chap 10

Two-sample Tests

USING STATISTICS @, BLK Foods

10.1 COMPARING THE MEANS OF TWOINDEPENDENT POPULATIONSZTest for the Difference Between Two MeansPooled-Variance I Test for the Difference Between

Two MeansConfidence Interval Estimate for the Difference

Between Two MeansSeparate-Variance t Test for the Difference

Between Two Means

10.2 COMPARING THE MEANS OF TWORELATED POPULATIONSPaired t TestConfidence Interval Estimate for the Mean

Difference

10.3 COMPARING TWO POPULATIONPROPORTIONSZTest for the Difference Between Two ProportionsConfidence Interval Estimate for the Difference

Between Two Proportions

10.4 F TEST FOR THE DIFFERENCEBETWEEN TWO VARIANCESFindins Lower-Tail Critical Values

EXCEL COMPANION TO CHAPTER 10E10.1 Using the ZTest for the Difference

Between Two Means (Unsummari zed Data)E10.2 Using the ZTest for the Difference

Between Two Means (Summarized Data)E10.3 Using the Pooled-Variance /Test

(UnsummarizedData)E10.4 Using the Pooled-Variance /Test

(Summarized Data)810.5 Using the Separate-Variance I Test

for the Difference Between Two Means(UnsummarizedData)

E 1 0.6 Using the Paired / Test for the DifferenceBetween TWo Means (Unsummarized Data)

E10.7 Using the ZTest for the DifferenceBetween Two Proportions (Summarized Data)

E10.8 Using the FTest for the DifferenceBetween Two Variances (Unsummarized Data)

E10.9 Using the FTest for the DifferenceBetween Two Variances (Summari zed D ata)

InI

t

I

I

this chapteq you learn how to use hypothesis testing for comparing the difference between:

The means of two independent populations

The means of two related populations

Two proportions

The variances oftwo independent populations

370 CHAPTERTEN Two-Sample Tests

Using Statistics @ BLK Foods

Does the type of display used in a supermarket affect the sales ofthe regional sales manager for BLK Foods, you want to compare the salesume of BLK cola when the product is placed in its the normal shelfto the sales volume when the product is featured in a special end-aisle diTo test the effectiveness of the end-aisle displays, you select 20 storesthe BLK supermarket chain that all experience similar storewide salesumes. You then randomly assign I 0 of the 20 stores to group I and l0 to2.The managers of the l0 stores in group I place the BLK cola in theshelflocation, alongside the other cola products. The l0 stores in groupthe special end-aisle promotional display. At the end of one week, theBLK cola are recorded. How can you determine whether sales of BLKusing the end-aisle displays are the same as those when the cola isthe normal shelf location? How can you decide if the variability in BLK

sales from store to store is the same for the two types of displays? How could you useanswers to these questions to improve sales of BLK colas?

Lfypothesis testing provides a confirmatory approach to data analysis. In Chapter 9,I llearned a variety of commonly used hypothesis-testing procedures that relate to a sisample of data selected from a single population. In this chapter, you will learn how tohypothesis testing to procedures that compare statistics from two samples of data taken

two populations. One such extension would be asking, 'Are the mean weekly sales of BLKwhen using an end-aisle display equal to the mean weekly sales of BLK cola when placed innormal shelf location?"

Suppose that you take a random sample of n, from one population and a random samplefrom a second population. The data collected in each sample are from a numerical varithe first population, the mean is represented by the symbol p,, and the standard devirepresented by the symbol or. In the second population, the mean is represented by thep2, and the standard deviation is represented by the symbol or.

The test statrstic used to determine the difference between the popu\ation means ison the difference between the sample means (X t - X z).If you assume that the samples aredomly and independently selected from populations that are normally distributed, thisfollows the standardized normal distribution. If the populations are not normally dithe Ztest is still appropriate if the sample sizes are large enough (typically n, andnr2the Central LimitTheorem in Section 7.4). Equation (10.1) defines the Ztest for thebetween two means.

ZTEST FOR THE DIFFERENCE BETWEEN TWO MEANS

1O.1 COMPARING THE MEANS OF TWO INDEPENDENTPOPULATIONSZ Test for the Difference Between Two Means

_ (X r -X ) - (p r -pz )- -la? o4

r / - T -

\ n t n 2

(10.1)

I

)I)

10.l : Comparing the Means of Two Indepcndent populat ions 3l I

where

X, : mean of the sample taken from population I

F1 : meon of population I

of : variance of population I

nr: size of the sample taken from population 1

X2: meanof the sample taken from population 2

Ir2: mean of population 2

o22: variance of population 2

n2: size of the sample taken from population 2

The test statistic Z follows a standardized normal distribution.

Pooled-Variance t Test for the Difference Between Two Means

In most cases, the variances of the two populations are not known. The only information you usu-ally have are the sample means and the sample variances. If you assume that the samples are ran-domly and independently selected from populations that are normally distributed and that the pop-ulation variances are equal (that is, oi - oi), you can use a pooled-variance t test to deterrninewhether there is a significant difference befween the means of the two populations. If the popula-tions are not normally distributed, the pooled-variance / test is still appropriate if the sample sizesare large enough (typically n, and nr> 30; see the Central Limit Theorem in Section 7.4).

To test the null hypothesis of no difference in the means of two independent populations:

H o i V t - F 2 o r p , - F z : 0

against the alternative that the rleans are not the same:

H r : p t , + F 2 o r p , - V 7 + 0

you use the pooled-variance l-test statistic shown in Equation ( 10.2). The pooled-variance / testgets its^name from the {act that the test statistic pools or combines the two sample variances Sfand S2r to compute Srj, the best estimate of the variance common to both populations underthe assumption that the two population variances are equal.l

POOLED-VARIANCE t TEST FOR THE DIFFERENCE BETWEEN TWO MEANS

lWhen the two sample sizes

are equal (that is, n, = n2),

the eguation for the pooled

variance can be simpltfied

- 2 S? +s4L a " p

2

,_ (Xr -X) - ( t r r - t r z ). - -

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

(10.2)

where

.s2

q 2* p

N1

s,2. - l

n l

- (n r - \S?+@z-DSj( r r - l ) + ( n 2 - l )

: pooled variance

: mean of the sample taken from population I

: variance of the sample taken from population I

: size of the sample taken from population I

X 2: mean of the sample taken from population 2

S22 : variance of the sample taken from population 2

nr: size of the sample taken from population 2

follows a 1 distribution with nt * ttt - 2 degrees of freedom.The test statistic /

372 CHAPTERTEN Two-samoleTests

FIGURE 10.1Regions of rejectionand nonrejection forthe oooled-variancet test for the differencebetween the means(two-tail test)

The pooled-variance /-test statistic follows a I distribution with n , * nt - 2 degrees ofFor a given level of significance, crt,, in a twotail test, you reject the null hypothesis if the/ test statistic is greater than the upper-tail critical value from the / distribution or ifthetest statistic is less than the lower-tail critical value from the / distribution. Fizure 10. Iregions of rejection. In a one-tail test in which the rejection region is in the lower tail, younull hypothesis if the computed test statistic is less than the lower-tail critical value from the tbution. [n a one-tail test in which the rejection region is in the upper tail. you reject the nullesis if the computed test statistic is greater than the upper-tail critical value from the /

To demonstrate the use of the pooled-variance I test, return to the Using Statisticson page 370. You want to determine whether the mean weekly sales of BLK cola are thewhen using a normal shelf location and when using an end-aisle display. There are twotions of interest. The first population is the set of all possible weekly sales of BLK colaif allBLK supermarkets used the normal shelf location. The second population is the set of allsible weekly sales of BLK cola if all the BLK supermarkets used the end-aisle displays.first sample contains the weekly sales of BLK cola from the 10 stores selected to use themal shelf location, and the second sample contains the weekly sales of BLK cola from thestores selected to use the end-aisle display. Table 10.1 contains the cola sales (in numbercases) for the two samples (see the !![ffr file).

Display Location

Normal

T A B L E 1 0 . 1Comparing BLK ColaWeekly Sales fromTwo Different DisplayLocat ions ( in Numberof Cases)

2240

5284

3059

5283

7 l66

5477

7690

6256

3464

The null and alternative hypotheses are

H o : h : p 2 o r p , - V z : 0

Hr: p, , + F2 or p, - V2+ 0

Assuming that the samples are from underlying normal populations having equalances, you can use the pooled-variance I test. The r test statistic follows a / distributionl0 + l0 - 2 : 18 desrees of freedom. Usine the cr : 0.05 level of sisnificance. vou dividerejection region into the two tails for this two-tail test (that is, two equal parts of 0.025Table E.3 shows that the critical values for this two-tail test are +2.1009 and -2.1009.

shown in Figure 10.2, the decision rule is

Reject Hoif t > rrs +2.1009or i f t < - l ra : -2 .1009;

otherwise, do not reject Hn.

10.2test ofis for thebetween the

at the 0.05 levelficance with

of freedom

10.3:ft Excel ttestfor the twolocations

Secrion E10.3 to create

10.1: Comparing the Means of Two Independent Populations

Critical \r-f;,Value

lrom Figue l0 3, the computa0 t statistla for this test is -3 .0446, and the p-value is 0.0070.

t -2.100erno f I:tion I

CriticalValue

Verlanccllypolhcdred tecn DlfierencoCTt StatPft<'{ orrrollCrldcal one{all

F{f<-{ tro{6ll

5ttJ 723flr,6rt8 r5t3333

t0 t025frxt66

0t8

30{{60J0351.73t10"0d?0?.rmt

Using Equation (10.2) on page 371 and the descriptive statistics provided in Figure 10.3,

- ( x t -x ) - ( tq -uz )-l ^ - , ( r l )t J : l - + - l

1 " \ ' t nz )

where

Therefore,

s3=(n r - \S? +@z-DSl( n r - 1 ) + ( n 2 - l )

9(3 s0.67 7 8) + 9(l 57.3333)= 254.00569 + 9

= # = 4.0446{50.801

You reject the null hypothesis because t : -3 .0446 < /r 8 -2.1009. The p-value (as computed

from Microsoft Excel) is 0.0070. In other words, the probability that t> 3.0446 or t < -3.0446is equal to 0.0070. Thisp-value indicates that if the population means are equal, the probability

Region of I Region of I Region ofReiection I Nonrejection lRejection

( s0 .3 -72 .0 \ -0 .0

zs+.ooso[! + l)( r0 r0 )

of observing a difference this large or larger in the two sample means is only 0.0070. Because

374 CHAPTER TEN Two-Samole Tests

the p-value is less than a : 0.05, there is sufficient evidence to reject the null hypothesis.'can conclude that the mean sales are different for the normal shelf location and the end-alocation. Based on these results, the sales are lower for the normal location (than forend-aisle location).

Example 10. I provides another application of the pooled-variance t-test.

EXAMPLE 10 .1 TESTING FOR THE DIFFERENCE IN THE MEAN DELTVERY TIMES

A local pizzarestaurant and a local branch ofa national chain are located across the streetfi'a college campus. The local pizza restaurant advertises that it delivers to the dormitories falthan the national chain. In order to determine whether this advertisement is valid, you and sofriends have decided to order 10 pizzas from the localpizza restaurant and l0 pizzas fromtnational chain, all at different times. The delivery times, in minutes (see the [[!!@l!fiIare shown in Table 10.2.

T A B L E 1 0 . 2Del ivery Times for LocalPizza Restaurant andNational Pizza Chain

Local Chain Local Chain

At the 0.05 level of significance, is there evidence that the mean delivery time for the locptzza restaurant is less than the mean delivery time for the national pizza chain?

SOLUTION Because you want to know whether the mean is lower for the local pizza restarant than for the national pizza chain, you have a one-tail test with the following null and altenative hypotheses:

1 6 . 811.'l1 5 . 616.7l 1 . 5

22.0t5.21 8 . 715.620.8

1 8 . 114.121.81 3 . 920.8

1 9 . 517.01 9 . 51 6 . 524.0

Ho: vt

Hi vr

) p, (The mean delivery time for the local pizza restaurant is equal toor greater than the mean delivery time for the national pizza chain.)

< F2 (The mean delivery time for the local pizza restaurant is less thanthe mean delivery time for the national pizza chain.)

FIGURE 10.4Microsoft Excel resultsof the pooled t testfor the pizza del iveryt ime data

See Section E10.3 to createthis.

Figure 10.4 displays Microsoft Excel results of the pooled I test for these data.

Local Cilein4 :Mean5 ,Variance6 Obseryallons7 ,Pooled VarlanceI Hypotheslzed Mean Olfierences jd f10 ,t Siat

l1 ;Pff.-0 ono-tail1_2. t Crltlcal one-tall13 rP[f<-$ two-tall

16.7 18.S9.5822 8.2151

10 108.8987

018

-1.63410.05981.73410.1196?.100914 rt Crltical two-tail

10. I : Comparing the Means of Two Independent Populations 37 5

Using Equat ion (10.2) on page 371,

(X t -X ) - ( t r r - pz )

where

( n r - l )S i+ ( r z - l )Sz2( r r - l ) + (n2 - l )

9 (9 .s822) + e (8 .2 l s l )= 8 .89879 + 9

Therefore.

( t 6 .7 - r 8 .88 ) -0 .0 = -1.6341

You do not reject the nul l hypothesis because t : -1.6341> /r8 -1.7341. Thep-value (ascomputed from Microsoft Excel) is 0.0598. This p-value indicates that the probability thatt < -1 .6341 is equal to 0.0598. In other words, if the population means are equal, the prob-ability that the sample mean delivery time for the local pizza restaurant is at least 2.18 min-utes faster than the national chain is 0.0598. Because the p-value is greater than cr : 0.05,there is insufficient evidence to reject the null hypothesis. Based on these results, there rsinsufficient evidence for the localpizza restaurant to make the advertising claim that it hasa faster deliverv time.

In testing for the difference between the means, you assume that the populations are nor-mally distribute4 with equal variances. For situations in which the two populations have equalvariances, the pooled-variance / test is robust (or not sensitive) to moderate departures fromthe assumption of normality, provided that the sample sizes are large. ln such situations, youcan use the pooled-variance I test without serious effects on its power. However, if you cannotassume that the data in each group are from normally distributed populations, you have twochoices. You can use a nonparametric procedure, such as the Wilcoxon rank sum test (coveredin Section 12.5), that does not depend on the assumption of normality for the two populations,or you can use a normalizing transformation (see reference 5) on each of the outcomes and thenuse the pooled-variance I test.

To check the assumption of normality in each of the two groups, observe the box-and-whisker plot of the sales for the two display locations in Figure 10.5. There appears to be onlymoderate departure from normality, so the assumption of normality needed for the / test is notseriously violated.

Frr. rl! " l n r nz )

. fromfastersomem thefile),

local

stau-rlter-

e 2 _

- 2 . 1 8

ffisqszf l + I I(10 r0 )

37 6 CHAPTER TEN Two-sample Tests

FIGURE 10.5Microsoft Excel box-and-whisker plot for the salesfor two aisle locations


Box-and-Whisker Plot for Sales Location

Confidence Interval Estimate of the Difference Between Two

Instead of, or in addition to, testing for the difference in the means of two independent

tions, you can use Equation (10.3) to develop a confidence interval estimate of the differencei

the means.

CONFIDENCE INTERVAL ESTIMATE OF THE DIFFERENCEBETWEEN TWO MEANS

tX, - x r l+,,,*,,-r./s;[, f . f ' lz - .1 r ' I r , nz )(10.3)

or

l A t - A 2 l - t r t + r t - 2 trr-r)\ * o \ , , n z )

where /,, +," -2 is the critical value of the I distribution with n, * nz - 2 degrees of freedom 1

for an area of alZ in the upper tail.

Using 95% confidence, the sample statistics reported in Figure 10.3 on page 373 andEquation ( 10.3),

nr = 50.3, nr = 10, Xz = 72. n2 = 10. S; = 254.0056, and r,, = 2.1009:

(50.3 - 72) ! (2 . t00e)

-2t .7 X ( 2. r 00ex 7. | 275)-21 .7 t 14 .97

-36.67 S pr - S t2 { -6 .73

2s4.oos6(! * Il( 10 t 0 )

10.1: Comparing the Means of Two Independent Populations 377

Therefore, you are 95% confident that the difference in mean sales between the normal shelflocation and the end-aisle location is between -36.67 cases of cola and -6.73 cases of cola. Inother words, the end-aisle location sells, on average, 6.73 to 36.67 cases more than the normalaisle location. From a hypothesis testing perspective, because the interval does not includezero, you reject the null hypothesis of no difference between the means of the rwo populations.

Separate-Variance t Test for the Difference Between Two MeansIn testing for the difference between the means of two independent populations when the pop-ulation variances are assumed to be equal, the sample variances are pooled together into a com-mon estimat", 53. However, if you cannot make this assumption, then the pooled-varianceI test is inappropriate. In this case, it is more appropriate to use the separate-variance I testdeveloped by Satterthwaite (see reference 4). In the Satterthwaite approximation procedure,you include the two separate sample variances in the computation of the /-test statistic-hence,the name separate-variance / test. The computations for the separate-variance / test are compli-cated but can be carried out by Microsoft Excel. Figure 10.6 presents the output from the sepa-rate-variance I test from Microsoft Excel for the cola data.

'e* Two-Scmplo Anmlng Vcrhnccr

10.6Excel results

separate-variancefor the display

data

E10.5 to create

Oiccrntlomllypottrdzed lcan DlfirrencrdfS t t

P(f<-! onc{allCrltlcal onc-tcll

P(f<-Q mo{allCrhlcal rwo.rall

cance cr:0.01?

350.6ilt0, tf/3EB10 100

153-0tr6OIIF9r:i{reio"mn,2.1$a

In Figure 10.6, the test statistic is t: -3.0446 and thep-value is 0.0077 < 0.05. Thus,the results for the separate-variance t test are almost exactly the same as those of the pooled-variance I test. The assumption of equality of population variances had no real effect on theresults. Sometimes, however, the results from the pooled- and separate-variance t tests conflictbecause the assumption of equal variances is violated. Therefore, it is important that you eval-uate the assumptions and use those results as a guide in appropriately selecting a test proce-dure. In Section 10.4. the F test is used to determine whether there is evidence of a differencein the two population variances. The results of that test can help you determine which of theI tests-pooled-variance or separate-variance-is more appropriate.

10.2 What is your decision in Problem l0.l ifyou are testing Ho: t\: F2 against the two-tailalternative Hl Fr + ;j, using the level of signifi-

the Basics

10.1 Given a sample of nr:40 from a popula-tion with known standard deviation o, : 20 andan independent sample of nr:50 from another

ion with known standard deviation o, : 10" whatrc value of the Z test statistic for testing Ho: trr : p2if=72 and Xr: ee' t

10.3 What is thep-value in Problem l0.l if youare testing Ho: l\: lr2 against the two-tail alter-native 11,: tt1 * 1tr?

378 CHAPTERTEN Two-SamoleTests

10.4 Assume that you have a sample of r, : 8,with the sample mean X1: 42 and a sample stan-dard deviation of ,S, : 4. and vou have an inde-

pendent sample oflr: l5 from another population with asample mean of X 2 : 34 and the sample standard devia-tion S, : 5.a. What is the value of the pooled-variance /-test statistic

for testing Ho: 1tr: 1t"r?b. In finding the critical value ofthe test statistic /, how

many degrees of freedom are there?c. Using the level of significance cx,:0.01, what is the crit-

ical value for a one-tail test of the hypothesis Ho: trt, t trt,against the alternative Hr: 1t r> 1tr?

d. What is your statistical decision?

10.5 What assumptions about the two populations arenecessary in Problem 10.4?

10.6 Referring to Problem 10.4, construct a 95o/o confi-dence interval estimate of the population differencebetween F1 and pr.

Applying the Concepts

10.7 The operations manager at a light bulb fac-tory wants to determine whether there is any dif-ference in the mean life expectancy of bulbs

manufactured on two different types of machines. The pop-ulation standard deviation of machine I is 110 hours and ofmachine II is 125 hours. A random sample of 25 light bulbsfrom machine I indicates a sample mean of 375 hours, anda similar sample of 25 from machine II indicates a samplemean of 362 hours.a. Using the 0.05 level of significance, is there any evi-

dence of a difference in the mean life of bulbs producedby the two types of machines?

b. Compute thep-value in (a) and interpret its meaning.

10.8 The purchasing director for an industrialparts factory is investigating the possibility ofpurchasing a new type of milling machine. She

determines that the new machine will be bought if there isevidence that the parts produced have a higher mean break-ing strength than those from the old machine. The popula-tion standard deviation ofthe breaking strength for the oldmachine is l0 kilograms and for the new machine is 9 kilo-grams. A sample of 100 parts taken from the old machineindicates a sample mean of 65 kilograms, and a similarsample of 100 from the new machine indicates a samplemean of 72 kilograms.a. Using the 0.01 level of significance, is there evidence that

the purchasing director should buy the new machine?b. Compute thep-value in (a) and interpret its meaning.

10.9 Millions of dollars are spent each year on diet foods.Trends such as the low-fat diet or the low-carb Atkins diet

have led to a host of new products. A study by Dr.Stern of the Philadelphia Veterans Administrationcompared weight loss between obese patients on adiet and obese patients on a low-carb diet (ExtractedR. Bazell. "Studv Casts Doubt on Advantases ofDiet," msnbc.com, May 17,2004). Let p,mean number of pounds obese patients on a low-fatlose in six months and p, represent the meanpounds obese patients on a low-carb diet lose imonths.a. State the null and alternative hypotheses ifyou

test whether the mean weisht loss between the twois equal.

b. In the context of this study, what is the meaningType I error?

c. In the context of this study, what is the meaningType II error?

d. Suppose that a sample of 100 obese patients on adiet lost a mean of 7.6 pounds in six months, withadard deviation of 3.2 pounds, while a sample ofobese patients on a low-carb diet lost a mean ofpounds in six months, with a standard deviation ofpounds. Assuming that the population variancesequal and using a 0.05 level of significance, is theredence of a difference in the mean weight loss ofpatients between the low-fat and low-carb diets?

10.10 When do children in the United States developerences for brand-name products? In a study reported inJournal of Consumer Psychology (Extracted from G.Achenreiner and D. R. John, "The Meaning of Brandto Children: A Developmental Investigation," JournalConsumer Psychology, 2003, 13(3), pp. 205-219),keters showed children identical pictures of athleticOne picture was labeled Nike, and one was labeled K-The children were asked to evaluate the shoes based onappearance, qualiry price, prestige, favorableness, anderence for owning. A score from 2 (highest producttion possible)to 1(lowest product evaluation possible)recorded for each child. The following table reportsresults of the study:

Age by SampleBrand Size

Age 8Nike 27K-Mart 22

Age 12Nike 39K-Mart 4l

Age 15Nike 35K-Mart 33

Sample Sample StandardMean Deviation

0.890.86

0.880.09

0.41-0.29

0.98r.07

1 . 0 11.08

0.810.92

' i l

a. Conduct a pooled-variance / test for the differencebetween two means for each of the three age groups.Use a level of significance of 0.05.

b. What assumptions are needed to conduct the tests in (a)?c. Write a brief summary of your findings.

10.11 According to a survey conducted inOctober 2001, consumers were trying to reducetheir credit card debt (Extracted from M. Price,

"Credit Debts Get Cut Down to Size," Newsday, November25,2001, p. F3). Based on a sample of 1,000 consumersin October 2001 and in October 2000, the mean creditcard debt was $2,41I in October 2001 as compared to$2,814 in October 2000. Suppose that the standard devia-tion was $847 .43 in October 2001 and $976.93 in October2000.a. Assuming that the population variances from both years

are equal, is there evidence that the mean credit carddebt was lower in October 2001 than in October 2000?(Use the cx : 0.05 level of significance.)

b. Determine thep-value in (a) and interpret its meaning.c. Assuming that the population variances from both years

are equal, construct and interpret a 95o/o confidenceinterval estimate of the difference between the popula-tion means in October 2001 and October 2000.

10,12 The Computer Anxiety Rating Scale(CARS) measures an individual's level of com-puter anxiety, on a scale from 20 (no anxiety) to

100 (highest level of anxiety). Researchers at Miamity administered CARS to l'12 business students.

of the objectives of the study was to determrnethere is a difference in the level of computer anxr-

experienced by female and male business students.found the following:

Males Females

l0.l: Comparing the Means of Two Independent Populations 319

Taper Locks Locking Pins

x 1.262,s 0.297n 2 0

0.5610.307

20

' ExtractedfromT Broome and D. Havelka, "Determinants

Computer Anxiety in Business Students," The Review of BusinessSystems, Spring 2002, 6(2), pp. 9-16

At the 0.05 level ofsignificance, is there evidence ofadifference in the mean computer anxiety experienced byfemale and male business students?Determine the p-value and interpret its meaning.What assumptions do you have to make about the twopopulations in order to justify the use ofthe r test?

13 A company making plastic optical components wasing inconsistencies in an optical measurement called

Two different types of pins used in the mold producedfollowing results:

Source: Extracted from J. Duncan, "Ghosts in Your Process? Who YaGoing to Call? " Quality Progress, May 2005, pp. 52-57.

a. Assuming that the population variances are equal andthe populations are normally distributed, at the 0.05level ofsignificance, is there evidence ofa difference inthe means between taper locks and locking pins?

b. Repeat (a), assuming that the population variances arenot equal.

c. Compare the results of (a) and (b).

10.14 A bank with a branch located in a commercial dis-trict of a city has developed an improved process for serv-ing customers during the noon-to-l p.m. lunch period. Thewaiting time (operationally defined as the time elapsedfrom when the customer enters the line until he or shereaches the teller window) of all customers during this houris recorded over a period of one week. A random sampleof 15 customers is 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

Suppose that another branch, located in a residential area,is also concerned with the noon-to-l p.m. lunch period. Arandom sample of l5 customers is selected (and stored inthe file EE&[D, and the results are 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. Assuming that the population variances from both banksare equal, is there evidence of a difference in the meanwaiting time between the two branches? (Use a: 0.05.)

b. Determine thep-value in (a) and interpret its meaning.c. What other assumption is necessary in (a)?d. Assuming that the population variances from both

branches are equal, construct and interpret a95o/o confi-dence interval estimate of the difference between thepopulation means in the two branches.

10.15 Repeat Problem 10.14 (a), assuming that the popu-lation variances in the two branches are not equal.Compare the results with those of Problem l0.la (a).

10.16 A problem with a telephone line that prevents acustomer from receiving or making calls is disconcerting toboth the customer and the telephone company. The data inthe file EfiftllE represent samples of 20 problemsreported to two different offices of a telephone companyand the time to clear these problems (in minutes) from thecustomers' lines:

40.2613.35

100

36.859.42

72

380 CHAPTERTEN Two-SamoleTests

Central Office I Time to Clear Problems (Minutes)

1 .48 1 .15 0 .78 2 .85 0 .52 1 .60 4 .15 3 .97 1 .48 3 .10

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. Assuming that the population variances from bothoffices are equal, is there evidence of a difference inthe mean waiting times between the two offices? (Useo( : 0.05.)

b. Determine thep-value in (a) and interpret its meaning.c. What other assumption is necessary in (a)?d. Assuming that the population variances from both

offices are equal, construct and interpret a 95Yo confi-dence interval estimate of the difference between thepopulation means in the two offices.

10.17 Repeat Problem 10.16 (a), assuming that the popu-lation variances in the two offices are not equal. Comparethe results with those of Problem 10.16 (a).

10.18 In intaglio printing, a design or figure is carvedbeneath the surface ofhard metal or stone. Suppose that anexperiment is designed to compare differences in meansurface hardness of steel plates used in intaglio printing(measured in indentation numbers), based on two differentsurface conditions-untreated and treated by lightly pol-ishing with emery paper. In the experiment, 40 steel platesare randomly assigned-20 that are untreated, and 20 thatare treated. The data are shown here and stored in the file

@EE'Untreated Treated

difference between the population means in theconditions.

10.19 Repeat Problem 10.18 (a), assuming that thetion variances from untreated and treated steel olates areequal. Compare the results with those of Problem 10.18

'lO.2O The director of trainins for an electronicment manufacturer is interested in determininsdifferent training methods have an effect on theity of assembly-line employees. She randomly assigns 42recently hired employees into two groups of 21. The fintgroup receives a computer-assisted, individual-based train-ing program, and the other receives a team-based trainingprogram. Upon completion of the training, the employeesare evaluated on the time (in seconds) it takes to assemble apart. The results are in the data filel[@f!!.a. Assuming that the variances in the populations of hain-

ing methods are equal, is there evidence of a differencebetween the mean assembly times (in seconds) ofemployees trained in a computer-assisted individual-based program and those trained in a team-based pro-gram? (Use a 0.05level of significance.)

b. What other assumption is necessary in (a)?c. Repeat (a), assuming that the population variances are

not equal.d. Compare the results of (a) and (c).e. Assuming equal variances, construct and interpret a 95%

confidence interval estimate of the difference betweenthe population means of the two training methods.

10.21 Nondestructive evaluation is a method that is usedto describe the properties of components or materialswithout causing any permanent physical change to theunits. It includes the determination of properties of materi-als and the classification of flaws by size, shape, type, andlocation. This method is most effective for detecting sur-face flaws and characterizing surface properties of electri-cally conductive materials. Recently, data were collectedthat classified each component as having a flaw or notbased on manual inspection and operator judgment andalso reported the size of the crack in the material. Do thecomponents classified as unflawed have a smaller meancrack size than components classified as flawed? Theresults in terms of crack size (in inches) are in the data file

@@Q (extracted from B. D. Olin and W. Q. Meeker,'Applications of Statistical Methods to NondestructiveEvaluation," Technometrics, 38, 1 996, p. 1 0 1.)a. Assuming that the population variances are equal, is

there evidence that the mean crack size is smaller for theunflawed specimens than for the flawed specimens?(Use a: 0.05.)

b. Repeat (a), assuming that the population variances arenot equal.

c. Compare the results of (a) and (b).

r64.3681 s 9 . 0 1 8153.871165.096157.184154.496160.920164.917169.09r175.276

177.t35163.903167.802160.818t67.433163.s38164.525t'71.230174.964166.31 I

r58.239r38.216168.006t49.654145.4561 68.1 78t54.321162.763161.020167.706

150.226155.620151.233158.653tst.204150.869t61.657r57.016156.670t47.920

a. Assuming that the population variances from both con-ditions are equal, is there evidence ofa difference in themean surface hardness between untreated and treatedsteel plates? (Use cr: 0.05.)

b. Determine thep-value in (a) and interpret its meaning.c. What other assumption is necessary in (a)?d. Assuming that the population variances from un-

treated and treated steel plates are equal, construct andinterpret a 95%o confidence interval estimate of the

10.2: Comparing the Means of Two Related Populations 381

1O.2 COMPARING THE MEANS OF TWO RELATED POPULATIONSThe hypothesis-testing procedures examined in Section 10.1 enable you to make comparisonsand examine differences in the means of two independent populations. In this section, you willlearn about a procedure for analyzing the difference between the means of two populationswhen you collect sample data from populations that are related-that is, when results of thefirst population are nol independent ofthe results ofthe second population.

There are two cases that involve related data between populations. In the first case, youtake repeated measurements from the same set of items or individuals. In the second case,items or individuals are matched according to some characteristic. In either case, the variableof interest becomes the dffirence between the values rather than the values themselves.

The first case for analyzingrelated samples involves taking repeated measurements on thesame items or individuals. Under the theory that the same items or individuals will behave alikeif treated alike, the objective is to show that any differences between two measurements of thesame items or individuals are due to different treatment conditions. For example, when per-forming a taste-testing experiment, you can use each person in the sample as his or her owncontrol so that you can have repeated measurements onthe same individual.

The second approach for analyzingrelated samples involves matching items or individualsaccording to some characteristic of interest. For example, in test marketing a product under twodifferent advertising campaigns, a sample of test markets can be matched on the basis of thetest market population size and/or demographic variables. By controlling these variables, youare better able to measure the effects of the two different advertising campaigns.

Regardless of whether you have matched samples or repeated measurements, the objec-tive is to study the difference between two measurements by reducing the effect of the vari-ability that is due to the items or individuals themselves. Table 10.3 shows the differences inthe individual values for two related populations. To read this table, let X1, Xt2, . . . , Xrrrep-resent the n values from a sample. And let X2t, X22, . . . , X2, represent either the correspond-ing n matched values from a second sample or the corresponding n repeated measurementsfrom the initial sample. Then, Dp D2, . . . , Dnwill represent the corresponding set of n differ-ence scores such that

Dt : X t t - X2 t , D2 : X12 - X22 , . . . , and Dr : X rn - X rn

Group

Value Difference

10 .3in inq the

Between TwoPopulations

sample size is large,

Centnl LimitTheorem

page 268) ensures youtt the sa m pli n g di stributi o n

Xztxzz

Dr: Xt t - Xr tDz: xr..z- Xn

Dr: xtt- xn":,

i,. Dn: Xrn- Xrn

To test for the mean difference between two related populations, you treat the differencescores, each D,, as values from a single sample. If you know the population standard deviationof the difference scores, you use the Ztest defined in Equation (10.q.2 This Ztest for the meandifference using samples from two related populations is equivalent to the one-sample Ztest forthe mean of the difference scores [see Equation (9.1) on page 334].

I2

xttxtz

i,,

Xz,

follows a normal

382 CHAPTER TEN Two-Samole Tesrs

Z TEST FOR THE MEAN DIFFERENCE

Z*D-po

6D-r,\l n

(10.4)

wheren

Sn..L/

-I

D : i = rn

Fp: hypothesized mean difference

or: population standard deviation ofthe difference scores

r : sample size

The test statistic Z follows a standardized normal distribution.

Paired t Test

In most cases, the population standard deviation is unknown. The only information youhave are the sample mean and the sample standard deviation.

lf you assume that the difference scores are randomly and independently selectedfiompopulation that is normally distributed you can use the paired I test for the mean differenrin related populations to determine whether there is a significant population mean diffeLike the one-sample I test developed in Section 9.4 [see Equation (9.2) on page 347], the tstatistic developed here follows the r distribution, with n - I degrees of freedom. Althoughmust assume that the population is normally distribute{ as long as the sample size is notsmall and the population is not highly skewed you can use the paired r test.

To test the null hypothesis that there is no difference in the means of two related

Hoi Fo- 0 (where Fn: \ -,ttz)

against the alternative that the means are not the same:

H r : P ' o + 0

you compute the I test statistic using Equation ( 10.5).

PAIRED t TEST FOR THE MEAN DIFF

(r0.s)

where

ERENCED -po

sp'""r1;

ns a . ^ : . ,) . \ u ; - D f

, - l

\ - ! -

/-/"i; - l

,D:

so:

The test statistic l follows a / distribution with n - l desrees of freedom.

llv

n alcece.est/ou3ry

N S :

10.2: Comparing the Means of Two Related Populations 383

For a two-tail test with a given level of significance, o, you reject the null hypothesis if thecomputed I test statistic is greater than the upper-tail critical value t r_, from the I distribution orif the computed test statistic is less than the lower-tail critical value -t, , from the / distribu-t ion. The decis ion ru le is

Reject Hoi f t>tn_lo r i f / < - / ,

, :otherwise, do not reject 11n.

The following example i l lustrates the use of the / test for the mean difference. TheAutomobile Assocation of America (AAA) conducted a mileage test to compare the gasolinemileage from real-life driving done by AAA members and results of city-highway driving doneaccording to current (as of 2005) government standards (extracted from J. Healey, "FuelEconomy Calculations to Be Altered," USA Today, January 11,2006, p. 1B).

What is the best way to design an experiment to compare the gasoline mileage from real-life driving done by AAA members and results of city-highway driving done according to cur-rent (as of 2005) government standards? One approach is to take two independent samples andthen use the hypothesis tests discussed in Section I 0. | . In this approach, you would use one setof automobiles to test the real-life driving done by AAA members. Then you would use a sec-ond set of different automobiles to test the results of city-highway driving done according tocurrent (as of2005) government standards.

However, because the first set of automobiles to test the real-l i fe driving done by AAAmembers may get lower or higher gasoline mileage than the second set of automobiles, thisis 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 atest of real - l i fe dr iv ing done by an AAA member and a test of c i ty-h ighway dr iv ingdone according to current (as of 2005) government standards. Measuring the two gaso-l ine mi leages for the same automobi les serves to reduce the var iabi l i ty in the gasol inemileages compared with what would occur if you used two independent sets of automobiles.This approach focuses on the differences between the real-l i fe driving done by an AAAmember and the city highway driving done according to current (as of 2005) governmentstandards.

Table 10.4 displays results (stored in the file EEE@EEffr) from a sample ofn : 9 automobiles from such an exoeriment.

Model Members CurrentT A B L E 1 0 . 4RepeatedMeasurements ofGasol ine Mi leage forReal-Life Driving byMA Members andCity-Highway DrivingDone According toCurrent (as of 2005)Government Standards

2005 Ford F-1502005 Chevrolet Silverado2002 Honda Accord LX2002 Honda Civic2004 Honda Civic Hybrid2002Ford Explorer2005 Toyota Camry2003 Toyota Corolla2005 Toyota Prius

14.31 5 . 027.827.948.81 6 . 823.732.837.3

1 6 . 81 7 . 826.2t J . z

47.61 8 . 328.53 3 . 144.0

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

384 CHAPTERTEN Two-SampleTests

the mean gasoline mileage is different between the two types of driving? Thus, the

alternative hypotheses are

Ho: Fn: 0 (There is no difference in mean gasoline mileage between the real-life dr

Uy an ene member and the city-highway driving done according to current fas of

ernment standards.)Ht: Fn* 0 (There is a difference in mean gasoline mileage between the real-life dr

Uy anlen member and the city-highway driving done according to current [as of

ernment standards.)

Choosing the level of significance of cr : 0.05 and assuming that the differencesmally distributed, you use the paired I test [Equation (10.5)]. For a sample of n=9there are n - | :8 degrees of freedom. Using Table E.3, the decision rule is

Reject Hoif t> tr :2.3060;or if r < -te: -2.3060;

otherwise, do not reject Ilo.

\sr{\ren--\t\\srursts\st\S$lt\\),$-t.rss$s,rsssN\\RsxsN\Rbls

YoL l '

a r l

D = i = t _ - L t ' t = _ 2 . 3 4 4 4n 9

and

n

\rn, - D)'

, - 1= 2.893575S D =

From Equation (10.5) on page 382,

_ - 2 . 3 4 4 4 - 0 = _ 2 . 4 3 0 12.893575

D -v,,SD

\ n

Because t: -2.4307 is less than -2.3060, you reject the null hypothesis, llo (t.. Figure

There is evidence of a difference in mean gasoline mileage between the real-life driving

by an AAA member and the city-highway driving done according to current (as of 2005)

ernment standards. Real life driving results in a lower mean gasoline mileage.

FIGURE 10.7Two-tail paired ttestat the 0.05 levelof significance withB degrees of freedom

Q / + 2 . 3 0 6 f f 8

Region of I Region of I Region ofRejection I Nonrejection lRejection

I

10.2: Comparing the Means ofTwo Related Populations 385

You can compute this test statistic along with the p-value by using Microsoft Excel (seeFigure 10.8). Because thep-value :0.0412 < o:0.05, you reject Ho.The p-value indicates thatif the two types of driving have the same mean gasoline mileage, the probability that one type ofdriving would have a mean that was 2.3444 miles per gallon less than the other type is 0.0412.Because this probability is less than cr:0.05, you conclude that the alternative hypothesis is true.

10.8A Microsoft Excel

ults of paired t testthe car mileage data

B Microsoft Excelisker plot

the car mi leage data

Condadonilocn Dlferencr

onc{allCrldcal onc.tcll

two{rll 0.0t1? 1Crldcal two{dl

Box-and-Whisker Plot for casolane Mileage Difierences

From Figure 10.8, Panel B, observe that the box-and-whisker plot shows approximatesymmetry. Thus, the data do not greatly contradict the underlying assumption of normality. Ifan exploratory data analysis reveals that the assumption of underlying normality in the popu-lation is severely violated" then the / test is inappropriate. Ifthis occurs, you can either use anonparametric procedure that does not make the stringent assumption of underlying normality(see References I and 2) or make a data transformation (see reference 5) and then recheck theassumptions to determine whether you should use the I test.

PAIRED I-TE5T OF PIZZA DELIVERY TIMES

Recall from Example l0.l on page 374 that a local pizzarestaurant located across the streetfrom a college campus advertises that it delivers to the dormitories faster than the local branchof a national pizza chain. In order to determine whether this advertisement is valid, you andsome friends have decided to order l0 pizzas from the local pizza restaurant and I 0 pizzas fromthe national chain. In fact, each time you ordered apizza from the local pizza restaurant, yourfriends ordered apizza from the national pizzachain. Thus, you have matched data. For each of

I0T613.

0e:

[email protected]||Il

XAMPLE 10 .2


T A B L E 1 0 . 5Del iverv Times for Localrrzza Kestaurant anoNational Pizza Chain

FIGURE 10.9Microsoft Excel oairedt test results for thepizza del ivery data

t-ll - l l

l-lSee Section E10.6 to createthis.

the ten times pizzas were ordered" you have one measurement from the local pizzaand one from the national chain. At the 0.05 level of significance, is the mean delivery timethe local pizza restaurant less than the mean delivery time for the national pizza chain?

SOLUTION Use the paired r test to analyze the data in Table 10.5 (see the file [@@@.

Time Local Chain Difference

I

2J

456789

1 0

1 6 . 8t t .71 5 . 616.717.51 8 . 1t 4 . l2 t . 813.920.8

22.015.21 8 . 71 5 . 620.81 9 . 517.019.51 6 . 524.0

-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.

Local Cltp'in4 Mean 16.7 18.S-5 Variance 9.58?2 8.21516 Observallors 10 107 rPearton Correlatlon 0.714'lI Hypotheslzed Mean Dlfference 0g i d f I10 t Stat 3.04481.'l tPff."q one-tail 0.007012. t Crltlcal one-tall 1.833113 Pfr.=q rro-rall 0.013914 rt Crlrlcal rwo-rail 2.2622


Ho: Vn > 0 (Mean delivery time for the local pizzarestaurant is greater than or equal tothemean delivery time for the national pizza chain.)Hi Fp < 0 (Mean delivery time for the local pizzarestaurant is less than the mean delivery timefor the national pizza chain.)

Choosing the level of significance o( : 0.05 and assuming that the differences are normally dis-tribute4 you use the paired / test [Equation (10.5) on page 382]. For a sample of r : 10 deliv-ery times, there are n - | -- 9 degrees of freedom. Using Table E.3, the decision rule is

R e j e c t H o i f t < t n : - 1 . 8 3 3 1 ;

otherwise, do not reject 11n.

For n : l0 differences (see Table 10.5). the sample mean difference is

tr ,- - 7 rR

D_ ,= , = - " " =_2 .13n10

.rant: for

10.2: Cornparing the Means of Two Relatecl Populat ions 387

and the sample standard deviat ion of the dif ference is

s1.' =

From Equation (10.5) on page 382,

D - V n - ) t R - 0

2.2641

tTt

Because t - -3.0448 is less than - I .833 I, you reject the null hypothesis Hn (the 7r-value is0.0070 < 0.05) . There is ev idence that the mean del ivery t ime is lower for the local p izzarestaurant than for the national pizza chain.

This conclusion is different from the one you reached when you used the pooled-variance rtest for these data. By pairing the delivery tirnes, you are able to focus on the differencesbetween the two pizza delivery services and not the variability created by ordering pizzas at dif-ferent t imes of day. The paired I test is a more powerful statistical procedure that is better ableto detect the difference between the two pizza delivery services.

Confidence Interval Estimate for the Mean Difference

Instead of, or in addition to, testing for the difference between the means of two related populations,you can use Equation 10.6 to construct a confidence interval estimate of the mean difference.

CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN DIFFERENCE

(10.6)

D - t r - t < D + t . - ,s^* U

G

Return to the exarnple comparing gasolinenileage generated by real-life driving and bygovernment standards. Using Equat ion (10.6). D - -2.3444.tD - 2.8936, n:9, and t :2.306(for 95% confidence and n - I : 8 degrees of freedom),

_2 .3444 t (2 .306) 2 '89 : ]6

r/q-2.3444 + 2.2242

- 4 . 5 6 8 6 ( U , S - 0 . 1 2 0 2

Thus, with 95% confidence, the mean difference in gasoline mileage between the real-life dri-ving done by an AAA member and the city highway driving done according to current (as of2005) government standards is between -4.5686 and -0.1202 miles per gallon. Because theinterval estirnate contains only values less than zero, you can conclude that there is a differencein the population means. The mean miles per gallon for the real-life driving done by an AAAmember is less than the mean rniles per gallon for the city-highway driving done according tocurrent (as of2005) government standards.

s,t;

P=u," !n

l is-liv-

388 CHAPTERTEN Tlvo-SamDleTestg

Learning the Basics

EnnFq rc.22 An experimental design for a paired / testlAsslsr I has, as a matched sample, 20 pairs of identical

twins. How many degrees of freedom are there inthis I test?

EE q 10.23 An experiment requires a measurementlAsslsT I before and after the presentation of a stimulus to

each of 15 subjects. In the analysis of the datacollected from this experiment, how many degrees of free-dom are there in the test?


10.24 The September issues of monthly mag-azines typically carry the most advertisingpages for any issue during the year. The follow-

ing data (stored in the file lilEltEIE) give the numberof advertising pages in September 2004 and September2005:

Magazine 2004 2005

fil. @!@ss[ (coded to maintain confireDresent measurements in-line that were collectedan analyzer during the production process and fromanalytical lab (extracted from M. Leitnaker, "Measurement Processes: In-line Versus AnalvtiMeasuremen ts," Qu a I i Q E n gin e erin g, | 3, 2000-2001,293-298).a. At the 0.05 level

difference in thean analytical lab?

of significance, is there evidence

b. What assumption is necessary to perform this test?c. Use a graphical method to evaluate the validity of

assumption in (a).d. Construct and interpret a 95o/o confidence interval

mate of the difference in the mean measurements inliand from an analytical lab.

10.26 Can students save money by buying theirbooks at Amazon.com? To investigate this possibility,random sample of 14 textbooks used during the 2006mer session at Miami Universitv was selected. The orifor these textbooks at both a local bookstore andAmazon.com were recorded. The orices for the texincluding all relevant taxes and shipping, are given(and are stored in the fileS!g!!pQ):

Textbook Book Store A

mean measurements in-line and

Martha Stewart LivingGood HousekeepingParentingGlamour (special issue)Popular MechanicsEbonyC o smop o litan (special issue)Ladies'Home JournalParentsVogueHarper s BazaarElleEsquireReal SimpleMen s HealthGQInStyleDetails

52 .14t t5 . t2r23.84184.7867.44

r22.32227.35125.21t39.14650.63261.09342.27165.58163 .10r39.76292.8s382.96206.97

75.25t49.411s8.37236.0085.02

141.77248.60136.99r49.68690.ss274.06346.94167.53163.80t40.16288.27376.00202.13

Concepts in Federal TAxafionIntermedi ate A cc ountin gThe Middle East and CentralAsiaWest s Business LawLeadership: Theory and PracticeMaking Choices for Multicultural

EducationDirect Instruction ReadingEssentials of EconomicsMarriage and FamilyAmerica and lts PeopleOceanographyCalculus : E arly T?ans cendental

Single VariableAccess to HealthWomen and G I o b alizati on

r38.2r 143.95.rst.92 rs2.7as2.06 53.00

1s9.31 143.9549.59 48.95

71.74 56.9598.12 97.35

102.12 99.64106.92 100.98100.44 9s.20105. l8 128.95

I I 5.00 133.5093.47 88.6029.54 18.48

a. At the 0.05 level of significance, is there evidence of adifference in the mean number of advertising pages inSeptember 2004 and September 2005?

b. What assumption is necessary to perform this test?c. Determine thep-value in (a) and interpret its meaning.d. Construct and interpret a 95o/o confidence interval esti-

mate of the difference in the mean number of advertis-ing pages in September 2004 and September 2005.

10.25 In industrial settings, alternative methods oftenexist for measuring variables of interest. The data in the

a. At the 0.01 level of significance, is there evidence ofdifference between the mean price of textbooks atlocal bookstorc and Amazon.com?

b. What assumption is necessary to perform this test?c. Construct a 99o/o confidence interval estimate of the

mean difference in price. Interpret the interval.d. Compare the results of (a) and (c).

Bee tuna. 6 oz. canSmith apples (l lb.)

DeCecco linguinisteak, I lb,

chicken, per pound

A newspaper article discussed the opening of a WholeMarket in the Time-Warner building in NewYork City.

following data (stored in the fileEEEEEE@ co--the prices of some kitchen staples at the new WholeMarket and at the Fairway supermarket located about

from the Time-Warner building:

Whole Foods Fairway

milk 2 . r9 1 .352.39 r.692.00 2.491.98 1.294.99 3.691.79 1.331.69 r.491.99 1.597.99 5.992, t9 1.49

eggsorange juice (64 oz.)

of Boston lettuceround, I lb.

10.2: Comparing the Means ofTwo Related Populations 389

immediately prior to the stem cell transplant and at the timeof the complete response:

Patient Before After

1 158 2842 189 2r43 202 lOt4 353 227s 416 2906 426 1767 44t 290

Source: Extractedfrom S. V Rajlamaa R. Fonseca, T E. llitzig,M. A. Gertz, and P R. Greipp, "Bone MarowAngiogenesis in PatientsAchieving Complete Response After Stem Cell TransplantationforMultiple Myeloma," Leukemi4 1999, 13, pp. 469472.

a. At the 0.05 level of significance, is there evidence that themean bone marrow microvessel density is higher beforethe stem cell transplant than after the stem cell transplant?

b. Interpret the meaning of thep-value in (a).c. Construct and interpret a 95o/o confidence interval esti-

mate of the mean difference in bone marrow microvesseldensity before and after the stem cell transplant.

d. What assumption is necessary to perform the test in (a)?

10.29 Over the past year, the vice president for humanresources at a latge medical center has run a series of three-month workshops aimed at increasing worker motivation andperformance. To check the effectiveness ofthe workshops, sheselected a random sample of 35 employees from the personnelfiles and recorded their most recent annual performance rat-ings, along with their ratings prior to attending the workshops.The data are stored in the file[@@. The Microsoft Excelresults in PanelsA and B provide both descriptive and inferen-tial information so that you can analyze the results and exam-ine the assumptions of the hypothesis test used:

State your findings and conclusions in a report to thevice president for human resources.

for Xcatr

: Extractedfrom W Grimes, "A Pleasure Palace Without the" The New York Times, February 1 8, 2004, pp. Fl , F5

tAt the 0.01 level of significance, is there evidence that'the mean price is higher at Whole Foods Market than atthe Fairway supermarket?Interpret the meaning of thep-value in (a).What assumption is necessary to perform the test in (a)?

Multiple myeloma, or blood plasma cancer, is char-by increased blood vessel formulation (angiogen-

in the bone marrow that is a prognostic factor in sur-. One treatment approach used for multiple myeloma iscell transplantation with the patient's own stem cells.

following data (stored in the file fi!$@l@ representbone marrow microvessel density for patients who had a

response to the stem cell transplant, as measuredblood and urine tests. The measurements were taken

Dlflslrlnce

SkawrerRangottnlmrm{3xlmumSumCounfllsq{0

7t5(BE0"gt25

Ito.r3{2

. . . 0i 3 {-2599no"arrt5sobios, r f f i .

37.16f735

DovladonVarlancr

sJtrfi1rslle

'5,-t0,

115,?321T2:t0,,g

1.10380.1103,

fl-l34iu,:

.:t&f ,3iln'aa:

OlrorallomPearon ConelatlonHypoitrcdrrd liin Dlfrerencrdfsr||

k-{ oncsftCdtlcal one{all

390 CHAPTERTEN Two-SampleTests

10.30 The data in the file@EErepresent the com-pressive strength, in thousands of pounds per square inch(psi), of 40 samples of concrete taken two and seven daysafter pouring.Source: Extracted from O. Caruillo-Gamboa and R. F Gunst," Measurement- Error- Mode I Col linearities," Technometrics, -34,1992, pp. 454-464.

At the 0.01 level of significance, is there evidencethe mean strength is lower at two days than at sevenWhat assumption is necessary to perform this test?Find the p-value in (a) and interpret its meaning.

a.

b.c.

10.3 COMPARING TWO POPULATION PROPORTIONSOften, you need to make comparisons and analyze differences between two population propor-tions. You can perform a test for the difference between two proportions selected from inde-pendent samples by using two different methods. This section presents a procedure whose teststatistic, Z, is approximated by a standardized normal distribution. In Section 12.1, a procedurewhose test statistic, 12, is approximated by a chi-square distribution is developed. As you willsee, the results from these two tests are equivalent.

Z Test for the Difference Between Two ProportionsIn evaluating differences between two population proportions, you can use a Z test for the dif.ference between two proportions. The test statistic Z is based on the difference between twosample proportions (pr- pz). This test statistic, given in Equation (10.7), approximatelyfol-lows a standardized normal distribution for large enough sample sizes.

ZTEST FOR THE DIFFERENCE BETWEEN TWO PROPORTIONS

(10.7)

with

- ( r 1 )P) l -+ -1

\ r t nz )

X r + X 1

\ * n 2

Y.l ) r = -

n ,' ' l

x)P 2 = -

n2

wnere

p1 : proportion of successes in sample I

X, : number of successes in sample 1

n,: sample size of sample 1

fil : proportion of successes in population 1

P2-- proportion of successes in sample 2

X, : number of successes in sample 2

nr: sample size of sample 2

fi2: proportion ofsuccesses in population 2

p : pooled estimate of the population proportion of successes

The test statistic Z approximately follows a standardized normal distribution.

10.3: Comparing Two Population Proportions 391

Under the null hypothesis, you assume that the two population proportions are equal(n, : n).Because the pooled estimate for the population proportion is based on the nullhypothesis, you combine, or pool, the two sample proportions to compute an overall esti-mate of the common population proportion. This estimate is equal to the number of suc-cesses in the two samples combined (Xt + X) divided by the total sample size from the twosample groups (nr+ n2).

As shown in the following table, you can use this Ztestfor the difference between popula-tion proportions to determine whether there is a difference in the proportion of successes in thetwo groups (two-tail test) or whether one group has a higher proportion of successes than theother group (one-tail test):

TWo-Tail Test One-Tail Test One-Tail Test

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

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

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

where

fil : proportion ofsuccesses in population I

7[2: proportion ofsuccesses in population 2

To test the null hypothesis that there is no difference between the proportions of two inde-pendent populations:

Ho: nr: n,

against the alternative that the two population proportions are not the same:

Hl T \+T.2

use the test statistic Z, given by Equation (10.7). For a given level of significance o, reject thenull hypothesis if the computed Z test statistic is greater than the upper-tail critical value fromthe standardized normal distribution or if the computed test statistic is less than the lower-tailcritical value from the standardized normal distribution.

To illustrate the use of the Z test for the equality of two proportions, suppose that you arethe manager of T.C. Resort Properties, a collection of five upscale resort hotels located on tworesort islands. On one of the islands, T.C. Resort Properties has two hotels, the Beachcomberand the Windsurfer. In tabulating the responses to the single question, 'Are you likely to choosethis hotel again?" 163 of 227 guests at the Beachcomber responded yes, and 154 of 262 guestsat the Windsurfer responded yes. At the 0.05 level of significance, is there evidence of a signif-icant difference in guest satisfaction (as measured by the likelihood to return to the hotel)between the two hotels?


H o : n , : r 2 o r n . - n r : 0

Hr: n, + Tc2or f i l - n2+ 0

Using the 0.05 level of significance, the critical values ne -1.96 and +1.96 (see Figure 10.10),and the decision rule is

Reject Hoif Z<-1.96or i f Z> +1.96;

otherwise, do not reject.Flo.

392 CHAPTERTEN Two-SamPle Tests

FIGURE 10.10Reqions of reiectionani nonrejec i ion whentesting a hyPothesis forthe difference betweentwo proportions at the0.05 level of significance

FIMfod itVh(P

lStt

Region ofRejection

Crit icalVa lue

Using Equation (10.7) on Pagelqo

Region of Crit icatNonrejection Value

Region ofRejection

p(r - . ; )- . ( ru'li

_o64sr(*.#)

where

and

so that

x1ny

163 -0 .7181227Pt n .= I z -154=0 .5878rL

f i2 262

P -

X r + X , _ 1 6 3 + 1 5 4 _ 3 1 1 = 0 . 6 4 g 3nr + n2 227 + 262 489

- 0.s878) - (0)

0.6483(l

0.1 303

( 0 . 7 1 8 1

l(urrsxo.oo8r)

- o '1303 = +3 .00880.0432

Using the 0.05 level of significance, reject the null hypothesis because z: +3'0088 > +1'96'

Thel-value is 0.0026 (cai-culated from iable E.2 or from the Microsoft Excel results of Figure

10.11). This means ttrat lf the null hypothesis is true, the probability that aZtest statisticis

less than -3.0088 is 0.0013, an4 similarly' the probability that a Z test statistic is greater than

+3.0088 is 0.0013. Thus, for this two-tai l test, thep-value is 0 '0013 + 0'0013:0'0026'

Because 0.0026 < cr : 0.05, you reject the null hypothesis. There is evidence to conclude that

the two hotels are significantly different with respect to guest satisfaction; a greater prop0r'

tion of guests are willing to return to the Beachcomber than to the Windsurfer'

0.1 303

1t?4b

iq9l 01 11 7t s1 4t 5

i617!81e.-20

,27324

FIGURE 10.11Microsoft Excel resultsfor the Z test for thedifference betweentwo proportions for thehotel guest satisfactionproblem


EXAMPLE 10 .3


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

-r{ORl{Slf{v{85,?}-xoRrstilv(l - 85.")=2' (1 - iloRHsDlsT{ABs(818D}-lF(823 < 85, 'RoJecl the oull hypothods",

"Do nol r.Jocl tho o{ll byporhGls)

TESTING FOR THE DIFFERENCE IN TWO PROPORTIONS

Money worries in the United States start at an early age. In a survey, 660 children (330 boysand 330 girls) ages 6 to 14 were asked the question, "Do you worry about having enoughmoney?" Of the boys surveyed20l (60.9%) said yes, and 178 (539%) of the girls surveyedsaid yes (extracted from D. Haralson and K. Simmons, "Snapshots," USA Today,May 24,2004,p. lB).At the 0.05 level of significance, is the proportion of boys who worry about havingenough money greater than the proportion of girls?

SOLUTION Because you want to know whether there is evidence that the proportion of boyswho worry about having enough money is greater than the proportion of girls, you have a one-tail test. The null and alternative hypotheses are

Hr: n, < n, (The proportion of boys who worry about having enough money is less than orequal to the proportion of girls.)H ,: n, > n, (The proportion of boys who worry about having enough money is greater than theproportion of girls.)

Using the 0.05 level of significance, for the one-tail test in the upper tail, the critical value is+1.645. The decision rule is

Re jec t Ho i f Z>+1.645;otherwise, do not reject F1o.

Using Equation (10.7) on page 390,

Z

96.ure: i s1an26.hatror- = X t -

n1

2ol = 0.609330

- 1 7 8 = 0 . 5 3 9330

Prx2

where

r zn2


and

n =Xr+X ,n l + n 2

201 + 178 379=... . . . . . . ' . . . . . . . . . . . . . . . . . ' -=_-n<1L"'

330 + 330 660

Z _

0.07

0.07= o r *

= + l ' 8 1 8

Using the 0.05 level of significance, reject the null hypothesis because Z : +1.818 > + 1.645,The p-value is 0.0344 (calculated from Table E.2). Therefore, if the null hypothesis is true, theprobability that a Z test statistic is greater than +l .818 is 0.0344 (which is less than o : 0.05).You conclude that there is evidence that the proportion ofboys who worry about having enoughmoney is greater than the proportion of girls.

Confidence Interval Estimate for the DifferenceBetween Two Proportions

Instead of, or in addition to, testing for the difference between the proportions of two indepen-dent populations, you can construct a confidence interval estimate of the difference betweenthe two proportions, using Equation ( 10.8).

CONFIDENCE INTERVAL ESTIMATE FOR THE DIFFERENCEBETWEEN TWO PROPORTIONS

(0 .60e-0 .s3e) - (0 )

\ \ "0 330 /

^l1o.z++s11o.00606 )0.07

ffi

( p 1 - p 2 ) + Z P r ( l - p ) , p 2 0 - p z )nr n2

(10.8)

tr:t

Lei

-

I A \

I . I

tb . (

+

l!!!tt-I A \-7 . t

I[ . r

A5

tfEl

mtaslmtL.

( n -pz ) -Z p,1(1 - nr) p2( l - p2)T - - 1 ! i 1 - i i 2

In1 n2

S(p r -p )+Z P{ r - p ) , p20 - p2 )T -

n1 n2

To construct a 95o/o confidence interval estimate of the population difference between thepercentages of guests who would return to the Beachcomber and who would return to theWindsurfer, you use the results on page 392 or from Figure 10.1 1 on page 393:

X, 163D r = 1 = _ = 0 . 7 1 8 1

n 1 z z l

x, 154D t = i = _ _ - _ = 0 . 5 8 7 8

n2 262


Using Equation (10.8),

(0 .7181 - 0 .5878) t (1 .96)" /L\l 227 262

0.1303 r (1.96x0.0426)

0.1303 r 0.0835

0.0468 ( (nr - TE) < 0.2138

Thus, you have 95o/o confidence that the difference between the population proportion ofguests who would return again to the Beachcomber and the Windsurfer is between 0.0468 and0.2138. In percentages, the difference is between 4.68% and,2l.35Yo. Guest satisfaction ishigher at the Beachcomber than at the Windsurfer.

the Basics

10.31 Assume that nr: 100, Xr: 50, nr: 100,andXr:30.

At the 0.05 level of significance, is there evidence ofa significant difference between the two populationproportions?Construct a95Yo confidence interval estimate of the dif-ference between the two population proportions.

10.32 Assume that nr: 100, Xr: 45, nr: 50,andXr:25.

At the 0.01 level of significance, is there evidence ofa significant difference between the two populationproportions?Construct a99o/o confidence interval estimate of the dif-ference between the two population proportions.

the Concepts

10.33 A sample of 500 shoppers was selected in alarge mefopolitan area to determine various infor-

ion concerning consumer behavior. Among the questionswas, "Do you enjoy shopping for clothing?" Of 240136 answered yes. Of 260 females, 224 answered yes.

Is there evidence of a significant difference betweenmales and females in the proportion who enjoy shoppingfor clothing at the 0.01 level of significance?Find thep-value in (a) and interpret its meaning.Construct and interpret a 99o/o confidence interval esti-mate of the difference between the proportion of malesand females who enjoy shopping for clothing.What are your answers to (a) through (c) if 206 malesenjoyed shopping for clothing?

An article referencing a survey conducted byServices claims that parents are more

confident than teachers that their schools will meet thestandards set by the No Child Left Behind Act. The surveyasked parents (and teachers), "How confident are you thatyour child's school (the school where you work) will meetthe standards by the deadline." The responses to that ques-tion are given in the following table:

Parents Teachers

Very confidentNot very confidentTotals

401684

1,085

162648810

Source: Adaptedfrom B. Feller "kachers More Likely Skepticsof No Child," The Cincinnati Enquirer, April 20, 2006, p. A4.

a. Set up the null and alternative hypotheses needed to tryto prove that the population proportion ofparents thatare very confident that their child's school will meet thestandards by the deadline is greater than the populationproportion ofteachers that are very confident that theschool where they work will meet standards by thedeadline.

b. Conduct the hypothesis test defined in (a), using a 0.05level ofsignificance.

c. Does the result of your test in (b) make it appropriate forthe article to claim that parents are more confident thanteachers?

10.35 The results of a study conducted as partof a yield-improvement effort at a semiconductor

manufacturing facility provided defect data for a sample of450 wafers. The following contingency table presents asunmary of the responses to two questions: "Was a particlefound on the die that produced the wafer?" and "Is thewafer good or bad?"-LOL Learning

396 cHAPTERTENTwo-Sample Tests

Quality of Wafer

PARTICLES Good Bad Totals

t4 36 50320 80 400334 116 450

a. Assume that 50 men and 50 women were rthe survey. At the 0.05 level of significance,evidence of a difference in the populationt ion of males and females who made gas mlpriority?

b. Assume that 500 men and 500 women were incthe survey. At the 0.05 level of significance, is theredence of a difference between males and females inproportion who made gas mileage a priority?

c. Discuss the effect of sample size on the Z test for theference between two proportions.

10.38 An experiment was conducted to studychoices made in mutual fund selection. Undeand MBA students were presented with different500 index funds that were identical except forSuppose 100 undergraduate students and 100 MBAdents were selected. Partial results are shown in thelowins table:

STUDENT GROUP

FUND Undergraduate

Highest-cost fund 27Not-highest-cost fund 73 |

Source: Extractedfrom J. Choi, D. Laibson, and B. Madrian,"Why Does the Law of One Practice Fail? An Experiment onMutual Funds," www. s o m.yale. edu/faculty/jj c8 3/fees, pdf.

a. At the 0.05 level of significance, is there evidencedifference between undergraduate and MBA studentsthe proportion who selected the highest-cost fund?

b. Find thep-value in (a) and interpret its meaning.

10.39 Where people turn for news is different forage groups (Extracted from P. Johnson, "YoungTurn to the Web for News." USA Todav. March 23.2006.9D). Suppose that a study conducted on this issuebased on 200 respondents who were between the ages ofand 50 and 200 respondents who were above age 50. Of200 respondents who were between the ages of 36 and82 got their news primarily from newspapers. Of therespondents who were above age 50, 104 got theirprimarily from newspapers.a. Is there evidence ofa sisnificant difference in the

portion who get their news primarily from nebetween those respondents 36 to 50 years old andabove 50 years old? (Use cr: 0.05.)

b. Determine thep-value in (a) and interpret its meaning.c. Construct and interpret,a95Yo confidence interval

mate of the difference between the populationtion of respondents who get their news primarilynewspapers between those respondents 36 to 50old and those above 50 years old.

YesNoTotals

U.S. TaxCode

Less Than$50,000

Source: Extractedfrom S.W Hall, "Analysis of Defectivity ofSemiconductor Wafers by Contingency Table," Proceedings InstituteofEnvironmental Sciences, Vol. I, 1994, pp. 177-183.

a. At the 0.05 level of significance, is there evidence of asignificant difference between the proportion of goodand bad wafers that have particles?

b. Determine thep-value in (a) and interpret its meaning.c. Construct and interpret a 95o/o confidence interval esti-

mate of the difference between the population propor-tion of good and bad wafers that contain particles.

d. What conclusions can you reach from this analysis?

10.36 According to an Ipsos poll, the perceptionof unfairness in the U.S. tax code is spread fairlyevenly across income groups, age groups, and

education levels. In an April 2006 survey of 1,005 adults,Ipsos reported that almost 600/o of allpeople said the code isunfair, while slightly more that 60% of those making morethan $50,000 viewed the code as unfair (Extracted from"People Cry Unfairness," The Cincinnati Enquirer, Aptll16,2006, p.A8). Suppose that the following contingencytable represents the specific breakdown ofresponses:

Income Level

More Than$50.000 Total

FairUnfairTotal

225 180280 320505 500

405600

I,005

a. At the 0.05 level of significance, is there evidence of adifference in the proportion of adults who think the U.S.tax code is unfair between the two income groups?

b. Find thep-value in (a) and interpret its meaning.

10,37 Is good gas mileage a priority for car shoppers? A sur-vey conducted by Progressive Insurance asked this questionto both men and women shopping for new cars. The data werereported as percentages, and no sample sizes were given:

Gender

GAS MILEAGEA PRIORITY? Men Women

YesNo

76%24%

84%t6%

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

10.4: F Tcst for the Difference Between Two Variances 397

1O.4 F TEST FOR THE DIFFERENCE BETWEEN TWO VARIANCESOften, you need to determine whether two independent populations have the same variability.This determination is made by testing variances. One important reason to test for the differencebetween the variances of 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 variances of two independent populations is basedon the ratio of the two sample variances. If you assume that each population is normally dis-tributed then the ratio Sf /S2' follows the F distribution (see Table E.5). The critical values ofthe F distribution in Table E.5 depend on two sets of degrees of freedom. The degrees of free-dom in the numerator of the ratio are for the first sample, and the degrees of freedom in thedenominator are for the second sample. Equation ( 10.9) defines the F test statistic for testingthe equality of two variances.

F TEST STATISTIC FOR TESTING THE EOUALITY OF TWO VARIANCESThe F-test statistic is equal to the variance of sample 1 divided by the variance of sample 2.

- )- J f' ^")r i

(10.e)

where

: variance of sample I

: variance of sample 2

: size of sample taken from population I

: size of sample taken from population 2

: degrees of freedom from sample 1 (that is, the numeratordegrees of freedom)

: degrees of freedom from sample 2 (that is, the denominatordegrees of freedom)

The test statistic F follows an F distribution with n, - 1 and n, - | degrees of freedom.

For a given level ofsignificance, o, to test the null hypothesis ofequality ofvariances:

nn: ol : ol

against the alternative hypothesis that the two population variances are not equal:

H , '" l '

you reject the null hypothesis if the computed tr test statistic is greater than the upper-tailcrit ical value, Fy, from the Fdistribution with n, - I degrees of freedom in the numeratorand n, - I degrees of freedom in the denominator, or if the computed Ftest statistic is lessthan the lower-tail crit ical value, F., from the F distribution with r, - I and n, - 1 degrees offreedom in the numerator and denominator, respectively. Thus, the decision rule is

Reject Hol f F> Fuo r i f F < t r , :

otherwise. do not relecr H,,.

This decision rule and rejection regions are displayed in Figure 10.12.

sir tn l

n2

n t - 1

n z - 1

)t

o l +o)

nS

398 cHAPTERTEN

FIGURE 10.12Regions of re ject ionand nonreject ion forthe two-tail F test

Two-Sample Tcsts

-t

lI

o i FL

Reg ion o fRejection

I . a -rLJ 1 r

1

Region o f Reg ion o fNonrejection Rejection

To illustrate how to use the F test to determine whether the two variances are equal, returnto the Using Statistics scenario concerning the sales of BLK cola in two different aisle loca-tions. To determine whether to use the pooled-variance I test or the separate-variance I test inSection 10. I , you can test the equality of the two population variances. The null and alternativehypotheses are

I t l ) .

t I I

Because this is a two-tail test, the rejection region is split into the lower and upper tails of thefdistribution. Using the level of significance cr : 0.05, each rejection region contains 0.025 ofthe distribution.

Because there are samples of l0 stores for each of the two display locations, there arel0 - I :9 degrees of freedom for group I and also for group 2. FL,, the upper-tail crit icalvalue of the Fdistribution, is found directly frorr-r Table E.5, a portion of which is presentedinTable 10.6. Because there are 9 degrees offreedom in the numerator and 9 degrees offreedonin the denominator, you find the upper-tail crit ical value, .Fr, by looking in the column labeled9 and the row labeled 9. Thus, the upper-tail crit ical value of this Fdistribution is 4.03.

Denominator Numerator d/,

o )

6 ;

)o l

)o l

TABLE 10 .6Finding F, , , the Upper-

f -t a i l Ln t | ca t va tue oT Fw i t h 9 a n d 9 D e g r e e sof Freedom for Upper-Tail Area of 0.025

641.80 799.s0 864.2038 .51 39 .00 39 .17t7 .44 16.04 1s.44

df1

I23

78

948.2039.3614.62

956.7039.3714.54

4.904.43

303947

8.077 .57

6.546.06

5 . 8 9< A ' '

4.994.53

Sottrce: Extnttted /iont Tctble E.5.

Finding Lower-Tail Crit ical Values

You compute Fr_, a lower-tailcrit ical value on the trdistribution with n, - I degrees of freedomin the numerator and r, - I degrees of lreedorn in the denominatoq by taking the reciprocal ofF u*, an upper-tail crit i ial value on the F distribution with degrees of freedom "switched" (thatis, nr- I degrees of freedorn in the numerator and n, - I degrees of freedom in the denomina-tor). This relationship is shown in Equation ( 10.10).

10.4: F Test for the Difference Between Two Variances 399

FINDING LOWER-TAIL CRITICAL VALUES FROM THE F DISTRIBUTION

(10.10)

where f'* is from an F distribution with n, - ! degrees of freedom in the numerator and

n, - I degrees of freedom in the denominator'

In the cola sales example, the degrees of freedom are 9 and 9 for both the numerator sam-

ple and denominator ru-pi., so there is no "switching" of degrees of freedom; you just take the

reciprocal. Therefore, to compute the lower-tail 0.025 critical value, you need to find the upper-

tail b.ozs critical value of F with 9 degrees of freedom in the numerator and 9 degrees of free-

dom in the denominator and take its reciprocal. As shown in Table 10.6 on page 398, this

upper-tail value is 4.03. Using Equation (10.10),

- 1 = 0.2484.03

As depicted in Figure 10.13, the decision rule is

Reject Hoif F> Fu:4.03o r i f F l F , : 0 . 2 4 8 ;

otherwise, do not reject Hn.

Region olRejection

.tI

f r = -L E '

t u *

lrNiL-; i nive

_ lf r = -

fU*

10.13ions of rejection andreiection for two-tail

for equality ofvariances at thelevel of significance9 and 9 degrees

Using Equation (10.9) on page 397 and,the cola sales data (see Table l0.l on page372),

the F test statistic is

F = {S;

350 '6778= G;d

= 2.228e

Because FL:0.248 < F:2.2289 . Fu:4.03, you do not reject 110. Thep-value is 0.2482 for

a two-tail iest (twice the p-value for ihe one-tail test shown in the Microsoft Excel results in

Figure 10.14). Because 0.2482 > 0.05, you conclude that there is no significant difference in

the variability of the sales of cola for the two display locations'


FTGURE 10.14Microsoft Excel Ftestresults for the BLK colasales data


Panel ATwo-tail test

Ho: cl, = 6f

H r : o l *o l

EXAMPLE 10 .4

o||G-tsll

4t0.6778 1t.33310 109 9

2.2,,'.90.12113.1789Crltlcal ono{.ll

In testins for a difference between two variances usins the F test described in thisyou assume that each of the two populations is normally distributed. The Ftest is veryto the normality assumption. lf box-and-whisker plots or normal probability plots suggesta mild departure from normality for either of the two populations, you should not use the,FIf this happens, a nonparametric approach is more appropriate (see references I and 2).

In testing for the equality of variances. as part of assessing the validity of thevariance r test procedure, the F test is a two-tail test. However, when you are interested invariability itself, the F test is often a one-tail test. Thus. in testing the equality of two variyou can use either a two-tail or one-tail test, depending on whether you are testing whethertwo population variances are dffirent or whether one variance is greater than the other variFigure 10.15 illustrates the three possible situations.

,At \

/ \

/ \o F r F

FIGURE 10.15 Determining the reject ion region for test ing the equal i ty of two populat ion var iances

Often, the sample sizes in the two groups differ. Example 10.4 demonstrates how to findalower-tail critical value from the F distribution in this situation.

FINDING THE LOWER-TAIL CRITICAL VALUE FROM THE F DISTRIBUTIONIN A TWO-TAIL TEST OF A HYPOTHESIS

You select a sample of n, : 8 from a normally distributed population. The variance for thissample Sf is 56.0. You select a sample of nr-- l0 from a second normal$ distributed popula-tion (independent of the first population). The variance for this sample S j is 24.0. Using thelevel of significance cr:0.05, test the null hypothesis of no difference in the two populationvariances against the two-tail alternative that there is evidence of a significant difference in thepopulation variances.

SOLUTION The null and alternative hypotheses are

Panel BOne-tai l test

Ho: ol>olH.,; ol < of

Panel COne-tail test

uo: ol < o22

H.,: ol> oj

* Region of RejectionRegion of Nonrejection

o7

otrol

6?


The F test statistic is given by Equation ( 10.9) on page 397 :

You use Table E.5 to find the upper and lower critical values of the F distribution. Withnt- l :7 degrees of f reedom in the numerator,nr- l :9 degrees of f reedom in the denom-inator, and o, : 0.05 split equally into the lower- and upper-tail rejection regions of 0.025each, the upper critical value, Fu, is 4.20 (see Table 10.7).

To find the lower critical va|ue, Fp with 7 degrees of freedom in the numerator and 9degrees of freedom in the denominator, you take the reciprocal of Fr* with degrees of freedomswitched to 9 in the numerator andT in the denominator. Thus, from Equation (10.10) on page399 and Table 10.7,

s?D - " 1t - - -

S;

F,7e

F ,= | - I =0 .207

" Fu* 4.82

Denominator Numerator df,T A B L E 1 0 . 7Finding Fu. and Fr,w i t h T a n d 9 D e o r e e sof Freedom, Using theLevel of Significanceu = 0.05

647.80 799.50 864.2038.51 39.00 39.1717.44 16.04 15.44

df1

II

2J

.30

.39

.47

7.57 4.43 4.364.10 4 .03

Source; Extrectedfrom Table E.5.

The decision rule is

Reject Hoif F > Fu: 4.20o r i f F . F L : 0 . 2 0 7 ;

otherwise, do not reject Ho.

From Equation (10.9) on page 397,the F test statistic is

-s iS;

=16 'q -21124.0

Because FL:0.207 < F:2.33 . Fu:4.20,you do not reject Ho. Using a 0.05 level of signif-icance, you conclude that there is no evidence ofa significant difference between the variancesin these two independent populations.

.20 956.70 9(

.36 39.37 3

.62 14.54 I


Learning the Basics

10.40 Determine Fu and Fr, the upper- andlower-tail critical values of F. in each of the fol-lowing two-tail tests:

a . c x ' : 0 . 1 0 , r , : 1 6 , n r : 2 1b . a : 0 . 0 5 , r , : 1 6 , n r : 2 1c . c [ : 0 .02 , n r : 16 , n r : 2 ld. cr : 0.01, n, : 16, nr-- 2l

10.41 Determine Fr,the upper-tail critical value of d ineach of the following one-tail tests:a . 0 : 0 . 0 5 , n r : 1 6 , n r : 2 lb . a : 0 .025, n r : 16 , n r : 21c . c r : 0 .01 , n , : 16 , n r :21d. c r : 0 .005, n , : 16 , n r :21

',O.42 Determine Fr,rhe lower-tail critical value of F, ineach of the following one-tail tests:a . c t , : 0 .05 , n , : 16 , n r :21b. c r : 0 .025, n r : 16 , n r :21c . c r : 0 . 0 1 , n r : 1 6 , n r : 2 1d. c r : 0 .005, n r : 16 , n r : 2 l

10.43 The following information is availablefor two samples drawn from independent nor-mally distributed populations :

nt = 25 S? = Bl.l nz = 25 S i = 16 l .g

What is the value of the F test statistic if you are testing thenull hypothesis //6: o? = a3t

@| 10.44 In Problem 10.43, how many degrees oflAsslsT I freedom are there in the numerator and denomi-

nator of the F test?

10.45 In Problems 10.43 and 10.44,what arethe critical values for F, and F, if the level ofsignificance, cr, is 0.05 and the alternative hypoth-

esis is .F11: of * oi?

'10.46 In Problems 10.43 through 10.45, what is your sta-tistical decision?

10.47 The following information is available for twosamples selected from independent but very right-skewedpopulations:

nr = 16 S? = 4l.t nz = 13 Sl = 36.q

Should you use the F test to test the null hypothesis ofequality of variances (Hs: ol = o3)? Discuss.

10.48 In Problem 10.47, assume that two samplesselected from independent normally distributed popua. At the 0.05 level of significance, is there evidence

difference between of and ai?

bc,

db. Suppose that you want to perform a one-tail test. At

0.05 level of significance, what is the upper-tail crivalue of the F test statistic to determine whether tevidence that of > o]Z Whut is your statistical decisi

c. Suppose that you want to perform a one-tail test. At0.05 level of significance, what is the lower-tail cvalue of the .F test statistic to determine whether thereevidence that of < c3? What is your statistical decisi


10.49 A professor in the accountingof a business school claims that there ismore variability in the final exam scores of

dents taking the introductory accounting course as ament than for students taking the course as part of a majaccounting. Random samples of 13 non-accounting(group l) and l0 accounting majors (group 2) are takenthe professor's class roster in his large lecture, and thelowing results are computed based on the final exam

nt = 13 S? = 210.2 nz = l0 Sl. = 3e.S

a. At the 0.05 level of significance, is there evidencesupport the professor's claim?

b. Interpret the p-value.c. What assumption do you need to make in (a) about

two populations in order to justify your use of

Source: ExtractedfromT. Broome and D. Havelka, "Determinants

of Computer Anxiety in Business Students," The Review of BusinessInformation Systems, Spring 2002, 6(2), pp. 9-16.

puter anxiety, on a scale from 20 (no anxiety) to100 (highest level of anxiety). Researchers at MiamiUniversity administered CARS to 172 business students.One of the objectives of the study was to determinewhether there is a difference between the level of computeranxiety experienced by female students and male students.They found the following:

Males Females

.F test?

l-lsELFl 10.50 The Computer Anxiety Rating ScaleG (CARS) measures an individual's level of com-

40.26 36.8513.35 9.42

100 72

1tir\)cI

asI

x^tn

the 0.05 level of significance, is there evidence of ace in the variability of the computer anxiety

assumption do you need to make about the twoions in order to justiff the use of the f'test?

Based on (a) and (b), which I test defined in SectionI should you use to test whether there is a significant

ifference in mean computer anxiety for female andstudents?

A bank with a branch located in a commercial dis-of a city has developed an improved process for serv-

customers during the noon-to-1 p.m. lunch period. Theing time (defined as the time elapsed from when the

enters the line until he or she reaches the teller) of all customers during this hour is recorded overof one week. A random sample of 15 customers is(and stored in the file !@), and the results (in

) are as follows:

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 another branch, located in a residential area,concerned with the noon-to-l p.m. lunch period. Asample of l5 customers is selected (and stored in

file @@), and the results (in minutes) are as

9.66 5.90 8.02 5.79 8.73 3.82 8.01 8.35

10,49 6.68 5.64 4.08 6.17 9.91 5.47

Is there evidence of a difference in the variability of thewaiting time between the two branches? (Use a:0.05.)Determine thep-value in (a) and interpret its meaning.What assumption is necessary in (a)? Is the assumptionmlid for these data?Based on the results of (a), is it appropriate to use thepooled-variance I test to compare the means of the twobranches?

10.52 A problem with a telephone line thatIJJ prevents a customer from receiving or makingis disconcerting to both the customer and the tele-

company. The following data (stored in the filerepresent samples of 20 problems reported to two

t offices of a telephone company and the time tothese problems (in minutes) from the customers'lines:

Office I Time to Clear Problems (Minutes)

1 .75 0 .78 2 .85 0 .52 1 .60 4 .15 3 .97 1 .48 3 .10

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 .75 0 .10 l . l0 0 .60 0 .52 3 .30 2 . t0 0 .58 4 .02

3.75 0.65 t .92 0.60 1.53 4.23 0.08 1.48 1.65 0.72

a. Is there evidence of a difference in the variabilityof the waiting times between the two offices? (Usecr : 0.05.)

b. Determine thep-value in (a) and interpret its meaning.c. What assumption is necessary in (a)? Is the assumption

valid for these data?d. Based on the results of (a), which / test defined in

Section 10.1 should you use to compare the means ofthe two offices?

10.53 The director of training for a company that manu-factures electronic equipment is interested in determiningwhether different training methods have an effect on theproductivity of assembly-line employees. She randomlyassigns 42 recently hired employees to two groups of 21.The first group receives a computer-assisted, individual-based training program, and the other receives a team-based training program. Upon completion of the training,the employees are evaluated on the time (in seconds) ittakes to assemble a part. The results are in the data file

@.a. Using a 0.05 level of significance, is there evidence of a

difference between the variances in assembly times (inseconds) of employees trained in a computer-assisted,individual-based program and those trained in a team-based program?

b. On the basis of the results in (a), which r test defined inSection 10.1 should you use to compare the means ofthe two groups? Discuss.

10.54 Is there a difference in the variation of the yield ofdifferent types of investment between banks? The follow-ing data, from the fil" El@![f[, represent the nation-wide highest yields for money market accounts and one-year CDs as ofJanuary 24,2006:

Money MarketAccounts I One-Year CD

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 0.05 level of significance, is there evidence of a dif-ference in the variance of the yield between money marketaccounts and one-year CDs? Assume that the populationyields are normally distributed.

404 CHAPTER TEN Two-Samolc Tests

TA

S u rt n \

In this chapter, you were introduced to a variety of two-sar.nple tests. For situations in which the samples are inde-pendent, you learned statistical test procedures for analyz-ing possib le d i f ferences between means, var iances. andproportions. ln addition, you learned a test proceclure thatis frequently used when analyzing differences between themeans of two related samples. Rcmernber that you need to

select thc test that is most appropriate for a given set ofconditions and to crit ically investigate the validity of theassumpt ions under ly ing each of the hypothesis- test ingprocedures.

The roadmap in Figurc 10. l6 i l lustrates the stensneeded in determining which two-sar.nple test of hypothesisto use: Thc fbllowing are the questions you need to consider,

Catego r i ca l

Ztest for thed i f ference

between twoproport ions

Centra ITendency

o? = otrz

Two-SampleTests

Yes

Focus

Yes

Pooled-Variancef Test

N u m e r i c a l

IndependentSamples?

Var iab i l i t y

IZ '

Typeof

Data

N o

FTestr o r a l = o l

N o Pairedt Test

Sepa rate-Va riancet Test

FIGURE '10. '16 Roadmap for select ing a two-sampre test of hypothesis

l . What typc of data do you lrave'/ If you are clealing withcategorical variables, use the Z rest for the differenccbetween two proportions. (This test assumes indeoen_dent samples. . l

2. lf you have a numerical variable, deten.nine whetheryou have independent sar.nples or related samples. lfyou have related samples, use the paired I test.

3. Ifyou have independent samples, is your focus on vari-abil ity or central tendency'? If the focus is variabil itv.use the, t r test .

If your focus is central tendency, determine whetheryou can assume that the variances of the two groupsare equal . (This assurnpt ion can be tested us ing theF test . )If yoLr can assume that the two groups have equal vari-ances, use the pooled-var iance I test . I f you cannotassume that the two groups have equal variances, usethe separate-variance / test.

4 .

Table 10.8 provides a l ist of topics covered in this chapter.

Kcy Equations 405

T A B L E 1 0 . 8

Summary of Topicsin Chapter 10

Types of Data

Type of Analysis Numerical CategoricalIcf:le1g

ps;is

Comparing twopopulations

Z and t tests for the differencein the means of twoindependent populations(Sect ion 10. I )

Pai red I test (Sect ion 10.2)

F test for differences betweentwo variances (Section 10.4)

Z test for the differencebetween two proportions(Sect ion 10.3)

ZTest for the Difference Betwee n Two Means

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

Confidence Interval Estimate for the Mean Difference

( r0 .1 )

( r 0.2)

( r 0.3)

( r 0.4)

D-t,,-r#=u, <D+,,-,h

(10.6)f , )l o i o ;

r l r! /7t n2

for the DifferencePooled-Variance I Test

Between Two Means

- Xt ) - ($r - pz)

Confidence Interval Estimate of the Difference

in the Means of Two Independent Populations

Z Test for the Difference Between Two Proportions

S ( X , - X r l + t , , , . , , . _ ,

ZTest for the Mean Difference

z =D _ l tn

9nJ;

, _ ( p t - p z \ - ( n r - / t . )( r0.7)

Confidence Interval Estimate for the Difference

Between Two Proporttont

( .p t - pz )+z^ lp ,o - p , l *pzo- p : t (10 .8 )

l n t n 2or

( p t - p ) - Zplt - p)

n 1< ( n r - n z )

a ( h - p z ) + Zp t ( l - p r ) p 2 ( l - p 2 l

t1l t7-t

F Test Statistic for Testing the Equality of Two Variances

s.'f = - ; (10.9)

s:'

Finding Lower-Tail Crit ical Values

liom the F Distribution

p t ( l - p t )

n2

FEBherrpsthe

lri-

not

useDifference

D - V n

s,^Fn

- 1r t = -r[]*

Paired t Test for the Mean

(r 0.s)(10.10)

406 CHAPTER TEN Two-Sample Tests

F distribution 397.Ftest statistic for testing the equality

of two variances 397matched 381paired I test for the mean difference

in related populations 382

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

Z test for difference betweentwo means 370

Ztestfor the difference betweentwo proportions 390

c. What type of statistical test should you use?d. What assumptions are needed to perform the test

selected?e. Repeat (a) through (d) for research hypothesis 2.

10.63 The pet-drug market is growing veryBefore new pet drugs can be introduced into theplace, they must be approved by the U.S. Food andAdministration (FDA). In 1999, the Novartis companytrying to get Anafranil, a drug to reduce dog anxiapproved. According to an article (E. Tanouye, "The 0wBowwow: With Growing Market in Pet Drugs,Revamp Clinical Trials," The Wqll Street Journal, Api,l1999). Novartis had to find a wav to translate a doe'sety symptoms into numbers that could be used to provethe FDA that the drug had a statistically significant eon the condition.a. What is meant by the phrase statistically si

effect?b. Consider an experiment in which dogs suffering

anxiety are divided into two groups. One group willgiven Anafranil, and the other will be given a(that is, a drug without active ingredients). How cantranslate a dog's anxiety symptoms into numbers?other words, define a continuous variable, Xr,thesurement of effectiveness of the drug Anafranil, andthe measurement of the effectiveness of the placebo.

c. Building on your answer to part (b), define the nullalternative hypotheses for this study.

10.64 In response to lawsuits filed against theindustry, many companies, such as Philip Morris, arening television advertisements that are supposed toteenagers about the dangers of smoking. Are theseindustry antismoking campaigns successful? Aresponsored antismoking commercials more effective?article (G. Fairclough, "Philip Morris's AntismokiCampaign Draws Fire," The Wall Street Journal, Apri.l1999, p. B I ) discussed a study in California thatcommercials made by the state of California andcials produced by Philip Morris. Researchers showed the

t(f(

tiI\nS

poa

Checking Your Understanding10.55 What are some of the criteria used in the selectionof a particular hypothesis-testing procedure?

10.56 Under what conditions should you use the pooled-variance / test to examine possible differences in the meansof two independent populations?'10.57 Under what conditions should you use the F test toexamine possible differences in the variances of two inde-pendent populations?

10.58 What is the distinction between two indeoendentpopulations and two related populations?

10.59 What is the distinction between repeated measure-ments and matched (or paired) items?

10.60 Under what conditions should you use the paired rtest for the mean difference between two related populations?

10.61 Explain the similarities and differences betweenthe test of hypothesis for the difference between the meansoftwo independent populations and the confidence intervalestimate of the difference between the means.

Applying the Concepts10.62 A study compared music compact disc prices forInternet-based retailers and traditional brick-and-mortarretailers (Extracted from L. Zoonky and S. Gosain, 'A

Longitudinal Price Comparison for Music CDs in Electronicand Brick-and-Mortar Markets: Pricing Strategies inEmergent Electronic Commerce," Journal of BusinessSt rat e gi e s, Spring 2002, | 9 (1), pp. 5 5-7 2). B efore collectin gthe data, the researchers carefully defined several researchhypotheses, including:l. The price dispersion on the Internet is lower than the

price dispersion in the brick-and-mortar market.2. Prices in electronic markets are lower than prices in

physical markets.a. Consider research hypothesis L Write the null and alter-

native hypotheses in terms of population parameters.Carefully define the population parameters used.

b. Define a Type I and Type II error for the hypotheses in (a). state ads and the Philip Morris ads to a group of Californi

and measured the effectiveness of both. Theconcluded that the state ads were more effec-

in relaying the dangers of smoking than the Philipis ads. The article suggests, however, that the study is

statistically reliable because the sample size was tooand because the study specifically selected partici-who are considered more likelv to start smokine than

How do you think the researchers measured effective-

Define the null and alternative hypotheses for this study.Explain the risks associated with Type I and Type IIenors in this study.What type of test is most appropriate in this situation?What do you think is meant by the phrase statisticallyreliable2

The FedEx St. Jude Classic professional golftour-t is held each year in Memphis, Tennessee. FedEx

this PGA tournament, and part of the proceeds gothe St. Jude Children's Research Hospital. In 2003, the

raised $679,115 for the hospital. This type ofsponsorship is known as cause-related marketing.

sample of spectators at the tournament were surveyedasked to respond to a series of statements on a 5-point

(l = Strongly Disagree, 2 : Disagree, 3 : Neutral,Agree, and 5 : Strongly Agree). The following areofthe questions asked:

Cause-related marketing creates a positive companyimage.I would be willing to pay more for a service that sup-ports a cause I care about.Cause-related marketing should be a standard part of acompany's activities.Based on its support of St. Jude, I will be more likely touse FedEx services.each question, the researchers tested the null hypothe-

that the mean response for males and females is equal.alternative hypothesis is that the mean response is dif-

for males and females. The followine table summa-the results:

Sample Mean

Female Male(nr=137) (2, =305) t p-Value

Chapter Review Problems 407

a. Interpret the results ofthe r test for question 1.b. Interpret the results ofthe t test for question 2.c. Interpret the results ofthe t test for question 3.d. Interpret the results ofthe I test for question 4.e. Write a short summary about the differences between

males and females concernins their views toward cause-related sponsorship.

10.66 Two professors wanted to study how students fromtheir two universities compared in their capabilities of usingExcel spreadsheets in undergraduate information systemscourses (Extracted from H. Howe, and M. G. Simkin,"Factors Affecting the Ability to Detect Spreadsheet Errors,'oDecision Sciences Journal of Innovative Education, January2006, pp. l0l-122). A comparison of the student demo-graphics was also performed. One school is a state universityin the Western United States. and the other school is a stateuniversity in the Eastern United States. The following tablecontains information regarding the ages of the students:

SampleSchool Size

MeanAge

StandardDeviation

WesternEastern

93 23.28r35 21.16

MeanYears

6,291.32

StandardDeviation

a. Using a 0.01 level of significance, is there evidence of adifference between the variances in age of students atthe Western school and at the Eastern school?

b. Discuss the practical implications of the test performedin (a). Address, specifically, the impact equal (orunequal) variances in age has on teaching an undergrad-uate information systems course.

c. To test for a difference in the mean age of students, is itmost appropriate to use the pooled-variance / test or theseparate-variance I test?

The following table contains information regarding theyears ofspreadsheet usage ofthe students:

SampleSchool Size

WesternEastern

93135

2.64.0

2.42 .1

: Extractedfrom R. L. Irwin, T Lachowetz, T. B. Cornwell,J. S. Cook, "Cause-Related Sport Sponsorship: AnAssessment

:Spectator Beliefs, Attitudes, and Behavioral Intentions," Sport

d. Using a 0.01 level of significance, is there evidence of adifference between the variances in years of spreadsheetusage of students at the Western school and at theEastern school?

e. Based on the results of (d), use the most appropriate testto determine, at the 0.01 level of significance, whetherthere is evidence of a difference in the mean years ofspreadsheet usage ofstudents at the Western school andat the Eastern school?

4.464.094.264.12

4.26 1.907 0.0s73.86 2.105 0.0353.91 3.258 0.0014.06 0.567 0.571

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

Accounting 9Research 4

3 8 ' , 71 3 1 0 9


10.67 The manager of computer operations of a large

company wants to study computer usage of two depart-ments within the company-the accounting departmentand the research department. A random sample of five jobs

from the accounting department in the past week and sixjobs from the research department in the past week areselected" and the processing time (in seconds) for each job

is recorded (and stored in the l@$!! file):

Department Processing Time (in Seconds)

computer majors in order to determine whether therehevidence that computer majors can write a Visual Basisprogram in less time than introductory students. Forthccomputer majors, the sample mean is 8.5 minutes, andthe sample standard deviation is 2.0 minutes. At the 0.05level of significance, completely analyze these data,What will you tell the professor?

e. A few days later, the professor calls again to tell youthata reviewer of her article wants her to include thep-valuefor the "correct" result in (a). In addition, the professorinquires about an unequal-variances problem, which thereviewer wants her to discuss in her article. In your ownwords, discuss the concept of p-value and describe theunequal-variances problem. Determine the p-value in(a) and discuss whether the unequal-variances problemhad any meaning in the professor's study.

10.69 An article in USA Today (D. Sharp, "CellphonesReveal Screaming Lack of Courtesy," USA Today,September 2001, p. 4,{) reported that according to a poll,the mean talking time per month for cell phones was 372minutes for men and 275 minutes for women, while themean talking time per month for traditional home phoneswas 334 minutes for men and 510 minutes for women.Suppose that the poll was based on a sample of 100 men and100 women, and that the standard deviation of the talkingtime per month for cell phones was 120 minutes for menand 100 minutes for women, while the standard deviation ofthe talking time per month for traditional home phones was100 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 andwomen?

c. Construct and interpret a 95o/o confidence interval esti-mate of the difference in the mean monthly talking timeon cell phones for men and women.

d. Construct and interpret a 95o/o confidence interval esti-mate of the difference in the mean monthly talking timeon traditional home phones for men and women.

e. Is there evidence of a difference in the variance ofthemonthly talking time on cell phones for men andwomen?

f. Is there evidence of a difference in the variance of themonthly talking time on traditional home phones formen and women?

g. Based on the results of (a) through (f , what conclusionscan you make concerning cell phone and traditionalhome phone usage between men and women?

'lO.7O A survey of 500 men and 500 women designed tostudy financial tensions between couples asked how likelythey were to hide purchases, cash, or investments fromtheir partners. The results were as follows:

l 29 6

Use a level of significance of 0.05.a. Is there evidence that the mean processing time in the

research department is greater than 6 seconds?b. Is there evidence of a difference between the variances

in the processing time of the two departments?c. Is there evidence of a difference between the mean pro-

cessing time of the accounting department and that ofthe research department?

d. Determine the p-values in (a) through (c) and interprettheir meanings.

e. Construct and interpret a 95o/o confidence interval esti-mate of the difference in the mean processing timesbetween the accounting and research departments.

10.68 A computer information systems professor is inter-ested in studying the amount of time it takes students enrolledin the introduction to computers course to write and run aprogram in Visual Basic. The professor hires you to analyzethe following results (in minutes) from a random sample ofnine students (the data are stored in the![ft! file):

1 0 t 3 9 1 5 1 2 1 3 1 1 1 3 1 2

a. At the 0.05 level of significance, is there evidence thatthe population mean amount is greater than 10 minutes?What will you tell the professor?

b. Suppose the computer professor, when checking herresults, realizes that the fourth student needed 5l min-utes rather than the recorded l5 minutes to write and runthe Visual Basic program. At the 0.05 level of signifi-cance, reanalyze the question posed in (a), using therevised data. What will you tell the professor now?

c. The professor is perplexed by these paradoxical resultsand requests an explanation from you regarding thejus-tification for the difference in your findings in (a) and(b). Discuss.

d. A few days later, the professor calls to tell you that thedilemma is completely resolved. The original number 15(the fourth data value) was correct, and therefore yourfindings in (a) are being used in the article she is writingfor a computer journal. Now she wants to hire you tocompare the results from that group of introduction tocomputers students against those from a sample of 1l

Likely to Hide

Extractedfrom L. Wei, "Your Money Manager as Financial" The Wall Street Journal, November 5-6, 2005, p. 84.

For each type of purchase, determine whether there is arence between men and women at the 0.05 level of

ificance.

As more Americans use cell phones, they questionit is okay to talk on cell phones. The following is a table

results, in percentages, for 2000 and2006 (extracted fromKoch, "Business Put a Lid on Chatterboxes i USA Tbday,

7, 2006, p. 3A). Suppose the survey was based onrespondents in 2000 and 100 respondents in 2006.

TO TALK ON A CEIL PHONE IN A 2OOO 2006

Chapter Review Problems 409

10.74 Management of a hotel was concerned withincreasing the return rate for hotel guests. One aspect offirst impressions by guests relates to the time it takes todeliver the guest's luggage to the room after check-in to thehotel. A random sample of 20 deliveries on a particular daywere selected in Wing A of the hotel, and a random sampleof 20 deliveries were selected in Wing B. The results arestored in the file lEtl*lElltl. Analyze the data and deter-mine whether there is a difference in the mean deliverytime in the two wings of the hotel (use o(: 0.05).'10.75 In manufacturing processes, the term work-in-process (often abbreviated WIP) is often used. In a bookmanufacturing plant, WIP represents the time it takes forsheets from a press to be folded, gathered, sewn, tipped onend sheets, and bound. The following data (stored in thefile @$) represent samples of 20 books at each of twoproduction plants and the processing time (operationallydefined as the time, in days, from when the books came offthe press to when they were packed in cartons):

Plant A

5.O 5.29 16.2s t0.92 1r.46 21.62 8.45 8.58 5.4r 11.42

rr.62 7.29 7.s0 7.96 4.42 10.s0 7.s8 9.29 7.54 8.92

Plant B

9.54 11.46 16.62 12.62 25.75 15.41 14.29 13.13 13.71 10.04

5.75 12.46 9.17 r3.2r 6.00 2.33 14.25 5.37 6.25 9.11

Completely analyze the differences between the processingtimes for the two plants, using o:0.05, and write a sum-mary of your findings to be presented to the vice presidentfor operations of the company.

10.76 Do marketing promotions, such as bobble-headgiveaways, increase attendance at Major League Baseballgames?An article reported on the effectiveness of marketingpromotions (extracted from T. C. Boyd and T. C. Krehbiel,'An Analysis of Specific Promotion Types on Attendance atMajor League Baseball Games," Mid-American Journal ofBusiness, 2006, 21, pp. 21,-32). The data file [!!@includes the following variables for the Kansas City Royalsduring |he2002 baseball season:

Game-Home games, in the order in which they wereplayedAttendance-Paid attendance for the gamePromotion-l : if a promotion was held; 0: if no pro-motion was held

a. At the 0.05 level of significance, is there evidence of adifference between the variances in the attendance atgames with promotions and games without promotions?

b. Based on the result of(a), conduct the appropriate test ofhypothesis to determine whether there is a difference inthe mean attendance at games with promotions andgames without promotions. (Use a: 0.05.)

c. Write a brief summarv of vour results.

Men

66t26627996747653

YEAR

39l l76605231

382

636645z l

For each type of location, determine whether there is afference between 2000 and 2006 in the proportion who

it is okay to talk on a cell phone (use the 0.05 level ofificance).

72 The lengths of life (in hours) of a sample of 40light bulbs produced by manufacturer A and a

e of 40 100-watt light bulbs produced by manufac-B are in the file![!$fr. Completely analyze the dif-

between the lengths of life of the bulbs producedtwo manufacturers (use cr: 0.05).

73 The data file [!!@@!E contains the ratings fordecor, service, and the price per person for a sample

50 restaurants located in an urban arca and,50 restau-located in a suburban area. Completely analyze thences between urban and suburban restaurants for the

iables food rating, decor rating, service rating, and priceperson, using c: 0.05.

: Extractedfrom Zagat Survey 2002: NewYork Cityand Zagat Survey 200 l-2002: Long Island Restaurants.


10,77 The manufacturer of Boston and Vermont asphaltshingles knows that product weight is a major factor in thecustomer's perception of quality. Moreover, the weight rep-resents the amount of raw materials being used and istherefore very important to the company from a cost stand-point. The last stage of the assembly-line packages theshingles before they are placed on wooden pallets. Once apallet is full (a pallet for most brands holds l6 squares ofshingles), it is weighed, and the measurement is recorded.The data filep@[!contains the weight (in pounds) froma sample of 368 pallets of Boston shingles and 330 palletsofVermont shingles. Completely analyze the differences inthe weights of the Boston and Vermont shingles, using cr :

0.0s.

10.78 The manufacturer of Boston and Vermont asphaltshingles provides its customers with a 2)-year warranty onmost of its products. To determine whether a shingle willlast as long as the warranty period, accelerated-life testingis conducted at the manufacturing plant. Accelerated-lifetesting exposes the shingle to the stresses it would be sub-ject to in a lifetime of normal use in a laboratory setting viaan experiment that takes only a few minutes to conduct. Inthis test, a shingle is repeatedly scraped with a brush for ashort period of time, and the shingle granules removed bythe brushing are weighed (in grams). Shingles that experi-ence low amounts of granule loss are expected to lastlonger in normal use than shingles that experience highamounts of granule loss. In this situation, a shingle shouldexperience no more than 0.8 grams of granule loss if it isexpected to last the length of the warranty period. The datafile !@ contains a sample of 170 measurementsmade on the company's Boston shingles and 140 measure-ments made on Vermont shingles. Completely analyze thedifferences in the granule loss of the Boston and Vermontshingles, using cr: 0.05.

Repoft Writing Exercise10.79 Referring to the results of Problems 10.77 and10.78 concerning the weight and granule loss of Bostonand Vermont shingles, write a report that summarizes yourconclusions.

Team ProjectThe data file @tIEslE contains information regard-ing nine variables from a sample of 838 mutual funds. Thevariables are

Category-Type of stocks comprising the mutual fund(small cap, mid cap, or large cap)Objective-Objective of stocks comprising the mutualfund (growth or value)Assets-In millions of dollarsFees-Sales charges (no or yes)Expense ratio-Ratio of expenses to net assets, inpercentage

2005 return-Twelve-month return in 2005Three-year return-Annualized return, 2003-2005Five-year return-Annualized return, 200 12005Risk-Risk-of-loss factor of the mutual fundaverage, or high)

10.80 Completely analyze the difference between mutualfunds without fees and mutual funds with fees in terms2005 return. three-vear return. five-vear return.expense ratio. Write a report summarizing your findings.

10.81 Completely analyze the difference between mutmlfunds that have a growth objective and mutual funds thathave a value objective in terms of 2005 return, three-yeureturn, five-year return, and expense ratio. Write a reportsummarizing your findings.

Student Survey Data Base10.82 Problem 1.27 on page 15 describes a survey of 50undergraduate students (see the file GEEEElilil$ED.For these data,a. at the 0.05 level of significance, is there evidence ofa

difference between males and females in grade pointaverage, expected starting salary, salary expected in fiveyears, age, and spending on textbooks and supplies?

b. at the 0.05 level of significance, is there evidence ofadifference between those students who plan to go tograduate school and those who do not plan to go to grad-uate school in grade point average. expected startingsalary, salary expected in five years, age, and spendingon textbooks and supplies?

10.83 Problem 1.27 on page 15 describes a survey of 50undergraduate students (see the file@![[[!@!!f$.a. Select a sample of 50 undergraduate students at your

school and conduct a similar survey for them.b. For the data collected in (a), repeat (a) and (b) of

Problem 10.82.c. Compare the results of (b) to those of Problem 10.82.

10.84 Problem 1.28 on page 15 describes a survey of50MBA students (see the file f.EEllllllElll$. For these data,at the 0.05 level of significance, is there evidence of a dif-ference between males and females in age, undergraduategrade point average, graduate grade point average,GMAT score, expected salary upon graduation, salaryexpected in five years, and spending on textbooks andsupplies?

10.85 Problem 1.28 on page 15 describes a survey of50MBA students (see the fileFEElliltEliE$.a. Select a sample of 50 graduate students in your MBA

program and conduct a similar survey for those students.b. For the data collected in (a), repeat Problem 10.84.c. Compare the results of (b) to those of Problem 10.84.

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40.738.043.643.834.935.741.4

'Managing the Springville Herald

A marketing department team is charged with improvingthe telemarketing process in order to increase the numberof home-delivery subscriptions sold. After several brain-$orming sessions, it was clear that the longer a callerspeaks to a respondent, the greater the chance that thecaller will sell a home-delivery subscription. Therefore, theteam decided to find ways to increase the length of thephone calls.

Initially, the team investigated the impact that the timeof a call mrght have on the length of the call. Under currentarrangements, calls were made in the evening hours,between 5:00 p.m. and 9:00 p.m., Monday through Friday.The team wanted to compare the length of calls made earlyin the evening (before 7:00 p.m.) with those made later inthe evening (after 7:00 p.m.) to determine whether one ofthese time periods is more conducive to lengthier calls and,

Web Case 411

correspondingly, to increased subscription sales. The teamselected a sample of 30 female callers who staff the tele-phone bank on Wednesday evenings and randomlyassigned 1 5 of them to the "early" group and l 5 to the"later" group. The callers knew that the team was observ-ing their efforts that evening but didn't know which callswere monitored. The callers had been trained to make theirtelephone presentations in a structured manner. They wereto read from a script, and their greeting was personal butinformal ("Hi, this is Mary Jones from the Springvil leHerald. May I speak to Bill Richards?").

Measurements were taken on the length of the call(defined as the difference, in seconds, between the time theperson answered the phone and the time he or she hungup). The results (stored in the file !fff[!) are presented inTab le SH l0 . l .

Time of Call

EarlyLate

ualhat'ear

)ort

' 50

D.

r f arintive

r f ar t O

ad-inging

50

f,ur

of

50rta,tif-ateg€,fiy

lnd

T A B L E S H 1 O . 1

Length of Cal ls, inSeconds, Based onr ' r - t lr i l i l g v t 9 o i l - L q t t y

Versus Late in theEvening

Time of Call

Early

EXERCISESSHIO.1 Analyze the data in Table SHl0. l and wr i te a

report to the rnarketing department team that indi-cates your findings. Include an attached appendixin which you discuss the reason you selected a par-ticular statistical test to compare the two indepen-dent groups ofcallers.

SH10.2 Suppose that instead of the research designdescribed here, there were only 15 callers sampled,

Web Case

Apply ltour knowledge about hypothesis testing in this WebCase, which continues the cereal-fill packaging disputeWeb Cuse fiom Chapters 7 and 9.

and each caller was to be monitored twice in theevening-once in the early time period and oncein the later t ime period. Suppose that in TableSH I 0. I, each pair of values represents a particularcaller's two measurements. Reanalyze these dataand write a report for presentation to the team thatindicates your findings.

SHl0.3 What other variables should be investisated next?whv?

Even after the recent public experiment about cerealbox weights, the Consumers Concerned About CerealCheaters (CCACC) remains convinced that Oxford Cereals

412 CHAPTER TEN Two-Sample Tests

has misled the public. The group has created and posted adocument in which it claims that cereal boxes produced atPlant Number 2 in Springville weigh less than the claimedmean of 368 grams. Visit the CCACC More Cheating pageat www.prenhall.com/Springville/MoreCheating.htm (oropen this Web page file from the text CD's Web Case folder)and then answer the following:

l . Do the CCACC's results prove that there is a statidifference in the mean weights of cereal boxesat Plant Numbers I and2?

) Perform the appropriate analysis to test the CChypothesis. What conclusions can you reach based ondata?

l. Conover, W. J., Practical Nonparametric Statistics,3rded. (New York: Wiley, 2000).

2. Daniel, W., Applied Nonparametric Statistics, 2nd ed.(Boston: Houghton Mifilin, 1990).

3. Microsoft Excel 2007 (Redmond, WA: Microsoft Corp.,2007).

4. Satterthwaite, F. E., 'An Approximate Distribution ofEstimates of Variance Components l' Biometrics Bulletin,2(1946): I 10-1 14.

5 . Snedecor. G. W." and W. G. Cochran. Statistical8th ed. (Ames. IA: Iowa State UniversiW Press, 1989).

6. Winer. B. J.. D. R. Brown. and K. M. Michels.Principles in Experimental Design,3rd ed. (NewMcGraw-Hill. 1989).

Excel Companion to Chapter 10 413

S

t ,

You can use Microsoft Excel to conduct all six two-samplehypothesis tests discussed in Chapter 10. Some tests requirethe use ofToolPak procedure. Others require that you usePHStat2 or one of the workbooks stored on the StudentCD-ROM. Some tests have Excel versions in which you canuse only unsummarized data, and others have Excel versionsin which you must use summarized data. (A few tests haveExcel versions for either unsummarized or summarized data.)

Use Table El0.l below to help choose the right test foryour data and as a guide for the rest of this Excel Companion.

Two-Sample Data Arrangements

Unsummarized data can be entered in a stacked orunstacked arrangement. In a stacked arrangement, all thevalues for a variable appear in a single column, next to acolumn that identifies the sample or group to which individ-ual values belong. In an unstacked arrangement, the valuesfor the samples or groups appear in separate columns. Forexample, Figure El0. I shows the stacked and unstackedversions of the cola display location sales analysis data ofTable l0 . l onpage372.

T A B L E E 1 0 . 1

Two-Sample Testsin Microsoft Excel

iEI 5511 i 59

527176

17 En&Aislete;fn*niste19 iEn&Aisle4-_En&Aisle2*1 'End,Aisle

FIGURE E10.1 Stacked and unstacked data

_. ._ A""r , * " -A_ i-1 JQIqllp Value2 rNormal 722 rNormal 223 : N o r m a l 3 4

6_, iNormal7 ,Normal8 : N o r m a l

406484

62s406'f84564q

54678366snB4

52a 1T I

7654678366w7784

2 t 7 23 i 3 4"-i-;

52; l3 d l

Gl 30

_9 :Normall 0 ,No rma l11 rNormaliz irno-nistei: ieno-gistetl lrn+ruste

Two-Sample TestUnsummarized Data(Excel Companion Section)

Summarized Data(Excel Companion Section)

Z test for the differencebetween two means

ToolPak z-Test: Two Samnlefor Means (E 10. I )

Z'lwo Means.xls or PHStat2Z Test for Differences inTwo Means (E10.2)

Pooled-variance / testfor the differencebetween two means

ToolPak t-Test: Two-SampleAssuming Equal Variances(E10.3)

Pooled-variance T.xls orPHStat2 t Test for DifferencesBetween Two Means (E10.4)

Separate-variance I testfor the differencebetween two means

ToolPak t-Test: Two-SampleAssuming Unequal Variances(Er0.s)

Not included

Paired I test ToolPak t-Test: Paired TwoSample for Means (E10.6)

Not included

Z test for the differencebetween twoproportions

Not included ZTwo Proportions.xls or PHStat2Z Test for Differences BetweenTwo Proportions (E 10.7)

,F test for the differencebetween two variances

ToolPak F-Test Two-Samplefor Variances (E10.8)

F Two Variances.xls orPHStat2 F Test for DifferencesBetween Two Variances t'E10.9wo Variances (E10.9)

-l

4I4 EXCEL COMPANION to Chanter 10

Specific statistical procedures or Excel worksheets foranalyses involving two or more groups require that the databe arranged either as stacked or unstacked data. The two-sample hypothesis tests for unsummarized data discussedin this E,xcel Companion require that your data beunstacked.

If you need to change stacked data into its unstackedequivalent, you can sort the data by sample or group andthen cut and paste the second sample's data to a new col-umn. Likewise, to stack unstacked data, you can copy thedata from the second sample directly below the first sampleand then add a column that identifies the group.

PHStat2 can automate these tasks by using either theUnstack Data or Stack Data procedures found in thePHStat2 Data Preparation submenu.

E10.1 USING THE Z TEST FOR THEDIFFERENCE BETWEEN TWOMEANS (UNSUMMARTZED DATA)

For unsummarized data, you conduct a Ztest for the differ-ence between two means by selecting the ToolPak z-Test:Two Sample for Means procedure.

Open to the worksheet that contains the unsummarizeddata for the two samples. Select Tools ) Data Analysis,select z-Test: Two Sample for Means from the DataAnalysis l ist, and click OK. In the procedure's dialog box(shown below), enter the cell range of one sample as theVariable 1 Range and the cell range of the other sample asthe Variable 2 Range. Enter the Hypothesized MeanDifference, the population variance of the first sample asthe Variable I Variance (known), and the population vari-ance of the second sample as the Variable 2 Variance(known). Select Labels and click OK. Results appear on anew worksheet.

l,.r r: f--;--l

variabb lRaf@: lLl L:=:Ii.--jJ

varribbaRerqc: lE t- c*d I

t"Urothsii:ed l"1a.n Ddffefcrcor I Hdt l

Va|l$lc 1Y6ri!nc6(knM);

Yafii?bh 2 vqiffie (krnm):

mL&b

Aloha: 0,05

i-uiiil! ifrtloftt

r. .l Qutp* nanga: 1"'tl*wqkgh8otflyl

i. ! Novr Uorkbsok

E1O.2 USING THE Z TEST FOR THEDIFFERENCE BETWEEN TWOM EANS (SUMMARTZED DATA)

For sunrmarized data, you conduct a Z test for the diffu-ence between two means by either selecting the PHStat2 ZTest for Differences in Two Means procedure or by makingentries in the !@fll@[! workbook.

Usinq Pl'{Stat2 Z Test for theDiffirence Betu/een Two Means

Select PHStat ) Two-Sample Tests ) Z Test forDifferences in Two Means. In the Z Test for theDifferences in Two Means dialog box (shown below), enterthe Hypothesized Difference and, if necessary, change theLevel of Significance. Enter the sample size, samplemean, and population standard deviation for the population1 sample and the population 2 sample. Click one of the te$options. enter a tit le as the Title, and click OK.

FIZ

Dat6

Hypothesized Dif f erenca :

L*vd of S[rficarrce;

Pogrlation X Sampb

Sam$e Siee:

Sam$e lvleanr

Fapdatbn Std. Deviation:

Pogrl$bn 2 Samde

Smph Sze:

Sam$e l*tem:

Poptdatbn 5td. Devi*imr

r*-tr*-i

II

i " " - - ' " '

1

f""*"--*" ''

Test Optbnsi; T$ro-Td Testi l.lppar-Tail Testi . Lorarer-Tail Te*

0r"rtp.rt O$icns

Ttlsr 1

Csrcel :

I

I!

l

l

!I

l!I

II

i

'er-

)_zi n o

forthelterthepleiontest

FIGURE E1O.2Z Two Means worksheet

Using Z Two Means.xls

Open to the ZTwo Means worksheet of theEIEEEEEEworkbook. This worksheet (see Figure E10.2) uses theNORMSINV?<h function to determine the lower andupper critical values and the NORMSDIST(Z value) func-tion to compute the p-values from the Z value calculated incel l B18. The worksheet conta ins the entr ies to solveProblem l0.l on page 377. To adapt this worksheet to otherproblerns, change, as is necessary, the hypothesized differ-ence and level of significance in cells 84 and B5 and thesample size, sample rnean and population standard deviationof each sample in the cell range 87:89 and B 1 1 :B 13.

lf you want to use this worksheet to show only one test,first make a copy of the worksheet (see the Excel Companionto Chapter l). For a two-tail test, select and delete rows 26through 34. For a lower-tail-test-only worksheet, select anddelete rows 31 through 34 and then select and delete rows 20through 25.For an upper-tail-test-on1y worksheet, select anddelete rows 20 through 30.

E10.3 USING THE POOLED-VARIANCEt TEST (UNSUMMARIZED DATA)

For unsummarized data, you conduct a pooled-variance Itest by selecting the ToolPak t-Test: Two-Sample AssumingEqual Variances procedure.

Open to the worksheet that contains the unsummarizeddata for the two samples. Select Tools t Data Analysis,select t-Test: Two-Sample Assuming Equal Variances

E I 0.4: Using the Pooled-Variance / Test ( Summarized Data) 41 5

-88 " 812-s0RT(89^ 2)rB7) + (813^ 2)/811).(816 - 8{f417

-NOftllSltlV{85,21-lroRlitstNv(l - (852)|-2' (1 - N0RtlSDlSr(ABs(Bl8))-lF(823 < 85, "Rojecr th. trull hypothesb".

'0o not r.jocl tho null hyporhosisJ

-NORHS|l{VG5}-iloRfls0tsT(Br8)-lF{828 < 85.

'Sejocr tlr€ n$ll hypothesls".

-Do oot roJect lho ntrll htpothGb]

-l{oRrsltlv{1 ,85)-1 - lroRusolsT(B18)-lF(833 < 85. "R.Joct tho troll lrypothods",

"Do not rqjoct ths null hypotheslr")

from the Data Analysis list and click OK. In the procedure'sdialog box (shown below), enter the cell range of one sampleas the Variable I Range and the cell range of the other sam-ple as the Variable 2 Range. Enter the Hypothesized MeanDifference, click Labels, and click OK. Results appear on anew worksheet. Figure 10.3 on page 373 shows the resultsfor the Table 10. I BLK cola sales data.

Xit*r*"*", tq1 m(llvebueaRane.: tE t- c"*d I

HypottErizdtvte.nutference: T E'eh --l

nt beb

Abh.: 0.05

- . lDut tFh' l r ' r

{-i: Eutprf nonqc: '!-

('.: Nes wwkshset Ply:

i-I Neei, gwkbook

E10.4 USING THE POOLED.VARIANCEt TEST (SUMMARTZED DATA)

For sumrnar ized data, you conduct a pooled-var ianceI test by either using the PHStat2 t Test for Differencesin Two Means procedure or by making entr ies in thennnFf,lElEil*inn workboo k.

1234567F

910t 112I J

1 11 5t o

1 71 81 92021n

x

27masJ I

a$

416 EXCEL CoMPANIoN to Chaoter lo

Usinq PHStat2 t Test for theDiffe-rences in Two Means

Select PHStat ) Two-Sample Tests ) t Test forDifferences in Two Means. In the t Test for Differences inTwo Means dialog box (shown at right), enter values for theHypothesized Difference and the Level of Significanceand then enter the Sample Size, Sample Mean, andSample Standard Deviation for the population I sampleand the population 2 sample. Click one of the test options,enter a tit le as the Title, and click OK. To include a confi-dence interval estimate of the difference between the twomeans (similar to one found in the PVt worksheetdescr ibed below), c l ick Conf idence Interval Est imatebefore you click OK.

Using Pooled-Variance T.xls

Open to the PVt worksheet of the fi!![!!f,lftfifiEEworkbook. This worksheet (see Figure E10.3) uses theTINY(l-confidence level, degrees of freedom) function todetermine the lower and upper critical values. To computethe p-values, this worksheet uses the TDIST(ABS(I),degrees offreedom,fails) function, in which ABS(/) is theabsolute value of the I test statistic , and tsils is either 1, fora one-tail test, or 2. for a two-tail test. The worksheet usesthe IF function in cells 83l and 836 to determine whichone of two values computed in a calculations area (notshown in Fisure E 10.3) to use.

FIGURE E10.3PVt worksheet

12J

5

7

I10l l't2

1 31 41 5'16

1a

1 81 9n2122

21x677aa303112

353517

- 8 7 - 1- 8 1 1 . 1-816 r 817-(8i6' 89^2) + (!17'813^2))i818-88 - 812-{820 - 81/SARTG19' {t/87 + 1lB11)}

--(nNV(85,818))-nNvG5,818)-TDIST(Atss{821}, 818, 2}-lF{826 < 85, 'Roj.cl lhe null hypothcair",

-Do rot feJoct tho ntrll hypolh$bJ

--rNV@'.85,810))-lF(821 < 0, E32. E33)-lF(B3l < 85, "Rejsd tho n{ll hypothGb-,

'Do not rolost th. null hypotheeir']

-Ftilve " 85 ,Bt8)l-lF(82t < 0, E33, E32l-lF(836 < 85, 'Rojocl the noll hyporh*b",

-Do tlot rejoct lhe n(ll hyporhosls'JNot shown

Cell E32: -TD|5T(ABS{B2i}.818, l)Cell F33r -l . E32

DSa

Hypothcsi:ed Of f ererre r

Lcvd of Sgrificmcel

Potriatbn I 5am*Sarnfle $zer

5anfle tban:

Sanrpb St*dard Deviathnl

Fotridion Z tumpb

Smdc Size:

5anpb l,!ean:

Sanpb Standard Deviatbnl

TEst OptloftE{; T$,o-Tal Test- upper-rr*l restd- Lor,vcr-Tdl Tcst

A*pt Options

Ttlcr l'**-*-- "

[- CorflJerre lr*sval Estinate

f"

H€b i li oK il cara i

t810.7: Using the ZTest for the Difference Between Two Proportions (Summarized Data) 417

The worksheet contains entries based on the Table I 0.1

BLK cofa sales data on page 312.To adapt this worksheet

to other problems, change, as necessary, the hypothesized

difference, level of significance, and sample statistics ofthe two samples in cel ls 84, 85, Bl :89, and Bl l :B13. I fyou do not want to include a confidence interval estimate

in your worksheet (see Figure E 10.4), select and delete the

cel l range D3:E16.

-816 + 817-Tltlv(l - F/, E10)-(E11' SoRT(819 " (t/87 + 1,'811)|l

-82{l " E12-820

" E12

FIGURE E10.4 Conf idence interval est imate area

E10.5 USING THE SEPARATE.VARIANCE t TEST FOR THEDIFFERENCE BETWEEN TWOMEANS (UNSUMMARIZED DATA)

For unsummarized data, you conduct a separate-variance /

test for the difference between two means by selecting the

ToolPak t-Test: Two-Sample Assuming Unequal Variancesprocedure.

Open to the worksheet that contains the unsummarizeddata for the two samples. Select Tools ) Data Analysis,

select t-Test: Two-Sample Assuming Unequal Variancesfrom the Data Analysis l ist, and click OK. In the proce-

dure's dialog 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 theHypothesized Mean Difference, click Labels, and click

0K. Figure 10.6 on page 377 shows the results of applyingthis procedure to the Table 10. 1 BLK cola sales data.

InFut F==;-".--lVdladelRffsc: lLl \sdnin@rj

lrdiadceRtrq.r tE t c"*"1 I

il@hgdzcdl,bfiDifferc'rrar l- H"h_l

trpactgPha: 0.6

Output oFtrorrs

Cguprrnrnc: L

O iFr wortdnct gfyr

O tlaw Watloot

E10.6 USING THE PAIRED tTEST FORTHE DIFFERENCE BETWEENTWO MEANS (UNSUMMARIZEDDATA)

For unsummarized data, you conduct a paired l test for thedifference between two means by selecting the ToolPakt-Test: Paired Two Sample for Mean procedure.

Open to the worksheet that contains the unsummarizeddata for the two samples. Select Tools ) Data Analysis,select t-Test: Paired Two Sample for Means from theData Analysis l ist and click OK. In the procedure's dialogbox (shown below), enter the cell range of one sample asthe Variable I Range and the cell range of the other sam-ple as the Variable 2 Range. Enter the HypothesizedMean Di f ference, c l ick Labels, and c l ick OK. Resul tsappear on a new worksheet. Figure 10.8 on page 385 showsthe results for the Table 10.4 car milease data.

rnrlur __=:-:]verirdJeiRarqe: tr =varraileaRar{r: tr [ c"*al

Hypoth6izrdmamDffarcrra T eb_l

trr*Abhar 0.6

lu lp,r f ( 'pfr i r ' i

|)gu$*nsrr: 1lo

,.1 New Wd.shcet gy:

i) ttar Wrtooor

E1O.7 USING THE Z TEST FOR THEDIFFERENCE BETWEEN TWOPROPORTTONS (SU M MARIZEDDATA)

For sunrmarized data, you conduct a Z test for the differencebetween two proportions by either selecting the PHStat2ZTest for Differences in Two Proportions procedure or bymaking entries in the E@E@[[E workbook.

Using PHStat2 Z Test forDifferences in Two Proportions

Select PHStat t Two-Sample Tests ) Z Test forDifferences in Two Proportions. In the Z Test forDifferences in Two Proportions dialog box (shown on page

418), enter values for the Hypothesized Difference and theLevel of Significance. Enter the Number of Successes andthe Sample Size for the population 1 sample and the popu-

lation 2 sample. Click one of the test options, enter a tit le asthe Title. and click OK. To include a confidence interval

estimate of the difference between the two proportions

418 EXCEL coMPANIoN to chanter lo

(simi lar to one shown in Figure El0.6 below), c l ickConfidence Interval Estimate before you click OK.

0Sa

Hlrpotha$zed Daffererrc:

Lcvd d ggnficmcc:

Populatbn 156nr*

lfumbcr d Srccsscsr

Sanple 5be:

Potrirtiln 2 sanplt

iflrber of Sr-acesasl

Sampla ftar

Test @tklnsi3 trr,n-tC test

r tffi-TdTest

f torarer-Teil Test

Qfrput Sions

r-lfis

f-

Ttle: i[* Corfldence lr*ervd E*inate

| e ' - - - - - ' - - - a l I

li ox il Csrcd i

FIGURE E10.5Zf P_Al l s ingle-tai ltest a rea

FIGURE E10.6ZTP_All confidenceinterval estimate area

I tst ? c

Iul a -

321FjI3 4 i

Using Z Two Proportions.xls

You open and use either the ZTP_TT or the ZTPworksheets of the EEEEEEEEEE workbook toa Z test for the difference between two proportions.worksheets use the NORMSINV@<$ function tomine the lower and upper crit ical values, andNORMSDIST(Z value) function to compute the p-from the Z value calculated in cell Bl8. To betterstand how a message gets displayed in theseread the 'About the IF function" part of Section E9.lpage 364.

The ZTP_TT worksheet (see Figure 10.11 on393) applies the two-tail Z test to the Section 10.3guest satisfaction example. The ZTP_All worksheetboth the upper-tail and lower-tail test (see Figure Eplus a confidence interval estimate of the diffebetween the two proportions (see Figure E10.6). Tothese worksheets to other problems, change, if nthe hypothesized difference, level of significance,number of successes and sample size for each samplece l l s 84 , 85 , 87 , 88 , B l0 , and B l1 .

lf you want the ZTP_Allworksheet to show onlyof the single-tail tests, f irst make a copy of thatsheet (see the Excel Companion to Chapter l ) . Forlower-tail-test-only worksheet, select and delete rowsthrough 34 and then select and delete rows 20 throughFor an upper-tail-test-only worksheet, select androws 20 through 30. If you do not want to include adence interval estimate in vour worksheet. delete colD and E.

-iloRrslilv(a5,-iORISDTSI/Atr,-rF(W < Ao, .R.l.c.

h. natt htt"oth.*-,-Do

not ,.1.d a/'. ,rutt ,rrrpottto&.1

-lloRilsll{v(iA5}-1 -l{oRllSDtST(Brsl-lF(833 < 85, -Rojecilho

null hypoth$fs,.,-Do not rorocl the null hyporhedr.)

-NoRISll{v(t . E7}z}-sORT((81a '(t . Btll/EB) . t8ts . n-ABS{E10'El t l

' 1t :

45

7R

atio1 1i2ti-1 4t-s15

8r5)/811)l

E I 0.9: Using the F Test for the Difference Between Two Variances (Summarized Data) 419

ilrt

r-IC

)S

L -

c

n

E10.8 USING THE F TEST FOR THEDIFFERENCE BETWEEN TWOVARTANCES (UNSUM MARTZEDDATA)

For unsummarized data, you conduct an -F test for the dif-ference between two variances by selecting the ToolPakF Test Two-Sample for Variances procedure.

Open to the worksheet that contains the unsummarizeddata for the two samples. Select Tools ) Data Analysis,select F Test Two-Sample for Variances from the DataAnalysis list, and click OK. In the procedure's dialog box(shown below), enter the cell range of one sample as theVariable I Range and the cell range of the other sample asthe Variable 2 Range. Click Labels and click OK. Resultsappear on a new worksheet. Figure 10. l4 on page 400 showsresults for the Table l0.l BLK cola sales data on page 372.

,l'"*r***, E f-Iil:ilv*trtlc3Raser tr Tc""d I

trL.bcb {- Hrb -l

Abha: 0.05

fiJtput options

O q*f'tf nr,ge: t

e Nall wortshcet Pbr

Om+Uctooor

E10.9 USING THE F TEST FOR THEDIFFERENCE BETWEEN TWOVAR|ANCES (SUMMARTZEDDATA)

For summarized data, you conduct an F test for the differ-ence between two variances by either selecting the PHStat2F Test for Differences in Two Variances procedure or bymaking entries in the EEEEEEIEEE workbook.

Using PHStat2 F Test forDifferences in Two Variances

Select PHStat ) Two-Sample Tests ) F Test forDifferences in Two Variances. In the F Test forDifferences in Two Variances dialog box (shown below),enter the Level of Significance and then enter the SampleSize and the Sample Standard Deviation for the popula-tion I sample and population 2 sample. Click one of thetest options, enter a title as the Title, and click OK.

Data

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Using F Two Variances.xls

Open to the F Two Var iances worksheet of the

tr@Eworkbook. This worksheet conducts anF test for the difference between two variances for theSec t i on 10 .4 BLK co la sa les examp le . The workshee t(see Figure E10.7) uses the function FINV(rpper-tailedp-value, numerstor degrees of freedom, denominatordegrees of freedom), in which upper-tuiled p-value isthe probabil ity that tr wil l be greater than the value, tocompute the upper and lower crit ical values and uses thefunction FDIST(F-/est statistic, nunterstor degrees of

freedom, denominutor degrees of freedom) to computethe p-values.

FIGURE ElO.7F Two Variancesworksheet

To adapt this worksheet to other problems, change, ifnecessary, the level of significance and the sample statis-tics for the two population samples in the tinted cells 84,86, B7 , 89, and B 10. If you want the worksheet to showonly a single test, first make a copy of that worksheet (seethe Excel Companion to Chapter l). For a two-tailtest-only worksheet, select and delete rows 23 through 31. Foralower-tail-test-only worksheet, select and delete rows 28through 3l and then select and delete cell range A17:821,For an upper-tail-test-only worksheet, select and deleterows 23 through 2T and then select and delete cell rangeAl7:821.

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-FltlV(l . Bl2, B'tl, 815l-FINV(81/2, B1{,81O-lF(8r3 > 1, 2 ' 817, 2 ' El8)'lF(820 < 85, 'Rejea rho null hypottads',

1)o nor rejed lhe null hypothcels"l

-Fltlvfl . B{, 811, 815)-El t-lF(825 < 85, "ReJect the null hypothcsls',

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-FlilV€r, 811, 815)-E17rlF{830 < 85, -Rojod rhe null hypothqds',

'Do not frlscl lh. null hypotheC6''lt{ol drown

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chap 10

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manager blk foods