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Hypothesis – Concept, Example, Meaning, Definition, Formulation & Types
Critical Regions – One tailed & two tailed
Types of errors – Type I & Type II
Parametric Statistical test – Meaning, Example & Types
Non-parametric Statistical test – Meaning, Example & Types
Differences between Parametric & Non-parametric tests

COMMON STATEMENTS WE SEE ACCROSS
(i) The refrigerator of certain brand
saves up to 20% electric bill,
(ii) The motorcycle of certain brand
gives 60 km/liter mileage,
(iii) A detergent of certain brand
produces the cleanest wash,
(iv) Ninety nine out of hundred
dentists recommend brand A
toothpaste for their patients to
save the teeth against cavity,
etc.
The technique of testing
such type of claims or
statements or
assumptions is known
as testing of
hypothesis.
The truth or falsity of a
claim or statement is
never known unless
we examine the entire
population.
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CONCEPT OF HYPOTHESIS TESTING
For example,
(i) a customer of motorcycle wants to test whether the claim of
motorcycle of certain brand gives the average mileage 60 km/liter
is true or false
(ii) the businessman of banana wants to test whether the average
weight of a banana of Kerala is more than 200 gm,
(iii) a doctor wants to test whether new medicine is really more
effective for controlling blood pressure than old medicine,
(iv) an economist wants to test whether the variability in incomes differ
in two populations,
(v) a psychologist wants to test whether the proportion of literates
between two groups of people is same, etc.
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Hypothesis(es) Meaning
◦ A hypothesis is a possible answer to a research question.
◦ It is a presumption or a hunch on the basis of which a study has to
be conducted.
◦ This hypothesis is tested for possible rejection or approval.
◦ Hypothesis is an assumption made about population parameter
(e.g., mean, median, variance, proportion, etc.).
◦ Research hypothesis is a predictive statement capable of
being tested by scientific methods that relates an IV to
some DV.
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Definition of Hypothesis
Goode and Hatt defined it as “ a proposition which can be put to test to
determined validity”.
Rummel “ a hypothesis is a statement capable of being tested and there
by verified or rejected”.
Grinnell – A hypothesis is written in such a way that it can be proven
or disproven by valid and reliable data, in order to obtain these
data that we perform our study.
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If you want to conduct a study on the effects of ‘Parental Depression on the
Academic Performance of Children’, you may like to conduct it without any
hypothesis but then you will have many dimensions to think upon and will be more
likely get distracted.
If you formulate a hypothesis that ‘Parental depression results in depression in
children too and this depression leads to low grades’ your research will get a
direction and you will not think about the broader effects of depression, everything
is well defined as on the grades of children, now you have to test the impact of
parental depression on the children’s depression and as well as on the grades
of children.
You may not need to test impacts on the extracurricular activities, class conduct
and other such things
How Does Hypothesis Works
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Types of hypothesis
Null Hypothesis(Hₒ)
◦ According to Prof. R.A. Fisher, “A null
hypothesis is a hypothesis which is
tested for possible rejection under
the assumption that it is true.”
◦ The hypothesis which we wish to test is
called as the null hypothesis.
◦ Ex : there is no relationship between
“student attendance” and “ student
result”
Alternative
Hypothesis(H₁)
◦ The hypothesis which
complements to the null
hypothesis is called
alternative hypothesis.
◦ Ex: there is relationship
between “ student attendance”
and “ student result”
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We state the null and alternative hypotheses in such a way that they cover all possibility of the
value of population parameter.

Formulation claims for Hypothesis(es)
Symbolic form of the
claims
(i) µ = 60 km / liter.
(ii) µ > 200 gm
(iii) µ1 > µ2
(iv) 𝝈₁² ⧣ 𝝈₂²
(v) P₁ = P₂ or P₁ –P₂ = 0
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For example,
(i) a customer of motorcycle wants to test whether
the claim of motorcycle of certain brand gives
the average mileage 60 km/liter is true or
false
(ii) the businessman of banana wants to test
whether the average weight of a banana of
Kerala is more than 200 gm,
(iii) a doctor wants to test whether new medicine is
really more effective for controlling blood
pressure than old medicine,
(iv) an economist wants to test whether the
variability in incomes differ in two populations,
(v) a psychologist wants to test whether the
proportion of literates between two groups of
people is same, etc.

Formulation of Null Hₒ & H₁ Hypothesis(es)
 “the motorcycle of certain brand gives the average mileage
60km/liter”. Symbolically (µ=60km/liter)
 The claim is µ = 60 km/liter then,
 Complement is µ ≠ 60km/liter.
 Null Hypothesis Hₒ: µ = 60 km/liter
 Alternative Hypothesis H₁: µ ≠ 60km/liter
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The thump rule is
that the statement
containing equality
is the null
hypothesis.

Formulation Null Hₒ & H₁ Hypothesis(es)
 “the average weight of a banana of Kerala is greater then
200gm”.
Symbolically ( µ > 200 gm).
 The claim is µ > 200gm
 Complement is µ ≤ 200 gm.
 Null Hypothesis Hₒ: µ ≤ 200 gm
 Alternative Hypothesis H₁: µ > 200 gm
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General Procedure for Hypothesis
Testing
Set up null Hₒ and Alternative H₁ Hypothesis
Decide the level of significance (α)
Choose an appropriate test statistic
Calculate the value of the test statistic
Obtain the critical (or cut-off) values
Compare the calculated value of test statistic
with critical value
Derive the conclusion
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Set up null Hₒ and Alternative H₁ Hypothesis
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Decide the level of significance (α)
◦ After setting the null and alternative hypotheses, we
establish a criteria for rejection or non-rejection of
null hypothesis, that is, decide the level of
significance (𝜶), at which we want to test our
hypothesis.
◦ Generally, it is taken as 5% or 1% (α = 0.05 or 0.01).
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Choose an appropriate test statistic
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Test statistic = statistic – Value of the parameter under
Standard error of statistic
Standard form like Z (standard normal), Chi Square.
t. F or any other well-known in literature.

Derive the conclusion
Do not Reject Hₒ
 If calculated value of test
statistic lies in non-rejection
region at α level of
significance then we do not
reject null hypothesis.
 If the calculated value is
lesser than Critical(Table)
Value then we do not reject
null hypothesis. CV<TV
Reject Hₒ
 If calculated value of test
statistic lies in rejection
region at α level of
significance then we reject
null hypothesis.
 If the calculated value is
greater than Critical
(Table) Value then we
reject null hypothesis.
CV>TV
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Critical Regions
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Null and Alternative Hypotheses and
Corresponding One-tailed and Two-tailed Tests
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TYPE-I AND TYPE-II ERRORS
◦ A faulty sample misleads the inference (or
conclusion) relating to the null hypothesis.
◦ For example, an engineer infers that a packet of
screws is substandard when actually it is not. It is an
error caused due to poor or inappropriate (faulty)
sample.
◦ Similarly, a packet of screws may infer good when
actually it is sub-standard.
◦ So we can commit two kinds of errors while testing a
hypothesis
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TYPE-I AND TYPE-II ERRORS
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Parametric Statistical test
Parametric statistic is a
branch of statistic,
which assumes that
sample data comes
from a population
that follows a
probability or normal
distribution.
When the assumption are
correct, parametric
methods will produce
more accurate and
precise estimates.
Assumptions
 The scores must be independent
 The observations must be drawn
from normally distributed
populations (Follow ND)
 The selected population is
representative of general
population
 The data is in Interval or Ratio
scale
 The populations (If comparing two
or more groups) must have the
same variances
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Types of Parametric test
Z- test t-test ANOVA F-test
Chi-
Square
test

Z- test
 A Z-test is given by Fisher.
 A Z-test is a type of hypothesis test or statistical test.
 It is used for testing the mean of a population versus a
standard or comparing the means of two population
 Z-test is the statistical test, used to analyze whether two
population means are different or not when the variances are
known and the sample size is large.
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Assumptions of Z-test
 Sample size is greater than 30.
 Data point should be independent from each other.
 Data should be randomly selected from a
population, where each item has an equal chance of
being selected.
 Data should follow normal distribution.
 The standard deviation of the populations is known.
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One-sample z-test
One-sample z-test we are
comparing the mean,
calculated on a single of
score (one sample) with
known standard
deviation.
Ex. The manager of a candy
manufacture wants to
know whether the mean
weight of batch of candy
boxes is equal to the
target value of 10 pounds
from historical data.
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Two-sample z-test
When testing for the differences between two groups can imagine two separate
situation.
Comparing the proportion of two population.
In two sample z-test both independent populations.
Ex: 1. Comparing the average engineering salaries of men versus women.
2. Comparing the fraction defectives from two production line.
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t-test
◦ It is developed by
Prof. W.S Gosset in
1908. It is also called
student t-test.
◦ A t-test statistical
significance indicates
whether or not the
difference between
two groups.
Assumption
 Samples must be random
and independent.
 When samples are small
n<30
 Standard deviation is not
known.
 Population is Normal
distributed.
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There are two ways to calculate T-test such as,
a. One sample t-test
b. Unpaired t-test.(independent two sample t-test)
c. Paired t-test.

One sample t-test
◦ In a one-sample t-test,
we compare the
average (or mean) of
one group against the
set average (or mean).
◦ This set average can be
any theoretical value (or
it can be the population
mean).
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t = t-statistic
m = mean of the group
µ = theoretical value or population mean
s = standard deviation of the group
n = group size or sample size

Unpaired t-test:
If there is no link between the data
then use the unpaired t-test.
When two separate set of
independent sample are obtain
one from each of the two
population being compared.
Ex:
1. Compare the height of girls
and boys.
2. Compare the 2 stress reduction
intervention. When one group
practiced mindfulness
meditation, while other learned
yoga.
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Paired t-test
Paired t-test consists of a sample of matched
pairs of similar units or one group of units
that has been tested.
If there is some link between the data then use
the paired t-test.(e.g. Before and after)
Ex: 1. where subject are tested prior to a
treatment say for high blood pressure,
and the same subject are tested again
after treatment with a blood pressure
lowering medication.
2. Test on person or any group before and
after training.
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ANOVA (Analysis of Variance)
The statistical technique known as “Analysis of
Variance”, commonly referred to by the
acronym ANOVA was developed by Prof.
R. A. Fisher in 1920’s.
The analysis of variance focuses on variability.
Variation is inherent in nature, so analysis of
variance means examining the variation
present in data or parts of data.
In other words, analysis of variance means to
find out the cause of variation in the data.
According to Professor R. A. Fisher, Analysis of
Variance (ANOVA) is "Separation of
variance ascribable to one group of
causes from the variance ascribable to
other group".
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Types of ANOVA
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If we consider, only
one independent
variable which
affects the
dependent variable.
One-way ANOVA
If the independent
variables/explanatory
variables are more than
one i.e. n (say) then it is
called n-way ANOVA. If n
is equal to two, then the
ANOVA is called Two-
way classified ANOVA
Two-way classified
ANOVA
is used when the
experimenter wants
to study the
interaction effects
among the
explanatory
variables
Factorial ANOVA
is used when the same
subjects
(experimental units)
are used for each
treatment (levels of
explanatory variable).
Repeated measure
ANOVA
Multivariate analysis of
variance (MANOVA)
is used when there
is more than one
response variable.

One-way ANOVA
◦ One-way analysis of
variance is a technique
where only one
independent variable at
different levels is
considered which affects
the response variable.
◦ Ex: You might be studying
the effect of tea on weight
loss, from three groups,
green tea, black tea, no
tea.
Assumptions
1) Dependent variable
measured on interval scale;
2) k sample are independently
and randomly drawn from
the population;
3) Population can be reasonably
to have a normal distribution;
4) Homogeneity of sample
variance.
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Two-way ANOVA
Two way ANOVA technique is
used when the data are
classified on the basis of two
factors.
And two way ANOVA analyzed
a 2 independent variable
and 1 dependent variable.
Ex: The agricultural output
may be classified on the
basis of different verities of
seeds and also on the basis
of different verities of
fertilizer used. 35

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ANOVA Table for One-way Classified Data

Chi-Square test
◦ It is drawn by Karl Pearson. Chi square test
is a statistical test used as a parametric for
testing for comparing variance .
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Non-parametric statistics test
Non-parametric statistics is the branch of statistics.
It refers to a statistical method in which the data is not required to fit a normal
distribution.
Nonparametric statistics uses data that is often ordinal, meaning it does not
rely on numbers, but rather a ranking or order of sorts.
For example: a survey conveying consumer preferences ranging from like to
dislike would be considered ordinal data.
This type of statistics can be used without the mean, sample size, standard
deviation or estimation of any other parameters.
The non-parametric test are called as “distribution-free” test since they make
no assumptions regarding the population distribution.
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Types of Non-parametric tests
 Chi-square test
 Rank correlation test
Rank sum test
◦
◦
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Chi-square test
◦ The chi-square (chi, the Greek letter pronounced "kye”)
◦ is officially known as the Pearson chi-square in homage to
its inventor, Karl Pearson.
◦ The chi-square test is one of the nonparametric tests for
testing three types of statistical tests:
▫ The goodness of fit,
▫ Test of Independence, and
▫ Homogeneity.
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Chi-square test
Test of Independence:
The independence of test is
difference between the
frequencies of occurrence in
two or more categories with two
or more groups.
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Goodness of fit.
Refers to whether a significant difference
exists between an observed number and
an expected number of responses, people
or other objects.
For example: suppose that we flip a coin 20
times and record the frequency of
occurrence of heads and tails. Then we
should expect 10 heads and 10 tails.

Spearman’s rank correlation test
◦ This method a measure of
association, that is based on
the rank of observation and
not on the numerical value of
the data
◦ It was developed by famous
Charles spearman in the early
1990s and such it is also known
as spearman’s rank correlation
co-efficient.
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Mann-Whitney U-test
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Mann-Whitney U-test
Assumptions
(i) The two samples are randomly and
independently drawn from their
respective populations.
(ii) The variable under study is
continuous.
(iii) The measurement scale is at least
ordinal.
(iv) The distributions of two populations
differ only with respect to location
parameter.
◦ Formula
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◦ the medians several
(more than two) populations the
same or not.
◦ non-parametric analogue to one
way ANOVA based on rank transformed data.
◦ The Kruskal-Wallis one way analysis of variance by ranks
(Kruskal, 1952) and (Kruskal and Wallis, 1952) is employed
with ordinal (rank order) data in a hypothesis testing
situation involving a design with two or more independent
samples.
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Assumptions
a) Each sample has been
randomly selected from the
population it represents;
b) The k samples are
independent of one another;
c) The variable under study is
continuous.
d) The measurement scale is at
least ordinal.
Kruskal-Wallis Test Statistics
Adjusted Factor for Ties
◦ Adjusted K-W Test Statistics
H=
𝐻
𝐶
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H =
12
𝑛(𝑛 + 1)
𝑖=1
𝑘
𝑅ᵢ²
𝑛ᵢ
− 3(n + 1)
C =
1
(𝑛³ − 𝑛)
𝑖=1
𝑟
𝑡ᵢ³ − 𝑡

Differences between Parametric & Non-parametric tests
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