Section 14.3 Comparing Two Populations: Independent Samples
Compare the two independent populations
Nonparametric test: Wilcoxon Rank Sum Test.
Hypothesis
Test Statistics
Rejection Region
T  TL or T  TU
Ho: D1 and D2 are identical
Ha: D1 is shift either to the left or right of D2
Ho: D1 and D2 are identical
Ha: D1 is shift to the right of D2
T1  TU , if n1  n2

T2  TL , if n2  n1
T , if n1  n2
T  1
T2 , if n2  n1
Ho: D1 and D2 are identical
Ha: D1 is shift to the left of D2
TL
and
TU
T1  TL , if n1  n2

T2  TU , if n2  n1
are obtained from Table XII
Assumptions:
1.
2.
The two samples are random and independent
The data is continuous (this ensures the probability of ties = 0)
Section 14.4 Comparing Two Populations: Dependent Samples (Paired Difference Experiment)
Analyze data from a paired difference experiment.
Nonparametric test: Wilcoxon Sign-Rank Test.
Hypothesis
Test Statistics
Ho: D1 and D2 are identical
Ha: D1 is shift either to the left or right of D2
T  min T , T 
T  T
Ho: D1 and D2 are identical
Ha: D1 is shift to the right of D2
Ho: D1 and D2 are identical
Ha: D1 is shift to the left of D2
Rejection Region
T  T0
T  T
T0
is given in Table XIII (pg 908)
Assumptions:
1. The sample of differences was randomly selected from the population of differences.
2. The probability distribution for the paired differences is continuous.
3. The population differences have a distribution that is symmetric.
Section 14.5 Comparing Three or More Populations: Completely Randomized Design
Compare the k population means of the treatments from CRD
Nonparametric test: Kruskal-Wallis H-test
Ho: The k probability distributions are identical
Ha: At least two of the k probability distributions differ in location
Test Statistics :
Where
12   Ri2
H
 
n(n  1)   ni

  3(n  1)


n  n1  n2  .....  nk
Rejection Region:
H   2
Assumptions:
1.
2.
3.
with df = k-1
All the samples are independent and randomly
selected.
Each sample has at least 5 observations.
The k probability distributions are continuous
Section 14.6 Comparing Three or More Populations: Completely Randomized “Block” Design
Compare the k population means of the treatments from CRBD
Nonparametric test: Friedman
Fr  Test .
Ho: The k probability distributions are identical
Ha: At least two of the k probability distributions differ in location
Test Statistic:
12
bk (k  1)
Fr   2
Fr 
Rejection Region:
 R  3b(k  1)
2
i
with df = k-1
Assumptions:
1.
2.
3.
4.
The treatments are randomly assigned to the blocks.
The probability distributions from which the samples
within each block are drawn are continuous.
There is no interaction between blocks and
treatments.
The observations in each block may be ranked.