CHAPTER

5
HYPOTHESIS TESTING
AND T-TEST
INTRODUCTION
The formulation of a hypothesis is a step towards formalising the research process. It is an
essential part of scientific method of research. The quality of hypothesis determines the value of the
results obtained from research. The value of hypothesis in research has been aptly stated by Claude
Bernard as follows, "The experimental method willnot give new and productive ideas to those who
do not have them, it willonly help in guiding the ideas to those who have them; and in developing
those so as to draw the best possible results. The ideas is the seed; the method is the soil which
provides it with the conditions to develop, to prosper and give better fruits following its nature. But
just as the soil willnever produce anything other than what has been sown, similarly only those ideas
which have been put to the experimental methods will be developed by the latte." Thus the ideas
stated in the form of hypothesis will determine the output of results. The value of hypothesis is high
in situations where information about the population parameter is difficult to get and sample data is
the only source of data which can be generalised through hypothesis testing. The current chapter
ntroduces the concept of hypothesis and discusses some parametric tests.

MEANING OF HYPOTHESIS
Hypothesis is an assumption made about a population parameter. This hypothesis is then proved
Or disproved by using the information from the sample to decide the likelihood of the hypothesized
POpulaion parameter to be correct or not. Hypothesis testing is a screening exercise. Hypothesis are
questions asked about the obiect of research and at the same time about the facts gathered by
oDServation and proposals for answers to these questions.! Hypothesis is sometimes a predictive
Datement that relates independent variable to a dependent variable and this relationship is open to
testing. The purpose of testing is not to question the findings obtained fromn sample but to judge the
un behind the difference between either two sample values or between a sample value and
Population parameter.
 The ,
prPROCEDURE
oHYPOTHESI
cedurTESTI
e NS G
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Determine
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Formulate
Fig. Collect Choose
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hesisHypothesis cancan
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esting Yes and
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edure critical Hypothesis
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 Hypothesis
5.8
Test ing and T
The earlier sections have familiarized us with various concepts of hypothesis testing. he T-basic
Test
testing is to judge whether a difference actually exists between the
objectiveof hypothesis Furtherthisissdone by proving or disapproving the null
ofsampleor
population. hypothesitws.ovalues
Hypothesis
O LFormulatethe
undertaking research starts with defining the problem clearly. Once that
The researcher
achieved has been
the researcherisin a position to define the null hypothesis and alternative hypothesis. Null
hypothesis is a statement of no change or no difference whereas alternative hypothesis is one in
which some difference or effect is expected to take place. The researcher is testing the null hypothesis
and it may be accepted or rejected based on evidence. The alternative hypothesis is the opposite of
alternative hypothesi
null hypothesis and rejection of null hypothesis leads to acceptance of
vice-versa.

Distribution
II. Select Relevant Test and Probability
The
In the next step, it is necessary to selecta statistical technique and probability distribution
choice of probability distribution depends on the purpose of hypothesis test. The researcher should
carefully see how the test statistic is computed and which sampling distribution is it following is
normal, t or distribution.

IIL. Choosing the critical Value

The next step involves deciding upon the criteria for accepting or rejecting the null hypothesis.
This involves a decision on (a) significance level (b) degree of freedom (c) one or two-tailed test.
Sigrificance level should be specified in percentage terms. It indicates the number of sample
means out of 100 that are outside the cut-off limits. A 5% significance level states that there are 5
chances out 100 of not accepting acorrect hypothesis. The choice of correct significance level depends
on the costs involved in committing an a and ß error.
The degrees of freedom refer to the free data used in calculating a sample or test Staus
Mathematically,the degree of freedom is calculated as
d.f. = n-k
where n= number of information items available
k= number of linear
constraints
For example in case of mean the degree of freedom is n since there are no constraints. However
when we calculate variance there is a constraint as mean has to be first calculated. Hence the degrees
of freedom are (1-1). It can be illustrated with an example of choosing five numbers whose sumis10.
There is no restriction in choosing the first four numbers. However, while selecting the fifth number
there is a constraint that it should satisfy the
sum total of 10.
Lastly it has to be decided whether atest is atwo-tailed test or one-tailed test. In case of atwo-
tailed test there are two rejection regions on either side of the curve whereas in case of a one-tailed
test the rejection region lies on one side of the curve only. The decision on one tail or two-tailis
influenced by the alternative hypothesis.
 Statistic
M.Collect Dataand Test
The next stage involves drawing a sample and collecting data using a data collection strategy that
iks the purpose of study. From this data the
suits
test statistic is computed. It is thís test statistic that
determines how close the sample is to null hypothesis.

V. Compare the Test Statistic and the Critical Value

This is the crucial stage where the test statistic computed earlier is compared with the critical
at 5% level
value specified. For example , if Z-distribution is being used then to test a two tailed test
of significance with (1-1) degree of freedom involves calculating the Z-value from the sample data
(called as test statistic). It is then compared to the Z-critical value at 5% i.e. |1.96|.

VI. Taking the Final Decision

null
In the last step, the researcher has to now make a decision of accepting or not accepting a
critical
nypothesis. e.g. (Refer to figure 11.1) If the Z-value calculated from the sample lies in the
hypothesis
egion 1.e. if this value is greater than |1.96| then it is in the rejection region and the null
acceptance
Wll be rejected. If however the Z test statistic value is less than |1.96| then it is in the
region and the null hypothesis will be accepted.
 UNIT-2: PARAMETRC AND NuN PARAMETRIC TESTS AND RESEARCH RPORT
Write briefly about the various sampling methods.
Answer: Model Paper-l, QI(b)
The various sampling methods are as follows,
Sampling Methods

Probability Sampling Non-Probability Sampling

Simple Random Complex Random Convenience Sampling Purposive Sampling

Sampling Sampling
Stratified Sampling Judgement Sampling|
Quota Sampling
Cluster Sampling
Systematic Sampling
Multistage Sampling

Figure: Sampling Methods

A Probability Sampling Method
Probability sampling methods are also known as random sampling methods. In these methods, sampling
pocess is random and the laws of probability are used for sampling. It is used when research is conclusive in
nature. These methods are classified into various types as follows,
1. Simple Random Sampling
In this technique, each and every item of the population is given an equal chance of being included in
the sample. The selection is thus free from personal bias. This method is also known as the method of
chance selection or unrestricted sampling. One of the example of simple random sampling is lottery
methods.
2 Complex Random Sampling
The various types of complex random sampling are as follows,
() Stratified Sampling
This process divides the population into homogenous groups or classes called 'strata'. Asample
is taken from each group by simple random method and the resulting sample is called a stratified
sample.
Astratified sample may be either proportionate or disproportionate. In aproportionate stratified
sampling plan, the number of items drawn from each stratum is proportional to the size of the
strata. While in adisproportionate stratified sampling, equal nunmber of items are taken from each
stratum irrespective of the size of the stratum.
(Ü) Cluster Sampling
This method of sampling is done for small groups or units. Whole population is taken into
consideration according to the given problem and is divided into sub-units which are known as
"clusters.From this sample of sub-units, each unit can be easily measured in the selected cluster
group.
The following points are to be considered for cluster sampling,
(a) Cluster sampling should be cost effective and within the limits of survey.
(b) Sampling units must be same for each cluster.
(ii) Systematic Sampling
This is used in thòse cases where a complete list of the population from which sample to be
drawn is available. The method is used to select every Kh item from the list, where Krefers to the
sampling interval. The starting point is selected at random.
ng:Xerox/Photocopying of this book is a CRIMINAL act. Anyone found guilty ls LIABLE to face LEGAL proceedings.
 SMA RES
54
For exanmple, If acomplete list of 1000
students of a college is available and if
asample of 200students is to be drawn,
then every Sh itenm (K= 5) must be taken.
Suppose the starting point is 3, then the 1%
item is 3rd student, second item would be
8th student (3 +5 8) the 3rd item would
be 13th student and so on.
(iv) Multistage Sampling
In this method, sampling is carried out
in several stages. It is mostly used when
population is very huge and simple
random sampling is not possible. For
example, for conducting a survey on
pre-clection opinion poll, the first stage is
choosing a state, second stage is choosing
town or city and third stage is choosing
the respondents.
B. Non-Probability Sampling Methods
Non-probability sampling methods are also
known as non-random sampling methods. In these
methods, probability is not considered for selecting
the sample. It is used when research is exploratory in
nature. These methods are classified into various types
as follows,
1. Convenience Sampling
Convenience sampling is also called 'chunk'.
A chunk is a fraction of one population taken
for investigation because of its convenient
availability. A sample obtained from readily
available lists such as telephone directories,
automobile registrations is a convenient
sample, even if the sample is drawn at random
from the lists.
2 Purposive Sampling
Sometimes sample cannot be obtained through
convenience sampling. The investigator may
wish to choose sample according to the research
type. In this case, purposive sampling is used.
This methods, under purposive sampling are
also follows,
(1) JudgementSampling
In this nethod, the choice of sampling
items exclusively depend upon the
judgement of the investigations. In
other words, the investigator exercises
his judgements in the choice of sample
items and includes those items in the
sample which he thinks are most typical
of the population with regards to the
aSIA PUBLISHERS AND DIST
 characteristics under investigation. The
success of this method depends on the
excellence in judgement. Ifthe individual
making decisions is knowledgeable
about the population and has a good
judgment, then resulting sample may be
representative.
(ii) Quota Sampling
It is one of the type of judgement
sampling. In this, quotas are set up
according to the given criteria, but within
the quotas the selection of sample items
depends on personal judgement.
6. define hypothesis and its types

Ans. Hypothesis Meaning: Asupposition or proposed explanation made on the basis of

limited evidence as astarting point for further investigation

The word hypothesis consists oftwo words Hvpo +thesis =Hynothesis

Hypo" means tentative or subject to the verification and "Thesis" means statement about
solution of a problem.
The world meaning of the term hypothesis is a tentative statement the solution of the problem
Hypothesis offers asolution of the problem that is to be verified empirically and based on
some rationale.

Definition of Hypothesis: Hypothesis is a tentative prediction or explanation of the

relationship between twovariables. it implies that there is a systematic relationship between
an independent and dependent variable".

TYPES OF HYPOTHESES
L Null hvpothesis: Anull hypothesis proposes no relationship between two variables. Denoted
by HO, it is a negative statement like "Attending physiotherapy sessions does not affect athletes'
on-field performance." Here, the author claims physiotherapy sessions have no effect on on-field
performances. Even if there is, it's only a coincidence.
2. Alternative hypothesis Considered to be the opposite of a null hypothesis, an alternative
hypothesis is donated as Hl or Ha. It explicitly states that the dependent variable affects the
independent variable. A good alternative hypothesis example is "Attending physiotherapy
sessions improves athletes' on-field performance." or "Water evaporates at
100C"
The alternative hypothesis further branches into directional and non-directional.
" Directional hypothesis: A hypothesis that states the result would be either positive or negative
is called directional hypothesis. It accompanies Hl with either the <'or >' sign.
"Non-directional hypothesis: A non-directional hypothesis only claims an effect on the
dependent variable. It does not clarify whether the result would be positive or negative. The sign
for a non-directional hypothesis is #!
3. Simple hypothesis Asimple hypothesis is a statement made to reflect the relation between
exactly two variables. One independent and one dependent. Consider the example, "Smo king is
aprominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the
independent variable, smoking.
4. Complex hvpothesis In contrast to a simple hypothesis, a complex hypothesis implies the
relationship between multiple independent and dependent variables. For instance, "Individuals
who eat more fruits tend to have higher immunity, lesser cholesterol, and high metabolism." The
independent variable is eating more fruits, while the dependent variables are higher immunity,
lesser cholesterol, and high metabolism.
5. Associative and casual hypothesis Associative and casual hypotheses dont exhibit how
many variables there will be. They define the relationship between the variables.
In an associative hypothesis, changing any one variable, dependent or independent, affects

others. In a casual hypothesis, the independent variable directly affects the dependent.
6. Empirical bypothesis Also referred to as the working hypothesis, an empirncal hypothesis
claims a theory's validation via experiments and observation.
This way, the statement appears justifiable and different from a wild guess.
Say, the hypothesis is "Women who take iron tablets face a lesser risk of anemia than those who
take vitamin B12," This is an example of an empirical hypothesis where the researcher the
statement after assessing a group of women who take iron tablets and charting the findings.
2. Statistical hypothesis The point of a statistical hypothesis is to test an already existing
hypothesis by studying a population sample. Hypothesis like "44% of the Indian population
belong in the age group of 22-27." leverage evidence to prove or disprove a particular statement.
T-test or T-Distribution
When population standard deviation (o)
is not known and the sample is of small size
(i.e., n s 30), we use r distribution (student's
distribution)for the sampling distribution of mean
and workout ' variable as,
t=
S

S (Sampling Standard Deviation),

=
E(X,-X)?
n-1

Where, X= Sample mean

=Population mean from which
sample is taken
n =Sample size
S= Sampling standard deviation.
Q20. DiscUss propertles and applic ations of
t-tests.
Answer:
Properties of T-tests
Theproperties of t-test student's t-distribution
are as follows,
1.
The probability curve of 7is symmetric, like in
standard normal distribution (Z).
2. The t-distribution ranges from -a to a just as
does a normal distribution.
3. The -distribution is bell shaped and symmetrical
around mean zero, like normal distribution.

4. The shapes of the t-distribution changes as the

sample size changes (the number of degrees
of freedom changes) whereas it is same for all
sample sizes in z-distribution.
5. The variance of t-distribution is always greater
than one and is defined only whenn>3.

6. The t-distribution is'more of platykurtic (less

packed at centre and higher in tails) than the
normal distribution.

7. The t-distribution has a greater dispersion than

the normal distribution. As n becomes larger,
the t-distribution approaches the standard
normal distribution.

8 There is a family of t-distribution one for each

sample size whereas, there is only one standard
normal distribution.
Standard normal distribution

t-distribution (say nr=15)

t-distribution (say n=7)

Figure
Applications of t-test
The following are some important applications
of t-test or t-distributions,
1. Test of hypothesis about the population mean.
2. Test ofhypothesis about the difference between
two means.

3. Test ofhypothesis about the difference between

two means with dependent samples.
4 Test of hypothesis about coefficient of
correlation.
 2.3F-TEST G
Q26. Explain F-Test, Its properties and
appllcations.
Answer : Model Puper-ll, Q3(b)
F-distributlon
F-Test or F-distribution is acontinuous
probability distribution used when two different
nomal population, are sampled. Consider S and S
as thesanmple variances of different random sample of
sizes n,and n, respectively. These samples are drawn
from two different normal population N(,, of) and N
(H,,o), where (,,o) and (,, o~) denotes the mean
and variances of S; and S; respectively.
F=
S;/a;
Inorder to determine whether the samples (S;,
S;) are drawn from two different populatíon, having
cqualvariances. It is necessary to compute the ratio of
variances related to two independent random sample.
This ratio is computed as,
If it is assumed that normal population have
equal variance, then,
F= if s; >S;

Therefore, the sampling distribution of F can

be written in the following form,
KFV-2)/2
f(F)
(VF+V,y,)Z
Where, K= JrFaF = 1,
Degree of freedom of n,, V=n, - 1
Degree of freedom of n,, V, =n, - 1
The value of F is
when two sample variancesapproximately equal to 1
are almost equal.
Properties of F-Distribution
Following are the properties ofF-distribution,
1. The F-distribution curve lies in only first
quadrant (2) and is unimodal.
2 The F-distribution is
population parameter andindependent
(free) of
depends only on the
degree freedom (1.e., Vand V) according
of
toits order.
3 The F-distribution mode is less than unity (i.e.,
mode < l).
 In the F-distribution
4. 1
figure, F
(K)=

F,(V, V,)

Figure: F-Distribution Curve

Where,
F(V, V) =Value of F
So, a is at the right of F, (V, V) under the
F-distribution curve.
Applications of F-Distribution
Following are the applications ofF-distribution,
1. It is used for testing the equality of many
population means.
2. It isused for comparing the sample variances.
3. It is used for performing analysis of variance.
4. It is used for testing the significance o!
regression equation.
5. It is used for whether the ratio
ratio
determining level
any
incrementally changes
chosen randomly.
from unity at