The document discusses hypothesis testing in statistics, covering both parametric and non-parametric tests. It explains the null and alternate hypotheses, provides examples, and details the types and applications of tests like the t-test, Mann-Whitney U test, and Kruskal-Wallis test. Furthermore, it highlights the distinctions between parametric and non-parametric tests based on their assumptions about population parameters and data distributions.

1. Hypothesis Testing- Parametric and Non-
Parametric Tests in Statistics
In hypothesis testing, Statistical tests are used to
check whether the null hypothesis is rejected or
not rejected. These Statistical tests assume a
null hypothesis of no relationship or no difference
between groups.
So, In this article, we will be discussing the
statistical test for hypothesis testing including both
parametric and non-parametric tests

2. Hypothesis Testing is a type of statistical analysis in which you
put your assumptions about a population parameter to the test. It
is used to estimate the relationship between 2 statistical
variables.
Let's discuss few examples of statistical hypothesis from real-life
-
A teacher assumes that 60% of his college's students come from
lower-middle-class families.
A doctor believes that 3D (Diet, Dose, and Discipline) is 90%
effective for diabetic patients.
Now that you know about hypothesis testing, look at the two
types of hypothesis testing in statistics.

3. Null Hypothesis and Alternate Hypothesis
The Null Hypothesis is the assumption that the
event will not occur. A null hypothesis has no
bearing on the study's outcome unless it is
rejected.
H0 is the symbol for it, and it is pronounced H-
naught.
The Alternate Hypothesis is the logical
opposite of the null hypothesis. The acceptance
of the alternative hypothesis follows the
rejection of the null hypothesis. H1 is the
symbol for it.

4. Let's understand this with an example.
A sanitizer manufacturer claims that its product kills 95
percent of germs on average.
To put this company's claim to the test, create a null and
alternate hypothesis.
H0 (Null Hypothesis): Average = 95%.
Alternative Hypothesis (H1): The average is less than 95%.
Another straightforward example to understand this concept
is determining whether or not a coin is fair and balanced. The
null hypothesis states that the probability of a show of heads
is equal to the likelihood of a show of tails. In contrast, the
alternate theory states that the probability of a show of heads
and tails would be very different.

7. Parametric Tests
Parametric tests are those tests for which we have prior knowledge of the
population distribution (i.e, normal), or if not then we can easily approximate it to a
normal distribution which is possible with the help of the Central Limit Theorem.
Parameters for using the normal distribution is –
Mean
Standard Deviation
Parametric tests usually assume certain properties of the parent
population from which we draw samples. Assumptions like
observations come from a normal population, sample size is large,
assumptions about the population parameters like mean, variance, etc.,
must hold good before parametric tests can be used.

8. But there are situations when the researcher cannot or does not want to make
such assumptions. In such situations we use statistical methods for testing
hypotheses which are called non-parametric tests because such tests do not
depend on any assumption about the parameters of the parent population.
Besides, most non-parametric tests assume only nominal or ordinal data, whereas
parametric tests require measurement equivalent to at least an interval
scale. As a result, non-parametric tests need more observations than
parametric tests to achieve the same size of Type I and Type II errors
IMPORTANT PARAMETRIC TESTS
The important parametric tests are:
1. z-test;
2. t-test;
3. χ2 -test, and
4. F-test.

11. The main difference between these two tests is that
one of them is dependent and the other is
independent to a certain extent from parameters like
mean, standard deviation, variation, and Central
Limit Theorem. All of these are different
parameters calculated on the data available.
Although every parametric test has a
nonparametric counterpart or equivalent.
Parametric: you know about the population
Non-Parametric: you don’t know about the population

12. Properties Parametric Non-parametric
Assumptions Yes No
central tendency Value Mean value Median value
Correlation Pearson Spearman
Probabilistic distribution Normal Arbitrary
Population knowledge Requires Does not require
Used for Interval data Nominal data
Applicability Variables Attributes & Variables
Examples z-test, t-test, etc.
Kruskal-Wallis, Mann-
Whitney

13. Parameters of
Comparison
Parametric Nonparametric
Definition
The test whose outcomes depend
on the distribution is called a
parametric test.
The test whose outcomes do not
depend on the distribution is
called a nonparametric test.
Statistical power
Parametric tests have higher
statistical power.
Nonparametric tests have lower
statistical power.
Versatility
Parametric tests are not applicable
to all situations.
Nonparametric tests are more
robust and can be applied to
different situations.
Central Tendency value
Mean value is the central tendency
value for this test.
Median value is the central
tendency value for this test.
Type of distribution
It is used on data that follows a
normal distribution.
It is used on data that follows any
arbitrary distribution.

14. What is Nonparametric Test?
Nonparametric tests are tests that aren’t dependent on any
assumptions of the data distribution or parameters to analyze
them. They are also sometimes referred to as “distribution-free
tests”. Nonparametric doesn’t necessarily mean that we know
nothing about the population, it means that the data is skewed
or “not normally distributed”.
The reasons why we use nonparametric tests are: *if the data doesn’t meet the
assumptions for the population sample or when data is skewed, *the population
sample size is too small, or *the data being analyzed is nominal or ordinal. The
different types of nonparametric tests are :
•Run Test for randomness of data,
•Mann Whitney U Test,
• Wilcoxon Matched Pairs Rank Test,
•Kruskul-Wallis Test,
• Kolmogorov-Smirnov Test

15. Run Test for randomness of data
•It is sometimes called the Geary test. It is test for randomness of dichotomous variables.
(A dichotomous variable is one that takes on one of only two possible values when observed
or measured)
What is a run?
Let's suppose we have observations of the random variable X, and observations of the random
variable Y.
Suppose we combine the two sets of independent observations into one larger collection of
observations, and then arrange the observations in increasing order of magnitude.
If we label from which set each of the ordered observations originally came, we might observe
something like this:
where x denotes an observation of the random variable X and y denotes an observation of
the random variable Y.
(We might observe this, for example, if the X values were 0.1, 0.4, 0.5, 0.6, 0.8, and 0.9, and
the Y values were 0.2, 0.3, 0.7, 1.0, 1.1, and 1.2).
That is, in this case, the smallest of all of the observations is an X value,
the second smallest of all of the observations is a Y value, the third smallest of all of the
observations is a Y value, the fourth smallest of all of the observations is an X value, and so on.
Now, each group of successive values of X and Y is what we call a run.

16. So, in this example, we have six runs. If we instead observed this ordered
arrangement:
we would have three runs. And, if we instead observed this ordered arrangement
:
we would have eight runs.

17. Why runs?
The next obvious question is in
what way might knowing the
number of runs be helpful in
testing the null hypothesis of the
equality of the distributions F(x)
and G(y). Let's investigate that
question by taking a look at a few
examples. Let's start with the case
in which the distributions are
equal. In that case, we might
observe something like this. In
this particular example, there are
eight runs. As you can see, this
kind of a picture suggests that
when the distributions are equal,
the number of runs will likely be
large.

18. Now, let's take a look at one way in which
the distribution functions could be
unequal. One possibility is that one of the
distribution functions is at least as great as
the other distribution function at all points
z. This situation might look something like
this. In this particular example, there are
only two runs. This kind of a situation
suggests that when one of the distribution
functions is at least as great as the other
distribution function, the number of runs
will likely be small. Note that this is what
the distribution functions might look like if
the median of Y was greater than the
median of X.

19. Man – Whitney U Test
•This test enables 2 independent samples by testing hypothesis on 2 population median.
also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test,
or
Wilcoxon–Mann–Whitney test)
•used to test whether two samples are likely to derive from the same population
(i.e., that the two populations have the same shape).
(Nonparametric tests used on two dependent samples are the Sign test and the
Wilcoxon signed-rank test)
(Non parametric test always talks about median, not by mean and variance)
The null and two-sided research hypotheses for the nonparametric test
are stated as follows:
H0: The two populations are equal versus
H1: The two populations are not equal.

20. Test Statistic for the Mann Whitney U Test
The test statistic for the Mann Whitney U Test is
denoted U and is the smaller of U1 and U2, defined
below.
where R1 = sum of the ranks for group 1 and R2 = sum
of the ranks for group 2.

21. Key Concept:
For any Mann-Whitney U test, the theoretical range of
U is from 0 (complete separation between groups, H0
most likely false and H1 most likely true) to n1*n2 (little
evidence in support of H1).
In every test, U1+U2 is always equal to n1*n2.

22. Wilcoxon Matched Pairs Rank Test
•The Wilcoxon signed-ranks test is a non-parametric equivalent of the paired t-
test. It is most commonly used to test for a difference in the mean (or median) of
paired observations - whether measurements on pairs of units or before and after
measurements on the same unit. It can also be used as a one-sample test to test
whether a particular sample came from a population with a specified median.
•The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test
used either to test the location of a population based on a sample of data, or to
compare the locations of two populations using two matched samples.
•The Wilcoxon test can be a good alternative to the t-test when population
means are not of interest; for example, when one wishes to test whether a
population's median is nonzero, or whether there is a better than 50% chance
that a sample from one population is greater than a sample from another
population.

23. Two slightly different versions of the test exist:
The Wilcoxon signed rank test compares your sample median
against a hypothetical median.
The Wilcoxon matched-pairs signed rank test computes the
difference between each set of matched pairs, then follows the
same procedure as the signed rank test to compare the sample
against some median.
The term “Wilcoxon” is often used for either test. This usually
isn’t confusing, as it should be obvious if the data is matched, or
not matched.
The null hypothesis for this test is that the medians of two
samples are equal. It is generally used:
As a non-parametric alternative to the one-sample t test or paired
t test.

24. •The Wilcoxon matched-pairs signed-rank test is a
nonparametric method to compare before-after, or matched
subjects. It is sometimes called simply the Wilcoxon matched-
pairs test.

25. The Kruskal-Wallis H test (sometimes also called the "one-way
ANOVA on ranks") is a rank-based nonparametric test that can be
used to determine if there are statistically significant differences
between two or more groups of an independent variable on a
continuous or ordinal dependent variable. It is considered the
nonparametric alternative to the one-way ANOVA, and an extension
of the Mann-Whitney U test to allow the comparison of more than
two independent groups.
Kruskal-Wallis (the nonparametric version of the ANOVA)
ANOVA allows a comparison of more than 2 groups at same time to determine whether
a relationship exists between them.

26. For example, you could use a Kruskal-Wallis H test to understand
whether exam performance, measured on a continuous scale from
0-100, differed based on test anxiety levels (i.e., your dependent
variable would be "exam performance" and your independent
variable would be "test anxiety level", which has three
independent groups: students with "low", "medium" and "high"
test anxiety levels). Alternately, you could use the Kruskal-Wallis
H test to understand whether attitudes towards pay discrimination,
where attitudes are measured on an ordinal scale, differed based on
job position (i.e., your dependent variable would be "attitudes
towards pay discrimination", measured on a 5-point scale from
"strongly agree" to "strongly disagree", and your independent
variable would be "job description", which has three independent
groups: "shop floor", "middle management" and "boardroom").

27. The Kruskal-Wallis test is one of the non parametric tests that is
used as a generalized form of the Mann Whitney U test. It is used
to test the null hypothesis which states that ‘k’ number of samples
has been drawn from the same population or the identical
population with the same or identical median. If Sj is the
population median for the jth group or sample in the Kruskal-
Wallis test, then the null hypothesis in mathematical form can be
written as S1 =S2= ….. = Sk. Obviously, the alternative
hypothesis would be that Si is not equal to Sj. This means that at
least one pair of groups or samples has different pairs.
In order to apply the Kruskal-Wallis test, one has to write the data
in a two way format in such a manner that each column
represents each successive sample. In the computation each of the
‘N’ observations is replaced in the form of ranks. This means that
all the values from the ‘k’ number of samples are combined
together and are ranked in a single series.

28. The Kruskal-Wallis test allows to compare three or more
groups. More precisely, it is used to compare three or more
groups in terms of a quantitative variable. It can be seen as the
extension to the Mann-Whitney test which allows to compare 2
groups under the non-normality assumption.
“Is the length of the flippers different between the 3 species of penguins?”
The null and alternative hypotheses of the Kruskal-Wallis test are:
•H0: The 3 species are equal in terms of flipper length
•H1: At least one species is different from the other 2 species in terms
of flipper length

29. Kolmogorov-Smirnov Test
The Kolmogorov-Smirnov Goodness of Fit Test (K-S test) compares
your data with a known distribution and lets you know if they
have the same distribution. Although the test is nonparametric — it
doesn’t assume any particular underlying distribution — it is
commonly used as a test for normality to see if your data is normally
distributed.It’s also used to check the assumption of normality in
Analysis of Variance.
More specifically, the test compares a known
hypothetical probability distribution (e.g. the normal
distribution) to the distribution generated by your data —
the empirical distribution function.

30. The hypotheses for the test are:
Null hypothesis (H0): the data comes from the specified
distribution.
Alternate Hypothesis (H1): at least one value does not match the
specified distribution.
That is,
H0: P = P0, H1: P ≠ P0.
Where P is the distribution of your sample , P0 is a specified
distribution
The Kolmogorov–Smirnov statistic quantifies a distance
between the empirical distribution function of the sample and
the cumulative distribution function of the reference
distribution, or between the empirical distribution functions of
two samples. The null distribution of this statistic is calculated
under the null hypothesis that the sample is drawn from the
reference distribution (in the one-sample case) or that the
samples are drawn from the same distribution (in the two-
sample case)

31. The Kolmogorov–Smirnov test can be modified to serve as a
goodness of fit test. In the special case of testing for normality of
the distribution, samples are standardized and compared with a
standard normal distribution. This is equivalent to setting the mean
and variance of the reference distribution equal to the sample
estimates, and it is known that using these to define the specific
reference distribution changes the null distribution of the test
statistic