Non-Parametric Test :
Mann-Whitney U Test
if we are interested in testing of difference between mean of two
independent population. In that case we use two sample T-Test is used when the
data follows assumptions of parametric T-Test. like two independent sample are
drawn from normal population & have equal variance. the variable are
measured in at least of an interval scale. But if the data are collected on
ordinal scale and sample drawn from population is not known. (that situation
aeries in the different filed like study of marketing, or biological studies,
etc. ). For that cases the parametric test cannot be used. in that situation a
Non- Parametric test are more appropriate. In such circumstances the
simple Non-Parametric test used is known as Mann-Whitney U test.
This Non-Parametric
Mann-Whitney U test developed by Mann, Whitney and Wilcoxon. therefore it name
is Mann-Whitney U test, and sometimes it is also called Wilcoxon's rank sum
test. The Mann-Whitney U test is alternative to the Parametric T-Test for
testing difference between means of two populations. if the assumptions of
T-Test is fulfill then Mann-Whitney U test gives weaker result than T-Test.
Assumptions: the test based on the following
assumptions.
1. The two samples are randomly
and independently drawn from population.
2.The variable measured in at
least ordinal scale.
3.The variable under study is
continuous.
the testing procedure of
Mann-Whitney U test is as follows.
Let X1, X2, .......Xn1, and Y1, Y2, .......Yn2, be a random and independent sample are drawn from two population
having median μ1 and μ2 respectively. here we want to test the hypothesis about the
median of two population is same or not therefore the null and alternative
hypothesis is
H0:
μ1 = μ2 V/S
H1: μ1 ≠ μ2
OR we also set the hypothesis for testing the
two sample are drawn from identical population repetitive to location of
population i.e. median of population. Therefore the null and alternative
hypothesis is
H0:
F1(X) = F2(X) V/S H1: F1(X) ≠ F2(X)
the following step consist for
testing:
Step I: Firstly we combining all observations from
two samples.
Step II: Then assign the rank to all combined
observations from smallest to largest observation that means we assign the rank
1 to smallest observation and rank 2 to next smallest observation and so on up
to last observation. and in data if the tie is occurs then we assign the
average rank to that observations.
Step III: Now we compute R1 and R2
R1 Is a sum of rank of
first sample of size n1
R2 Is a sum of rank of
Second sample of size n2
Step IV :
the test statistics is
U1= n1 n2 + {( n1(n1+1))/2} - R1
U2= n1 n2 +{( n2(n2+1))/2} – R2
Test Statistics is
U= min (U1, U2)
And taking decision about the null hypothesis
we firstly obtain the critical value of test statistics at 𝜶 % level of significance. and it can be obtained from the table of
critical values of Mann- Whitney U test.
Decision Rule: if the calculated value of test
statistics U is less than or equal to critical value(i.e Ucal ≤ Utab) then we reject the
null hypothesis at 𝜶 % level of significance. other wise
we accept the null hypothesis. (if sample is less
than 20 use above test statistics U )
if the sample size is greater
than 20 then we use large sample test.
for Large sample
if either n1 or n2 are greater than 20 the test
statistics U is approximately normally distributed with mean E(U) and variance
var(U) .
E(U) = (n1 x n2) /2 and Var(U) = {(n1x n2(n1 + n2 +1))}/12
now the normal approximation test statistics is
Z = {U - E(U)}/√var(U)
Decision Rule: if the calculated value of test
statistics Z is less than or equal to critical value(i.e Zcal > Ztab) then we reject the
null hypothesis at 𝜶 % level of significance. otherwise
we accept the null hypothesis. (i.e Zcal ≤ Ztab)
also read the assumptions for
parametric and Non-Parametric test.
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