Nonpara Tests

8/8/2019 Nonpara Tests

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NonparametricNonparametric MethodsMethods

Sign TestSign Test Wilcoxon Signed Rank TestWilcoxon Signed Rank Test

MannMann--WhitneyWhitney--Wilcoxon Rank Sum TestWilcoxon Rank Sum Test

KruskalKruskal--Wallis TestWallis Test

Rank CorrelationRank Correlation


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Nonparametric MethodsNonparametric Methods

Most of the statistical methods referred to as parametricMost of the statistical methods referred to as parametricrequire the use ofrequire the use of intervalinterval-- oror ratioratio--scaled datascaled data..

Nonparametric methods are often the only way toNonparametric methods are often the only way toanalyzeanalyze nominalnominal oror ordinal dataordinal data and draw statisticaland draw statistical

conclusions.conclusions. Nonparametric methods require no assumptions aboutNonparametric methods require no assumptions about

the population probability distributions.the population probability distributions.

Nonparametric methods are often calledNonparametric methods are often called distributiondistribution--

free methodsfree methods..


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Nonparametric MethodsNonparametric Methods

In general, for a statistical method to be classified asIn general, for a statistical method to be classified asnonparametric, it must satisfy at least one of thenonparametric, it must satisfy at least one of thefollowing conditions.following conditions.

The method can be used with nominal data.The method can be used with nominal data.

The method can be used with ordinal data.The method can be used with ordinal data. The method can be used with interval or ratio dataThe method can be used with interval or ratio data

when no assumption can be made about thewhen no assumption can be made about thepopulation probability distribution.population probability distribution.


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Sign TestSign Test

A common application of theA common application of the sign testsign test involves usinginvolves usinga sample ofa sample of nn potential customers to identify apotential customers to identify apreference for one of two brands of a product.preference for one of two brands of a product.

The objective is to determine whether there is aThe objective is to determine whether there is adifference in preference between the two items beingdifference in preference between the two items beingcompared.compared.

To record the preference data, we use a plus sign ifTo record the preference data, we use a plus sign ifthe individual prefers one brand and a minus sign ifthe individual prefers one brand and a minus sign ifthe individual prefers the other brand.the individual prefers the other brand.

Because the data are recorded as plus and minusBecause the data are recorded as plus and minussigns, this test is called the sign test.signs, this test is called the sign test.


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Sign Test: SmallSign Test: Small--Sample CaseSample Case

The smallThe small--sample case for the sign test should besample case for the sign test should beused wheneverused whenever nn


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Sign Test: LargeSign Test: Large--Sample CaseSample Case

UsingUsing HH00::pp = .5 and= .5 and nn > 20, the sampling distribution> 20, the sampling distributionfor the number of plus signs can be approximated byfor the number of plus signs can be approximated bya normal distribution.a normal distribution.

When no preference is stated (When no preference is stated (HH00::pp = .5), the sampling= .5), the samplingdistribution will have:distribution will have:

The test statistic is:The test statistic is:

HH00 is rejected if theis rejected if the pp--valuevalue


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Example: Ketchup Taste TestExample: Ketchup Taste Test


AAAABBBB

As part of a market research study, aAs part of a market research study, a

sample of 80 consumers were asked to tastesample of 80 consumers were asked to taste

two brands of ketchup and indicate atwo brands of ketchup and indicate a

preference. Do the data shown on the nextpreference. Do the data shown on the nextslide indicate a significant difference in theslide indicate a significant difference in the

consumer preferences for the two brands?consumer preferences for the two brands?


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45 preferred Brand A Ketchup45 preferred Brand A Ketchup(+ sign recorded)(+ sign recorded)

27

preferred Brand B Ketchup27

preferred Brand B Ketchup((__sign recorded)sign recorded)

8 had no preference8 had no preference


Example: Ketchup Taste TestExample: Ketchup Taste Test

AAAABBBB

The analysis will be based onThe analysis will be based ona sample size of 45 + 27 = 72.a sample size of 45 + 27 = 72.The analysis will be based onThe analysis will be based ona sample size of 45 + 27 = 72.a sample size of 45 + 27 = 72.


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HypothesesHypotheses

50H p {

AAAA

BBBB


0 : 50H p !

A preference for one brand over the other existsA preference for one brand over the other exists

No preference for one brand over the other existsNo preference for one brand over the other exists


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Sampling Distribution for Number of Plus SignsSampling Distribution for Number of Plus Signs

QQ = .5(72) = 36= .5(72) = 36

AAAA

BBBB


7 ) 4 43nW


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pp--Value = 2(1.0000Value = 2(1.0000 -- .9830) = .034.9830) = .034

Rejection RuleRejection RuleAAAA

BBBB


pp--ValueValue

zz = (= (xx QQ)/)/WW= (45= (45 -- 36)/4.243 = 2.1236)/4.243 = 2.12

Test StatisticTest Statistic

Using .05 level of significance:Using .05 level of significance:

RejectReject HH00 ififpp--valuevalue


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AAAA

BBBB


ConclusionConclusionBecause theBecause thepp--value


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Hypothesis Test About a MedianHypothesis Test About a Median

We can apply the sign test by:We can apply the sign test by: Using aUsing a plus signplus sign whenever the data in the samplewhenever the data in the sample

are above the hypothesized value of the medianare above the hypothesized value of the median

Using aUsing a minus signminus sign whenever the data in thewhenever the data in the

sample are below the hypothesized value of thesample are below the hypothesized value of themedianmedian

Discarding any data exactly equal to theDiscarding any data exactly equal to thehypothesized medianhypothesized median


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4 yea s!HH00: Median Age: Median Age

ea{HHaa: Median Age: Median Age

Example: Trim Fitness CenterExample: Trim Fitness CenterA hypothesis test is being conductedA hypothesis test is being conducted

about the median age of female membersabout the median age of female members

of the Trimof the Trim Fitness Center.Fitness Center.

In a sample of 40 female members, 25 are olderIn a sample of 40 female members, 25 are older

than 34, 14 are younger than 34, and 1 is 34.than 34, 14 are younger than 34, and 1 is 34. Is thereIs theresufficient evidence to rejectsufficient evidence to reject HH00? Assume? Assume EE = .05.= .05.


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pp--Value = 2(1.0000Value = 2(1.0000 .9608) = .0784.9608) = .0784

QQ = .5(= .5(nn) = .5(39) = 19.5) = .5(39) = 19.5

.25 .25(39) 3.1225nW ! ! !


pp--ValueValue

zz = (= (xx QQ)/)/WW= (25= (25 19.5)/3.1225 = 1.7619.5)/3.1225 = 1.76


Mean and Standard DeviationMean and Standard Deviation


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Rejection RuleRejection Rule

ConclusionConclusion

Do not rejectDo not reject HH00. The. Thepp--value for this twovalue for this two--tailtailtest is .0784. There is insufficient evidence in thetest is .0784. There is insufficient evidence in thesample to conclude that the median age issample to conclude that the median age is notnot 34 for34 forfemale members of Trimfemale members of Trim Fitness Center.Fitness Center.

Using .05 level of significance:Using .05 level of significance:



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Wilcoxon SignedWilcoxon Signed--Rank TestRank Test

This test is the nonparametric alternative to theThis test is the nonparametric alternative to theparametric matchedparametric matched--sample test presented insample test presented in

Chapter 10.Chapter 10. The methodology of the parametric matchedThe methodology of the parametric matched--samplesample

analysis requires:analysis requires:

interval data, andinterval data, and the assumption that the population of differencesthe assumption that the population of differences

between the pairs of observations is normallybetween the pairs of observations is normallydistributed.distributed.

If the assumption of normally distributed differencesIf the assumption of normally distributed differencesis not appropriate, the Wilcoxon signedis not appropriate, the Wilcoxon signed--rank test canrank test canbe used.be used.


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Example: Express DeliveriesExample: Express Deliveries


A firm has decided to select oneA firm has decided to select one

of two express delivery services toof two express delivery services to

provide nextprovide next--day deliveries to itsday deliveries to its

district offices.district offices.To test the delivery times of the two services, theTo test the delivery times of the two services, the

firm sends two reports to a sample of 10 districtfirm sends two reports to a sample of 10 district

offices, with one report carried by one service and theoffices, with one report carried by one service and the

other report carried by the second service. Do the dataother report carried by the second service. Do the dataon the next slide indicate a difference in the twoon the next slide indicate a difference in the two

services?services?


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SeattleSeattle

Los AngelesLos Angeles

BostonBoston

ClevelandClevelandNew YorkNew York

HoustonHouston

AtlantaAtlanta

St. LouisSt. Louis

MilwaukeeMilwaukee

DenverDenver

32 hrs.32 hrs.

3030

1919

16

16

1515

1818

1414

1010

77

1616

25 hrs.25 hrs.

2424

1515

15151313

1515

1515

88

99

1111

District OfficeDistrict Office OverNightOverNight NiteFliteNiteFlite


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Preliminary Steps of the TestPreliminary Steps of the Test Compute the differences between the pairedCompute the differences between the pairedobservations.observations.

Discard any differences of zero.Discard any differences of zero.

Rank the absolute value of the differences fromRank the absolute value of the differences from

lowest to highest. Tied differences are assignedlowest to highest. Tied differences are assignedthe average ranking of their positions.the average ranking of their positions.

Give the ranks the sign of the original differenceGive the ranks the sign of the original differencein the data.in the data.

Sum the signed ranks.Sum the signed ranks.

. . . next we will determine whether the. . . next we will determine whether the

sum is significantly different from zero.sum is significantly different from zero.


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SeattleSeattle

Los AngelesLos Angeles

BostonBoston

ClevelandClevelandNew YorkNew York

HoustonHouston

AtlantaAtlanta

St. LouisSt. Louis

MilwaukeeMilwaukee

DenverDenver

77

66

44

1122

33

11

22

22

55

District OfficeDistrict Office Differ.Differ. |Diff.| Rank Sign. Rank|Diff.| Rank Sign. Rank1010

99

77

1.51.544

66

1.51.5

44

88

+10+10

+9+9

+7+7

+1.5+1.5+4+4

+6+6

1.51.5

+4+4

+8+8+44+44


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HypothesesHypothesesHH00: The delivery times of the two services are the: The delivery times of the two services are the

same; neither offers faster service than the other.same; neither offers faster service than the other.

HHaa: Delivery times differ between the two services;: Delivery times differ between the two services;

recommend the one with the smaller times.recommend the one with the smaller times.


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Sampling Distribution ofSampling Distribution ofTT

for Identical Populationsfor Identical Populations

QQTT= 0= 0

( 1 ( 1 10(11 ( 119.6

6 6T

n n nW

! ! !


TT


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Rejection RuleRejection RuleUsing .05 level of significance,Using .05 level of significance,



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ConclusionConclusionRejectReject HH00. The. Thepp--value for this twovalue for this two--tail test istail test is

.025. There is sufficient evidence in the sample to.025. There is sufficient evidence in the sample toconclude that a difference exists in the delivery timesconclude that a difference exists in the delivery timesprovided by the two services.provided by the two services.



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MannMann--WhitneyWhitney--Wilcoxon TestWilcoxon Test

This test is another nonparametric method forThis test is another nonparametric method fordetermining whether there is a difference betweendetermining whether there is a difference betweentwo populations.two populations.

This test, unlike the Wilcoxon signedThis test, unlike the Wilcoxon signed--rank test, isrank test, is notnotbased on a matched sample.based on a matched sample.

This test doesThis test does notnot require interval data or therequire interval data or theassumption that both populations are normallyassumption that both populations are normallydistributed.distributed.

The only requirement is that the measurement scaleThe only requirement is that the measurement scale

for the data is at least ordinal.for the data is at least ordinal.


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HHaa: The two populations are not identical: The two populations are not identical

HH00: The two populations are identical: The two populations are identical

Instead of testing for the difference between theInstead of testing for the difference between themeans of two populations, this method tests tomeans of two populations, this method tests todetermine whether the two populations are identical.determine whether the two populations are identical.

The hypotheses are:The hypotheses are:


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Example: Westin FreezersExample: Westin FreezersManufacturer labels indicate theManufacturer labels indicate the

annual energy cost associated withannual energy cost associated with

operating home appliances such asoperating home appliances such as

freezers.freezers.The energy costs for a sample ofThe energy costs for a sample of

10 Westin freezers and a sample of 1010 Westin freezers and a sample of 10

Easton Freezers are shown on the next slide. Do theEaston Freezers are shown on the next slide. Do the

data indicate, usingdata indicate, using EE = .05, that a difference exists in= .05, that a difference exists inthe annual energy costs for the two brands of freezers?the annual energy costs for the two brands of freezers?


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$55.10$55.10

54.5054.50

53.2053.20

53.0053.00

55.5055.50

54.9054.90

55.8055.80

54.0054.00

54.2054.2055.2055.20

$56.10$56.10

54.7054.70

54.4054.40

55.4055.40

54.1054.10

56.0056.00

55.5055.50

55.0055.00

54.3054.3057.0057.00

W

estin FreezersW

estin Freezers Easton FreezersEaston Freezers


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HHaa: Annual energy costs differ for the two: Annual energy costs differ for the two

brands of freezers.brands of freezers.

HH00: Annual energy costs for Westin freezers: Annual energy costs for Westin freezers

and Easton freezers are the same.and Easton freezers are the same.


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MannMann--WhitneyWhitney--Wilcoxon Test:Wilcoxon Test:LargeLarge--Sample CaseSample Case

First, rank theFirst, rank the combinedcombined data from the lowest todata from the lowest tothe highest values, with tied values being assignedthe highest values, with tied values being assignedthe average of the tied rankings.the average of the tied rankings.

Then, computeThen, compute TT, the sum of the ranks for the first, the sum of the ranks for the first

sample.sample. Then, compare the observed value ofThen, compare the observed value of TTto theto the

sampling distribution ofsampling distribution of TTfor identical populations.for identical populations.The value of the standardized test statisticThe value of the standardized test statistic zz willwillprovide the basis for deciding whether to rejectprovide the basis for deciding whether to reject HH00..


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MannMann--WhitneyWhitney--Wilcoxon Test:Wilcoxon Test:LargeLarge--Sample CaseSample Case

2 22T n n n nW !

Approximately normal, providedApproximately normal, providednn11 >> 10 and10 and nn22 >> 1010

QQTT== nn11((nn11 ++ nn22 + 1)+ 1)


for Identical Populationsfor Identical Populations MeanMean

Standard DeviationStandard Deviation

Distribution FormDistribution Form


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$55.10$55.10

54.5054.50

53.2053.20

53.0053.00

55.5055.50

54.9054.90

55.8055.80

54.0054.00

54.2054.2055.2055.20

$56.10$56.10

54.7054.70

54.4054.40

55.4055.40

54.1054.10

56.0056.00

55.5055.50

55.0055.00

54.3054.3057.0057.00

W

estin FreezersW

estin Freezers Easton FreezersEaston Freezers

Sum of RanksSum of Ranks Sum of RanksSum of Ranks

RankRank RankRank

86.586.5 123.5123.5

11

22

1212

88

15.515.5

1010

1717

33

551313

1919

99

77

1414

44

1818

15.515.5

1111

66

2020


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for Identical Populationsfor Identical Populations

QQTT= (10)(21) = 105= (10)(21) = 105


1 2 1 2

1( 1)12

1 (10)(10)(21)12

1 2

Tn n n nW !

!

!

TT


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Rejection RuleRejection RuleUsing .05 level of significance,Using .05 level of significance,



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ConclusionConclusionDo not rejectDo not reject HH00. The. Thepp--value >value > EE. There is. There is

insufficient evidence in the sample data to concludeinsufficient evidence in the sample data to concludethat there is a difference in the annual energy costthat there is a difference in the annual energy costassociated with the two brands of freezers.associated with the two brands of freezers.


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The MannThe Mann--WhitneyWhitney--Wilcoxon test has been extendedWilcoxon test has been extendedby Kruskal and Wallis for cases of three or moreby Kruskal and Wallis for cases of three or morepopulations.populations.

The KruskalThe Kruskal--Wallis test can be used with ordinal dataWallis test can be used with ordinal dataas well as with interval or ratio data.as well as with interval or ratio data.

Also, the KruskalAlso, the Kruskal--Wallis test does not require theWallis test does not require theassumption of normally distributed populations.assumption of normally distributed populations.

HHaa: Not all populations are identical: Not all populations are identical

HH00: All populations are identical: All populations are identical


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!

!

-

2

1

123( 1

( 1

k

i

T

iT T i

RW n

n n n

where:where: kk = number of populations= number of populations

nnii = number of items in sample= number of items in sample ii

nnTT

== 77nnii = total number of items in all samples= total number of items in all samples

RRii = sum of the ranks for sample= sum of the ranks for sample ii


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When the populations are identical, the samplingWhen the populations are identical, the samplingdistribution of the test statisticdistribution of the test statistic WWcan be approximatedcan be approximatedby a chiby a chi--square distribution withsquare distribution with kk 1 degrees of1 degrees offreedom.freedom.

This approximation is acceptable if each of the sampleThis approximation is acceptable if each of the samplesizessizes nnii isis >> 5.5.

The rejection rule is:The rejection rule is: RejectReject HH00 ifif pp--valuevalue


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The Pearson correlation coefficient,The Pearson correlation coefficient,rr

, is a measure of, is a measure ofthe linear association between two variables forthe linear association between two variables forwhich interval or ratio data are available.which interval or ratio data are available.

TheThe Spearman rankSpearman rank--correlation coefficientcorrelation coefficient,, rrss, is a, is ameasure of association between two variables whenmeasure of association between two variables whenonly ordinal data are available.only ordinal data are available.

Values ofValues of rrss can range fromcan range from 1.0 to +1.0, where1.0 to +1.0, where

values near 1.0 indicate a strong positivevalues near 1.0 indicate a strong positiveassociation between the rankings, andassociation between the rankings, and

values nearvalues near --1.0 indicate a strong negative1.0 indicate a strong negativeassociation between the rankings.association between the rankings.


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Spearman RankSpearman Rank--Correlation Coefficient,Correlation Coefficient,rr

ss

2

2

61

( 1)

i

s

dr

n n!

where:where: nn = number of items being ranked= number of items being ranked

xxii = rank of item= rank of item ii with respect to one variablewith respect to one variable

yyii = rank of item= rank of item ii with respect to a second variablewith respect to a second variable

ddii== xx

ii-- yy

ii


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Test for Significant Rank CorrelationTest for Significant Rank Correlation

s

H p !

a: 0sH p {

We may want to use sample results to make anWe may want to use sample results to make aninference about the population rank correlationinference about the population rank correlationppss..

To do so, we must test the hypotheses:To do so, we must test the hypotheses:

(No rank correlation exists)(No rank correlation exists)

(Rank correlation exists)(Rank correlation exists)


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sr

Q

1

1sr

n

W !

Approximately normal, providedApproximately normal, provided nn >> 1010

Sampling Distribution ofSampling Distribution ofrr

ss whenwhenppss = 0= 0 MeanMean

Standard DeviationStandard Deviation

Distribution FormDistribution Form


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Example: Crennor InvestorsExample: Crennor InvestorsCrennor Investors providesCrennor Investors provides

a portfolio management servicea portfolio management service

for its clients. Two of Crennorsfor its clients. Two of Crennors

analysts ranked ten investmentsanalysts ranked ten investmentsas shown on the next slide. Useas shown on the next slide. Use

rank correlation, withrank correlation, with EE = .10, to= .10, to

comment on the agreement of the two analystscomment on the agreement of the two analysts

rankings.rankings.


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Analyst #2Analyst #2 1 5 6 2 9 7 3 10 4 81 5 6 2 9 7 3 10 4 8

Analyst #1

Analyst #1 1 4 9 8 6 3 5 7 2 101 4 9 8 6 3 5 7 2 10

InvestmentInvestment A B C D E F G H I JA B C D E F G H I J

Example: Crennor InvestorsExample: Crennor Investors

: sH p !

a: 0sH p {

(No rank correlation exists)(No rank correlation exists)

(Rank correlation exists)(Rank correlation exists)

Analysts RankingsAnalysts Rankings



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AABBCC

DDEEFFGG

HHIIJJ

114499

8866

335577

221010

115566

2299

77

33

10104488

00--1133

66

--33--4422

--33--2222

001199

363699

16164499

4444

Sum =Sum =9292

InvestmentInvestment Analyst #1Analyst #1RankingRanking Analyst #2Analyst #2RankingRanking Differ.Differ. (Differ.)(Differ.)22


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Sampling Distribution ofr

sAssuming No Rank Correlation


srW

QQrr = 0= 0rrss


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6 6 9 44 4is

dr

n n! ! !


zz = (= (rrss -- QQrr)/)/WWrr = (.4424= (.4424 -- 0)/.3333 = 1.330)/.3333 = 1.33

Rejection RuleRejection Rule

With .10 level of significance:With .10 level of significance:



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Do no rejectDo no reject HH00. The. The pp--value >value > EE. There is not a. There is not a

significant rank correlation. The two analysts are notsignificant rank correlation. The two analysts are not

showing agreement in their ranking of the riskshowing agreement in their ranking of the risk

associated with the different investments.associated with the different investments.


ConclusionConclusion


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End of Chapter 19End of Chapter 19

Nonpara Tests

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