AK/ECON 3480 M & N WINTER 2006

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

AK/ECON 3480 M & NAK/ECON 3480 M & NWINTER 2006WINTER 2006

Power Point Presentation Power Point Presentation

Professor Ying KongProfessor Ying Kong

School of Analytic Studies and Information School of Analytic Studies and Information TechnologyTechnology

Atkinson Faculty of Liberal and Professional Atkinson Faculty of Liberal and Professional StudiesStudies

York UniversityYork University

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Chapter 19Chapter 19Nonparametric Methods Nonparametric Methods

Sign TestSign Test Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test Kruskal-Wallis TestKruskal-Wallis Test Rank CorrelationRank Correlation

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Nonparametric MethodsNonparametric Methods

Most of the statistical methods referred to as parametric Most of the statistical methods referred to as parametric require the use of require the use of intervalinterval- or - or ratio-scaled dataratio-scaled data..

Nonparametric methods are often the only way Nonparametric methods are often the only way to analyze to analyze nominalnominal or or ordinal dataordinal data and draw and draw statistical conclusions.statistical conclusions.

Nonparametric methods require no assumptions Nonparametric methods require no assumptions about the population probability distributions.about the population probability distributions.

Nonparametric methods are often called Nonparametric methods are often called distribution-free methodsdistribution-free methods..

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Nonparametric MethodsNonparametric Methods

In general, for a statistical method to be In general, for a statistical method to be classified as nonparametric, it must satisfy at classified as nonparametric, it must satisfy at least one of the following conditions.least one of the 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 The method can be used with interval or ratio data when no assumption can be made ratio data when no assumption can be made about the population probability distribution.about the population probability distribution.

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Sign TestSign Test

A common application of the A common application of the sign testsign test involves involves using a sample of using a sample of n n potential customers to identify potential customers to identify a preference for one of two brands of a product.a preference for one of two brands of a product.

The objective is to determine whether there is The objective is to determine whether there is a difference in preference between the two a difference in preference between the two items being compared.items being compared.

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

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

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Sign Test: Small-Sample CaseSign Test: Small-Sample Case

The small-sample case for the sign test should The small-sample case for the sign test should be used whenever be used whenever nn << 20. 20.

The hypotheses areThe hypotheses are

a : .50H pa : .50H p

0 : .50H p0 : .50H p

A preference for one brandA preference for one brandover the other exists.over the other exists.

No preference for one brandNo preference for one brandover the other exists.over the other exists.

The number of plus signs is our test statistic.The number of plus signs is our test statistic. Assuming Assuming HH00 is true, the sampling distribution for the is true, the sampling distribution for the

test statistic is a binomial distribution with test statistic is a binomial distribution with pp = .5. = .5.

HH00 is rejected if the is rejected if the pp-value -value << level of significance, level of significance, ..

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Sign Test: Large-Sample CaseSign Test: Large-Sample Case

Using Using HH00: : pp = .5 and = .5 and nn > 20, the sampling > 20, the sampling distribution for the number of plus signs can distribution for the number of plus signs can be approximated by a normal distribution.be approximated by a normal distribution.

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

The test statistic is:The test statistic is:

HH00 is rejected if the is rejected if the pp-value -value << level of significance, level of significance, ..

Mean: Mean: = .50 = .50nnStandard Deviation: Standard Deviation: .25n .25n

xz

x

z

((xx is the number is the number of plus signs)of plus signs)

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Example: Ketchup Taste TestExample: Ketchup Taste Test


AAAA BBBB

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

sample of 36 consumers were asked to tastesample of 36 consumers were asked to taste

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

preference. Do the data shown on the nextpreference. Do the data shown on the next

slide 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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1818 preferred Brand A Ketchuppreferred Brand A Ketchup (+ sign recorded)(+ sign recorded)

1212 preferred Brand B Ketchuppreferred Brand B Ketchup ((__ sign recorded) sign recorded) 6 had no preference6 had no preference


Example: Ketchup Taste TestExample: Ketchup Taste Test

AAAA BBBBThe analysis will be based The analysis will be based

onon

a sample size of 18 + 12 = a sample size of 18 + 12 = 30.30.

The analysis will be based The analysis will be based onon

a sample size of 18 + 12 = a sample size of 18 + 12 = 30.30.

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HypothesesHypotheses

a : .50H pa : .50H p

AAAA BBBB


0 : .50H p0 : .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

= .5(30) = 15= .5(30) = 15

AAAA BBBB


.25 .25(30) 2.74n .25 .25(30) 2.74n

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Rejection RuleRejection Rule AAAA BBBB


pp-Value = 2(.5000 - .3643) = .2714-Value = 2(.5000 - .3643) = .2714

pp-Value-Value

z z = ( = (xx – – )/)/ = (18 - 15)/2.74 = 3/2.74 = 1.10 = (18 - 15)/2.74 = 3/2.74 = 1.10

Test StatisticTest Statistic

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

Reject Reject HH00 if if pp-value -value << .05 .05

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AAAA BBBB


ConclusionConclusion

Because the Because the pp-value > -value > , we cannot reject , we cannot reject HH00. There is insufficient evidence in the sample . There is insufficient evidence in the sample to conclude that a difference in preference exists to conclude that a difference in preference exists for the two brands of ketchup. for the two brands of ketchup.

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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 a plus sign whenever the data in the sample Using a plus sign whenever the data in the sample

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

• Using a minus sign whenever the data in Using a minus sign whenever the data in the sample are below the hypothesized the sample are below the hypothesized value of the medianvalue of the median

• Discarding any data exactly equal to the Discarding any data exactly equal to the hypothesized medianhypothesized median

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34 years34 yearsHH00: Median Age: Median Age34 years34 yearsHHaa: Median Age: Median Age

Example: Trim Fitness CenterExample: Trim Fitness Center

A 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 there

sufficient evidence to reject sufficient evidence to reject HH00? Assume ? Assume = .05. = .05.

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pp-Value = 2(.5000 -Value = 2(.5000 .4608) = .0784.4608) = .0784

= .5(39) = 19.5= .5(39) = 19.5

.25 .25(39) 3.12n .25 .25(39) 3.12n


pp-Value-Value

z z = ( = (xx – – )/)/ = (25 – 19.5)/3.12 = 1.76 = (25 – 19.5)/3.12 = 1.76


Mean and Standard DeviationMean and Standard Deviation

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Rejection RuleRejection Rule


Do not reject Do not reject HH00. The . The pp-value for this two-tail test -value for this two-tail test is .0784. There is insufficient evidence in the is .0784. There is insufficient evidence in the sample to conclude that the median age is sample to conclude that the median age is notnot 34 34 for female members of Trim for female members of Trim Fitness Center.Fitness Center.

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


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Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test

This test is the nonparametric alternative to This test is the nonparametric alternative to the parametric matched-sample test the parametric matched-sample test presented in Chapter 10.presented in Chapter 10.

The methodology of the parametric matched-The methodology of the parametric matched-sample analysis requires:sample analysis requires:• interval data, andinterval data, and• the assumption that the population of the assumption that the population of

differences between the pairs of differences between the pairs of observations is normally distributed.observations is normally distributed.

If the assumption of normally distributed If the assumption of normally distributed differences is not appropriate, the Wilcoxon differences is not appropriate, the Wilcoxon signed-rank test can be used.signed-rank test can 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 next-day deliveries to itsprovide next-day 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 district firm 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 data

on the next slide indicate a difference in the twoon the next slide indicate a difference in the two

services?services?

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SeattleSeattleLos AngelesLos Angeles

BostonBostonClevelandClevelandNew YorkNew YorkHoustonHoustonAtlantaAtlantaSt. LouisSt. LouisMilwaukeeMilwaukeeDenverDenver

32 hrs.32 hrs.3030191916161515181814141010 771616

25 hrs.25 hrs.242415151515131315151515 88 991111

District OfficeDistrict Office OverNightOverNight NiteFliteNiteFlite

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Preliminary Steps of the TestPreliminary Steps of the Test• Compute the differences between the Compute the differences between the

paired observations.paired observations.• Discard any differences of zero.Discard any differences of zero.• Rank the absolute value of the differences Rank the absolute value of the differences

from lowest to highest. Tied differences from lowest to highest. Tied differences are assigned the average ranking of their are assigned the average ranking of their positions.positions.• Give the ranks the sign of the original Give the ranks the sign of the original difference in the data.difference 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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SeattleSeattleLos AngelesLos Angeles

BostonBostonClevelandClevelandNew YorkNew YorkHoustonHoustonAtlantaAtlantaSt. LouisSt. LouisMilwaukeeMilwaukeeDenverDenver

77 66 44 11 22 3311 2222 55

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

10109977

1.51.54466

1.51.54488

+10+10+9+9+7+7

+1.5+1.5+4+4+6+61.51.5+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 of Sampling Distribution of TT for Identical Populations for Identical Populations

TT = 0 = 0

( 1)(2 1) 10(11)(21)19.62

6 6T

n n n

( 1)(2 1) 10(11)(21)19.62

6 6T

n n n


TT

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Using .05 level of significance,Using .05 level of significance,

Reject Reject HH00 if if pp-value -value << .05 .05 Test StatisticTest Statistic

pp-Value-Value

zz = ( = (TT - - T T )/)/TT = (44 - 0)/19.62 = 2.24 = (44 - 0)/19.62 = 2.24

pp-Value = 2(.5000 - .4875) = .025-Value = 2(.5000 - .4875) = .025

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Reject Reject HH00. The . The pp-value for this two-tail -value for this two-tail test is .025. There is sufficient evidence in the test is .025. There is sufficient evidence in the sample to conclude that a difference exists in sample to conclude that a difference exists in the delivery times provided by the two services. the delivery times provided by the two services.


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Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test

This test is another nonparametric method for This test is another nonparametric method for determining whether there is a difference determining whether there is a difference between two populations.between two populations.

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

This test does This test does notnot require interval data or the require interval data or the assumption that both populations are normally assumption that both populations are normally distributed.distributed.

The only requirement is that the measurement The only requirement is that the measurement scale for the data is at least ordinal.scale for the data is at least ordinal.

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HHaa: The two populations are not identical: The two populations are not identicalHH00: The two populations are identical: The two populations are identical

Instead of testing for the difference between the Instead of testing for the difference between the means of two populations, this method tests to means of two populations, this method tests to determine 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, using data indicate, using = .05, that a difference exists in = .05, that a difference exists in

the 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 54.2054.20 55.2055.20

$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 54.3054.30 57.0057.00

Westin FreezersWestin FreezersEaston FreezersEaston Freezers

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HypothesesHypotheses


HHaa: Annual energy costs differ for : Annual energy costs differ for

the twothe 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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Mann-Whitney-Wilcoxon Test:Mann-Whitney-Wilcoxon Test:Large-Sample CaseLarge-Sample Case

First, rank the First, rank the combinedcombined data from the lowest to data from the lowest to

the highest values, with tied values being the highest values, with tied values being assigned the average of the tied rankings.assigned the average of the tied rankings.

Then, compute Then, compute TT, the sum of the ranks for the , the sum of the ranks for the first sample.first sample.

Then, compare the observed value of Then, compare the observed value of TT to the to the sampling distribution of sampling distribution of TT for identical populations. for identical populations. The value of the standardized test statistic The value of the standardized test statistic zz will will provide the basis for deciding whether to reject provide the basis for deciding whether to reject HH00..

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Mann-Whitney-Wilcoxon Test:Mann-Whitney-Wilcoxon Test:Large-Sample CaseLarge-Sample Case

1 2 1 21 ( 1)12T n n n n 1 2 1 21 ( 1)12T n n n n

Approximately normal, providedApproximately normal, provided

nn11 >> 10 and 10 and nn22 >> 10 10

TT = = nn11((nn11 + + nn22 + 1) + 1)


• 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 54.2054.20 55.2055.20

$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 54.3054.30 57.0057.00

Westin FreezersWestin Freezers Easton FreezersEaston Freezers

Sum of RanksSum of Ranks Sum of RanksSum of Ranks

RankRank RankRank

86.586.5 123.5123.5

1122

121288

15.15.551010

17173355

1313

19199977

141444

181815.15.55111166

2020

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TT = ½(10)(21) = 105 = ½(10)(21) = 105


1 2 1 21 ( 1)12

1 (10)(10)(21)12 13.23

T n n n n

1 2 1 21 ( 1)12

1 (10)(10)(21)12 13.23

T n n n n

TT

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Using .05 level of significance,Using .05 level of significance,

Reject Reject HH00 if if pp-value -value << .05 .05 Test StatisticTest Statistic

pp-Value-Value

zz = ( = (TT - - T T )/)/TT = (86.5 = (86.5 105)/13.23 = -1.40 105)/13.23 = -1.40

pp-Value = 2(.5000 - .4192) = .1616-Value = 2(.5000 - .4192) = .1616


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ConclusionConclusionDo not reject Do not reject HH00. The . The pp-value > -value > . There is . There is

insufficient evidence in the sample data to conclude insufficient evidence in the sample data to conclude that there is a difference in the annual energy cost that there is a difference in the annual energy cost associated with the two brands of freezers.associated with the two brands of freezers.

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Kruskal-Wallis TestKruskal-Wallis Test

The Mann-Whitney-Wilcoxon test has been The Mann-Whitney-Wilcoxon test has been extended by Kruskal and Wallis for cases of extended by Kruskal and Wallis for cases of three or more populations.three or more populations.

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

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

HHaa: Not all populations are identical: Not all populations are identicalHH00: All populations are identical: All populations are identical

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2

1

123( 1)

( 1)

ki

TiT T i

RW n

n n n

2

1

123( 1)

( 1)

ki

TiT 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 = = nnii = 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 When the populations are identical, the sampling distribution of the test statistic sampling distribution of the test statistic WW can can be approximated by a chi-square distribution be approximated by a chi-square distribution with with kk – 1 degrees of freedom. – 1 degrees of freedom.

This approximation is acceptable if each of the This approximation is acceptable if each of the sample sizes sample sizes nnii is is >> 5. 5.

The rejection rule is: The rejection rule is: Reject Reject HH00 if if pp-value -value <<

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Rank CorrelationRank Correlation

The Pearson correlation coefficient, The Pearson correlation coefficient, rr, is a measure of , is a measure of the linear association between two variables for the linear association between two variables for which interval or ratio data are available.which interval or ratio data are available.

The The Spearman rank-correlation coefficientSpearman rank-correlation coefficient, , rrs s , , is a measure of association between two is a measure of association between two variables when only ordinal data are available.variables when only ordinal data are available.

Values of Values of rrss can range from –1.0 to +1.0, where can range from –1.0 to +1.0, where

• values near 1.0 indicate a strong positive values near 1.0 indicate a strong positive association between the rankings, andassociation between the rankings, and

• values near -1.0 indicate a strong negative values near -1.0 indicate a strong negative association between the rankings.association between the rankings.

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Spearman Rank-Correlation Coefficient, Spearman Rank-Correlation Coefficient, rrss

2

2

61

( 1)i

s

dr

n n

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 variable with respect to one variable

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

ddii = = xxii - - yyii

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Test for Significant Rank CorrelationTest for Significant Rank Correlation

0 : 0sH p 0 : 0sH p

a : 0sH p a : 0sH p

We may want to use sample results to make an We may want to use sample results to make an inference about the population rank correlation inference about the population rank correlation ppss..

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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0sr

0sr

11sr n

1

1sr n

Approximately normal, provided Approximately normal, provided nn >> 10 10

Sampling Distribution ofSampling Distribution of rrss when when ppss = 0 = 0

• MeanMean

• Standard DeviationStandard Deviation

• Distribution FormDistribution Form

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Example: Crennor InvestorsExample: Crennor Investors Crennor Investors provides Crennor Investors provides

a portfolio management servicea portfolio management service

for its clients. Two of Crennor’sfor its clients. Two of Crennor’s

analysts ranked ten investmentsanalysts ranked ten investments

as shown on the next slide. Useas shown on the next slide. Use

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

comment on the agreement of the two analysts’comment 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 #1Analyst #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

0 : 0sH p 0 : 0sH p

a : 0sH p a : 0sH p (No rank correlation exists)(No rank correlation exists)

(Rank correlation exists)(Rank correlation exists)

• Analysts’ RankingsAnalysts’ Rankings

• HypothesesHypotheses

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AABBCCDDEEFFGGHHIIJJ

114499886633557722

1010

11556622997733

10104488

00-1-13366-3-3-4-422-3-3-2-222

001199

363699

161644994444

Sum =Sum = 9292

InvestmentInvestmentAnalyst #1Analyst #1

RankingRankingAnalyst #2Analyst #2

RankingRanking Differ.Differ. (Differ.(Differ.))22

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Sampling Distribution of rs

Assuming No Rank Correlation


1.333

10 1sr

1

.33310 1sr

rr = 0 = 0rrss

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Test StatisticTest Statistic2

2

6 6(92)1 1 0.4424

( 1) 10(100 1)i

s

dr

n n

2

2

6 6(92)1 1 0.4424

( 1) 10(100 1)i

s

dr

n n


zz = ( = (rrss - - r r )/)/rr = (.4424 - 0)/.3333 = 1.33 = (.4424 - 0)/.3333 = 1.33


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


pp-Value-Value

pp-Value = 2(.5000 - .4082) = .1836-Value = 2(.5000 - .4082) = .1836

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Do no reject Do no reject HH00. The . The pp-value > -value > . There is . There is

not a significant rank correlation. The two analysts not a significant rank correlation. The two analysts are not showing agreement in their ranking of the are not showing agreement in their ranking of the risk associated with the different investments.risk associated with the different investments.



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End of Chapter 19End of Chapter 19

Documents

AK/ECON 3480 M & N WINTER 2006