying thisprocedureto the Table 10.1 BLK cola salesdata.

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E1O.7 USINGTHE Z TESTFOR THE DIFFERENCEBETWEENTWO (SUM MARIZED PROPORTTONS DATA) Forsunrmarized data,you conducta Z testfor thedifference betweentwo proportionsby eitherselectingthe PHStat2 ZTest for Differencesin Two Proportionsprocedureor by workbook. makingentriesin theE@E@[[E

Using PHStat2 Z Test for Differences in Two Proportions Select PHStat t Two-Sample Tests ) Z Test for Differences in Two Proportions. In the Z Test for Differences in Two Proportions dialog box (shown on page 418), enter values for the Hypothesized Difference and the and Level of Significance.Enter the Number of Successes the Sample Size for the population 1 sample and the population 2 sample.Click one of the test options,entera title as the Title. and click OK. To include a confidenceinterval estimate of the difference between the two proportions

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EXCELcoMPANIoN to chanter lo

(similar to one shown in Figure El0.6 below), click ConfidenceInterval Estimatebeforeyou click OK.

Using Z Two Proportions.xls You open and use eitherthe ZTP_TT or the ZTP

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E I 0.9: Using the F Testfor the DifferenceBetweenTwo Variances(SummarizedData)

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E10.8USINGTHEF TESTFORTHE DIFFERENCE BETWEEN TWO (UNSUMMARTZED VARTANCES DATA) Forunsummarizeddata,you conduct an -Ftest for the differencebetween two variances by selecting the ToolPak F TestTwo-Sample for Variances procedure. Opento the worksheetthat containsthe unsummarized datafor the two samples. Select Tools ) Data Analysis, selectF Test Two-Sample for Variances from the Data Analysislist, and click OK. In the procedure'sdialog box (shownbelow), enter the cell range of one sample as the VariableI Range and the cell range of the other sample as theVariable 2 Range. Click Labels and click OK. Results appear on a new worksheet.Figure 10.l4 on page 400 shows results for the Table l0.l BLK cola salesdata on page372.

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Select PHStat ) Two-SampleTests ) F Test for Differences in Two Variances. In the F Test for Differencesin Two Variances dialogbox (shownbelow), enterthe Level of Significanceandthenenterthe Sample Sizeand the SampleStandard Deviationfor the population I sampleand population2 sample.Click one of the testoptions,entera title astheTitle, andclick OK.

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E10.9USINGTHEF TESTFORTHE DIFFERENCE BETWEEN TWO (SUMMARTZED VAR|ANCES DATA) Forsummarizeddata, you conduct an F test for the differencebetweentwo variancesby either selectingthe PHStat2 F Testfor Differences in Two Variances procedure or by

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Using F Two Variances.xls Open to the F Two Variances worksheet of the This worksheetconductsan tr@Eworkbook. F test for the difference between two variances for the S e c t i o n 1 0 . 4 B L K c o l a s a l e se x a m p l e .T h e w o r k s h e e t (see Figure E10.7) uses the function FINV(rpper-tailed p-value, numerstor degrees of freedom, denominator degrees of freedom), in which upper-tuiled p-value is the probability that tr will be greater than the value, to compute the upper and lower critical values and uses the function FDIST(F-/est statistic, nunterstor degrees of freedom, denominutor degrees of freedom) to compute t h ep - v a l u e s .

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To adapt this worksheet to other problems, change, if necessary,the level of significance and the samplestatistics for the two population samples in the tinted cells84, 86, B7 , 89, and B 10. If you want the worksheetto show only a single test, first make a copy of that worksheet(see the Excel Companion to Chapter l). For a two-tailtestonly worksheet,selectand deleterows 23 through31.Fora lower-tail-test-onlyworksheet,select and deleterows28 through 3l and then select and delete cell range A17:821, For an upper-tail-test-onlyworksheet, select and delete rows 23 through 2T and then select and deletecell range Al7:821.

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