The document discusses the Wilcoxon Rank-Sum test, a non-parametric hypothesis test that checks if two populations are identically distributed. It assumes the two populations are identically distributed and their ordering would be evenly intermixed. The test ranks all observations from both groups as if they come from one large group, then sums the ranks for at least one population's sample. It also discusses Type I and Type II errors in hypothesis testing, where a Type I error rejects the null hypothesis when it is true, and a Type II error accepts the null hypothesis when it is false. The probabilities of each are denoted by alpha and beta, respectively.

1. Wilcoxon Rank-Sum Test
and
Type I and Type II Errors
Lakshmi. M. B
Sr. No. 9219708
S3, MTech CSE
Sahrdaya College of Engineering and Technology, Kodakara
October 6, 2020
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2. Wilcoxon Rank-Sum Test
Non-parametric hypothesis test
Checks whether two populations are identically distributed.
Assumption: 2 populations are identically distributed.
Expectation: Ordering would be evenly intermixed among
themselves.
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3. Steps involved in Wilcoxon rank-sum test are:-
1 Rank the set of observations from the 2 groups as if they
come from one large group.
2 The assigned ranks are summed for atleast one population’s
sample.
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4. Example:
Let 2 populations be pop1 and pop2 with
independently random samples of size n1 and n2
respectively.
Total no:of observations, N = n1 + n2
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5. Step 1:
Smallest observation receives rank 1
Second smallest observation receives rank 2
.
.
.
Largest observation receives rank N.
Ties among observation receives rank equal to the average
ranks they span.
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6. Step 2:
Ranks are used to specific assumptions about the shape of the
distribution.
If the distribution of pop1 is shifted to the right of pop2, then
the rank-sum of pop1 sample should be larger than the
rank-sum of pop2.
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7. Wilcoxon rank-sum test determines the significance of the
observed rank-sums.
wilcox.test() – ranks the observations, determines the
respective rank-sums, and then determines the probability of
such rank-sums of such magnitude.
Example: wilcox.test(x, y, conf.int = TRUE)
More robust than the t-test.
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8. Type I and Type II Errors
There are 2 types of errors in a hypothesis test.
They are:-
1 Type I Error
2 Type II Error
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11. Type I Error:
Rejection of null hypothesis when the null hypothesis is TRUE.
Probability is denoted by α
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12. Type II Error:
Acceptance of null hypothesis when the null hypothesis is
False.
Probability is denoted by β
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13. By selecting an appropriate significance level, the probability
of committing a type I error can be defined before any data is
collected or analyzed.
Probability of committing a Type II error is more difficult to
determine.
If 2 population means are truly not equal, the probability of
committing a type II error will depend on how far apart the
means truly are.
To reduce the probability of a type II error to a reasonable
level; increase the sample size.
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