cd-rom chapter 15 introduction to nonparametric statistics
TRANSCRIPT
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CD-ROM Chapter 15CD-ROM Chapter 15
Introduction to Introduction to Nonparametric Nonparametric
StatisticsStatistics
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Chapter 15 - Chapter 15 - Chapter Chapter OutcomesOutcomesAfter studying the material in this chapter, you should be able to:Recognize when and how to use the runs test and testing for randomness.Know when and how to perform a Mann-Whitney U test.Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it in a decision-making context.Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA.
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Nonparametric StatisticsNonparametric Statistics
Nonparametric statistical Nonparametric statistical proceduresprocedures are those statistical methods that do not concern themselves with population distributions and/or parameters.
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The Runs TestThe Runs Test
The runs testruns test is a statistical procedure used to determine whether the pattern of occurrences of two types of observations is determined by a random process.
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The Runs TestThe Runs Test
A runrun is a succession of occurrences of a certain type preceded and followed by occurrences of the alternate type or by no occurrences at all.
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The Runs TestThe Runs Test(Table 15-1)(Table 15-1)
Sequence Number Code Sequence Number Code1 0.34561 - 11 0.67201 +2 0.42789 - 12 0.23790 -3 0.36925 - 13 0.24509 -4 0.89563 + 14 0.01467 -5 0.25679 - 15 0.78345 +6 0.92001 + 16 0.69112 +7 0.58345 + 17 0.46023 -8 0.23114 - 18 0.38633 -9 0.12672 - 19 0.60914 +
10 0.88569 + 20 0.95234 +
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The Runs TestThe Runs Test(Small Sample Example)(Small Sample Example)
H0: Computer-generated numbers are random between 0.0 and 1.0.
HA: Computer-generated numbers are not random .
--- + - ++ -- ++ --- ++ -- ++Runs: 1 2 3 4 5 6 7 8 9 10
There are r = 10 runsFrom runs table (Appendix K) with n1 = 9 and n2 = 11, the
critical value of r is 6
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The Runs TestThe Runs Test(Small Sample Example)(Small Sample Example)
Test Statistic:
r = 10 runs
Critical Values from Runs Table:
Possible
Runs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Reject HReject H00Reject HReject H00Do not reject HDo not reject H00
Decision:
Since r = 10, we do not reject the null hypothesis.
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Large Sample Runs TestLarge Sample Runs Test
MEAN AND STANDARD DEVIATION FOR MEAN AND STANDARD DEVIATION FOR rr
where:n1 = Number of occurrences of first type
n2 = Number of occurrences of second type
12
21
21
nn
nnr
)1()(
)2)(2(
212
21
212121
nnnn
nnnnnnr
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Large Sample Runs TestLarge Sample Runs Test
TEST STATISTIC FOR LARGE TEST STATISTIC FOR LARGE SAMPLE RUNS TESTSAMPLE RUNS TEST
r
rrz
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Large Sample Runs TestLarge Sample Runs Test(Example 15-2)(Example 15-2)
OOOUOOUOUUOOUUOOOOUUOUUOOO
UUUOOOOUUOOUUUOUUOOUUUUU
OOOUOUUOOOUOOOOUUUOUUOOOU
OOUUOUOOUUUOUUOOOOUUUOOO
Table 15-2
n1 = 53 “O’s” n2 = 47 “U’s”
r = 45 runs
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96.1025. z0
Large Sample Runs Test Large Sample Runs Test (Example 15-2)(Example 15-2)
Rejection Region /2 = 0.025
Since z= -1.174 > -1.96 and < 1.96, we do not reject H0,
96.1025. z
Rejection Region /2 = 0.025
H0: Yogurt fill amounts are randomly distributed above and below 24-ounce level.H1: Yogurt fill amounts are not randomly distributed above and below 24-ounce level.
= 0.05
174.195659.4
82.5045
r
rrz
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Mann-Whitney U TestMann-Whitney U Test
The Mann Whitney U test can be used to compare two samples from two populations if the following assumptions are satisfied:
• The two samples are independent and random.
• The value measured is a continuous variable.
• The measurement scale used is at least ordinal.
• If they differ, the distributions of the two populations will differ only with respect to the central location.
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Mann-Whitney U TestMann-Whitney U Test
U-STATISTICSU-STATISTICS
where:n1 and n2 are the two sample sizes
R1 and R2 = Sum of ranks for samples
1 and 2
111
211 2
)1(R
nnnnU
222
212 2
)1(R
nnnnU
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Mann-Whitney U TestMann-Whitney U Test- Large Samples -- Large Samples -
MEAN AND STANDARD DEVIATION FOR MEAN AND STANDARD DEVIATION FOR THE THE UU-STATISTIC-STATISTIC
where:n1 and n2 = Sample sizes from
populations 1 and 2
221nn
12
)1)()(( 2121
nnnn
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Mann-Whitney U TestMann-Whitney U Test- Large Samples -- Large Samples -
MANN-WHITNEY U-TEST STATISTICMANN-WHITNEY U-TEST STATISTIC
12)1)()((
2
2121
21
nnnn
nnU
z
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0~~21
Mann-Whitney U TestMann-Whitney U Test(Example 15-4)(Example 15-4)
Since z= -1.027 > -1.645, we do not reject H0,
645.1z
Rejection Region = 0.05
05.0
0~~:
0~~:
21
210
AH
H
027.1
12)1404144)(404)(144(
088,29412,27
12)1)()((
2
2121
21
nnnn
nnU
z
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Wilcoxon Matched-Pairs Wilcoxon Matched-Pairs TestTest
The Wilcoxon matched pairs signed rank test can be used in those cases where the following assumptions are satisfied:
• The differences are measured on a continuous variable.
• The measurement scale used is at least interval.
• The distribution of the population differences is symmetric about their median.
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Wilcoxon Matched-Pairs Wilcoxon Matched-Pairs TestTest
WILCOXON MEAN AND STANDARD WILCOXON MEAN AND STANDARD DEVIATIONDEVIATION
where:n = Number of paired values
4
)1(
nn
24
)12)(1(
nnn
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Wilcoxon Matched-Pairs Wilcoxon Matched-Pairs TestTest
WILCOXON TEST STATISTICWILCOXON TEST STATISTIC
24)12)(1(
4)1(
nnn
nnT
z
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Kruskal-Wallis One-Way Kruskal-Wallis One-Way Analysis of VarianceAnalysis of Variance
Kruskal-Wallis one-way analysis of variance can be used in one-way analysis of variance if the variables satisfy the following:
• They have a continuous distribution.• The data are at least ordinal.• The samples are independent.• The samples come from populations
whose only possible difference is that at least one may have a different central location than the others.
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Kruskal-Wallis One-Way Kruskal-Wallis One-Way Analysis of VarianceAnalysis of Variance
H-STATISTICH-STATISTIC
where:N = Sum of sample sizes in all samplesk = Number of samplesRi = Sum of ranks in the ith sample
ni = Size of the ith sample
1),1(3)1(
12
1
2
kdfwithNn
R
NNH
k
i i
i
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Kruskal-Wallis One-Way Kruskal-Wallis One-Way Analysis of VarianceAnalysis of Variance
CORRECTION FOR TIED RANKINGSCORRECTION FOR TIED RANKINGS
where:g = Number of different groups of tiesti = Number of tied observations in the
ith tied group of scoresN = Total number of observations
NN
ttg
iii
31
3 )(1
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Kruskal-Wallis One-Way Kruskal-Wallis One-Way Analysis of VarianceAnalysis of Variance
H-STATISTIC CORRECTED FOR TIED H-STATISTIC CORRECTED FOR TIED RANKINGSRANKINGS
NN
tt
NnR
NNH g
iii
k
i i
i
31
3
1
2
)(1
)1(3)1(
12
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Key TermsKey Terms
• Kruskal-Wallis One-Way Analysis of Variance
• Mann-Whitney U Test
• Nonparametric Statistical Procedure
• Run
• Runs Test
• Wilcoxon Test