Microsoft Power Point - Week8

8/8/2019 Microsoft Power Point - Week8

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Week 8Week 8Week 8Week 8 ---- 9999

Non- parametric Test


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Non-parametric test

One that makes no assumptionsabout the specific shape of thepopulation from which a sample is

drawn.

How to differentia the parametrictest vs non-parametric test??


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Parametric testHo : 1 = 2 = 3

H1 : Not all are equalvs

Non parametric testHo : M1 = M2 = M3

H1 : Not all median are equal


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nonparametric:

One Sample Two Samples More Than Two

Samples

Wilcoxon

Signed RankTest

Wilcoxon

Rank SumTest

Kruskal-Wallis

Test

Wilcoxon

Signed RankTest

Dependent Independent

parametric counterpart:

t-test,

one sample

t-test,

paired sample

t-test,

two Independent

samples

One-way

ANOVA


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A non-parametric test should be usedinstead of its parametric counterpartwhenever:

1)Data are of the nominal or ordinal scale ofmeasurement

2)Data are of the interval or ratio scale ofmeasurement but one or more other

assumptions, such as the normality of theunderlying population distribution, are notmet.


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Advantages

1. Fewer assumptions about the

population

2. The techniques can be applied when

sample sizes are very small3. Samples with data of the nominal or

ordinal scales of measurement can be

tested


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Disadvantages

1. Compared to a parametric test, the

information in the data is used lessefficiently, and the power of the test willbe lower. For this reason, a parametrictest is preferable whenever itsassumptions have been met.

2. Non-parametric testing places greaterreliance on statistical tables.


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WilcoxonWilcoxonWilcoxonWilcoxon SignedSignedSignedSigned

Rank TestRank TestRank TestRank Test

(0ne sample)(0ne sample)(0ne sample)(0ne sample)

First section:First section:First section:First section:


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For one sample, the Wilcoxon signedrank method tests whether the samplecould have been drawn from apopulation having a hypothesized valueas its median.

Assumptions:

Data are assumed to be continuous and ofthe interval or ratio scales of measurement.


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Hypothesis and Decision Rule The Research Question (H1): Test the value of a single

population median, m{ , >, M0

H0: M M0H1: M < M0

Two-Tail Test Left-Tail Test Right-Tail Test

Reject

WL WU

RejectDo NotReject

Reject

WL

Do Not Reject

WU

RejectDo Not Reject

Reject H0 if W < WL

or if W > WU

Reject H0

if W < WL

Reject H0

if W > WU


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Uncle Chuah have 9grandchildren. He believed

that each of hisgrandchildren could get the

Ang Pow at least RM250. Atthe 0.05 level of

significance, is thereevidence to prove Uncle

Chuahs claim.


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16.519.5

8840-402109

6625252758

22992597

3312-122386

7735352855

4.54.520202704

114-42463

~002502

4.54.520-202301Rank -Rank +Rankdx-mMoney, xGrandchildren

Test statistics = R+ = 19.5


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1.Ho : M RM2502.H1 : M > RM2503. = 0.05, n = 8 (The total of d which are

not equal to 0), critical value = 30.4.Reject Ho, if the test statistics > 30.

Otherwise, do not reject Ho.

5.Test statistics = R+ = 19.5.6.Do not reject Ho since R+ = 19.5


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W (The Normal Approximation)

When the number of observation for which

di = 0 is n > 20, a z-test will be a closeapproximation to the Wilcoxon signed ranktest.

W distribution approaches a normal curvesas n becomes larger.


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Z-test approximation to the Wilcoxon signedrank test:

Test Statistics:

24)12)(1(

4

)1(

++

+

=

nnn

nnw

z

Where, W = sum of the R+ ranks

n = number of observations

for which di = 0

(continued)


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WilcoxonWilcoxonWilcoxonWilcoxon SignedSignedSignedSigned

Rank TestRank TestRank TestRank Test

(paired(paired(paired(paired----test)test)test)test)

2nd section:2nd section:2nd section:2nd section:


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nonparametric:

One Sample Two Samples More Than Two

Samples

Wilcoxon

Signed Rank

Test

Wilcoxon

Rank Sum

Test

Kruskal-Wallis

Test

Wilcoxon

Signed Rank

Test

Dependent Independent

parametric counterpart:

t-test,

one sample

t-test,

paired sample

t-test,

two Independent

samples

One-way

ANOVA


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Wilcoxon Signed Rank Test forComparing Paired Samples

The Wilcoxon Signed Rank test can also beused for paired samples

Use if assumption of normality is violated for

the paired-t

test. Assumptions:

Data are assumed to be continuous and of the

interval or ratio scales of measurement. The observations must be related or dependent.


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Hypothesis and Decision Rule

H0: Md = 0H1: Md 0

H0: Md 0H1: Md > 0

H0: Md 0H1: Md < 0

Two-Tail Test Left-Tail Test Right-Tail Test

Reject

WL WU

RejectDo NotReject

Reject

WL

Do Not Reject

WU

RejectDo Not Reject

Reject H0 if W < WLor if W > WU

Reject H0 if W < WL Reject H0 if W > WU

The Research Question (H1): Test the difference

in two population medians, paired samples, md {,>,


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Compare the sum of

positive rank with theWvalue.

Compute thedifferences between

related

observations.

Rank the absolutedifferences from

low to high.

Return the signs to

the ranks and sumpositive and negative

ranks.

Wilcoxon SignedWilcoxon Signed--Rank TestRank Test Continued


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Are they paid equally?


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57.58.5

885.6-5.625.519.9

996.4-6.435.429

~0015.515.5

11118.1-8.12617.9

2.52.53-340.337.3

10106.9-6.921.814.9

774.7-4.736.131.4

664.6-4.642.137.52.52.53320.723.7

553.93.917.721.6

443.7-3.720.416.7

110.90.923.124

Rank-Rank+Rankdd = x - yHusband, y

(RM10,000)Wife, x

(RM10,000)

Test statistics = R+ = 8.5


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1.Ho : Md = 0

2.H1 : Md 03. = 0.10, n = 11, lower critical value =

14, upper critical value = 52.

4.Reject Ho, if the test statistics 52. Otherwise, donot reject Ho.

5.Test statistics = R+ = 8.5.6.Reject Ho since the test statistics =8.5

< 14.7.There is enough evidence to concludethat wife and husband are not paid

equally.


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The z-test described in the preceding

section can be used for paired samples. When the number of observation for which

di

= 0 is n > 20, a z-test will be a closeapproximation to the Wilcoxon signed ranktest.


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Z-test approximation to the Wilcoxon signed

rank test:

Test Statistics:

24

)12)(1(

4

)1(

++

+

=

nnn

nnw

z

Where, W = sum of the R+ ranks

n = number of observations

for which di = 0

(continued)

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