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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W (The Normal Approximation)
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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W (The Normal Approximation)
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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W (The Normal Approximation)
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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