Unformatted text preview: Research Methodology Unit 3 Unit 3 Hypothesis Structure
3.1 Introduction
Objectives 3.2 Meaning and Examples of Hypothesis
3.2.1
3.2.2 Nature of Hypothesis 3.2.3 The need for having Hypothesis 3.2.4
3.3 Criteria for constructing of hypothesis Characteristics of good hypothesis Types of hypothesis
3.3.1. Null Hypothesis and alternative hypothesis 3.4 Concepts of Hypothesis
3.4.1
3.4.2 Decision rule of testing hypothesis 3.4.3 Type I and Type II Errors 3.4.4
3.5 The level of Significance Two Tailed and One Tailed Test Procedures for testing hypothesis
3.5.1 Making formal statement 3.5.2 Selecting a significant level 3.5.3 Deciding the distribution to use 3.5.4 Selecting a Random Sample and computing am approximate
value 3.5.5
3.5.6
3.6 Calculation of Probability
Comparing the Probability Testing of Hypothesis
3.6.1 Important Parametric Tests Self Assessment Questions
3.7 Summary Sikkim Manipal University Page No. 27 Research Methodology 3.8 Terminal Questions 3.9 Unit 3 Answers to SAQs and TQs 3.1 Introduction
A hypothesis is an assumption about relations between variables. It is a
tentative explanation of the research problem or a guess about the research
outcome. Before starting the research, the researcher has a rather general,
diffused, even confused notion of the problem. It may take long time for the
researcher to say what questions he had been seeking answers to. Hence,
an adequate statement about the research problem is very important. What
is a good problem statement? It is an interrogative statement that asks: what
relationship exists between two or more variables? It then further asks
questions like: Is A related to B or not? How are A and B related to C? Is A
related to B under conditions X and Y? Proposing a statement pertaining to
relationship between A and B is called a hypothesis.
Objectives:
After studying this lesson you should be able to understand: Meaning and Examples of Hypothesis Criteria for constructing of hypothesis Nature of Hypothesis the need for having Hypothesis Characteristics of good hypothesis Types of hypothesis Null Hypothesis and alternative hypothesis Concepts of Hypothesis The level of Significance Decision rule of testing hypothesis Sikkim Manipal University Page No. 28 Research Methodology Type I and Type II Errors Two Tailed and One Tailed Test Procedures for Testing hypothesis Unit 3 Testing of Hypothesis 3.2 Meaning and Examples of Hypothesis
According to Theodorson and Theodorson, “a hypothesis is a tentative
statement asserting a relationship between certain facts. Kerlinger describes
it as “a conjectural statement of the relationship between two or more
variables”. Black and Champion have described it as “a tentative statement
about something, the validity of which is usually unknown”. This statement is
intended to be tested empirically and is either verified or rejected. It the
statement is not sufficiently established, it is not considered a scientific law.
In other words, a hypothesis carries clear implications for testing the stated
relationship, i.e., it contains variables that are measurable and specifying
how they are related. A statement that lacks variables or that does not
explain how the variables are related to each other is no hypothesis in
scientific sense.
3.2.1 Criteria for Hypothesis Construction
Hypothesis is never formulated in the form of a question. The standards to
be met in formulating a hypothesis: It should be empirically testable, whether it is right or wrong. It should be specific and precise. The statements in the hypothesis should not be contradictory. It should specify variables between which the relationship is to be
established. It should describe one issue only. Sikkim Manipal University Page No. 29 Research Methodology Unit 3 3.2.2 Nature of Hypothesis
A scientifically justified hypothesis must meet the following criteria: It must accurately reflect the relevant sociological fact. It must not be in contradiction with approved relevant statements of
other scientific disciplines. It must consider the experience of other researchers. 3.2.3 The Need for having Working Hypothesis A hypothesis gives a definite point to the investigation, and it guides the
direction on the study. A hypothesis specifies the sources of data, which shall be studied, and
in what context they shall be studied. It determines the data needs. A hypothesis suggests which type of research is likely to be most
appropriate. It determines the most appropriate technique of analysis. A hypothesis contributes to the development of theory 3.2.4 Characteristics of Good Hypothesis
1. Conceptual Clarity
2. Specificity
3. Testability
4. Availability of Techniques
5. Theoretical relevance
6. Consistency
7. Objectivity
8. Simplicity Sikkim Manipal University Page No. 30 Research Methodology Unit 3 3.3 Types of Hypothesis
There are many kinds of hypothesis the researcher has to be working with.
One type of hypothesis asserts that something is the case in a given
instance; that a particular object, person or situation has particular
characteristics. Another type of hypothesis deals with the frequency of
occurrence or of association among variables; this type of hypothesis may
state that X is associated with Y. A certain Y proportion of items e.g.
urbanism tends to be accompanied by mental disease or than something
are greater or lesser than some other thing in specific settings. Yet another
type of hypothesis asserts that a particular characteristics is one of the
factors which determine another characteristic, i.e. X is the producer of Y.
hypothesis of this type are called causal hypothesis.
3.3.1 Null Hypothesis and Alternative Hypothesis
In the context of statistical analysis, we often talk null and alternative
hypothesis. If we are to compare method A with method B about its
superiority and if we proceed on the assumption that both methods are
equally good, then this assumption is termed as null hypothesis. As against
this, we may think that the method A is superior, it is alternative hypothesis.
Symbolically presented as:
Null hypothesis = H0 and Alternative hypothesis = Ha
Suppose we want to test the hypothesis that the population mean is equal to
the hypothesis mean (µ H0) = 100. Then we would say that the null
hypotheses are that the population mean is equal to the hypothesized mean
100 and symbolical we can express as: H0: µ= µ H0=100
If our sample results do not support these null hypotheses, we should
conclude that something else is true. What we conclude rejecting the null
hypothesis is known as alternative hypothesis. If we accept H0, then we are
rejecting Ha and if we reject H0, then we are accepting Ha. For H0: µ= µ
H0=100, we may consider three possible alternative hypotheses as follows: Sikkim Manipal University Page No. 31 Research Methodology Unit 3 Alternative Hypothesis To be read as follows Ha: µ≠µ H0 (The alternative hypothesis is that the population mean is not equal to 100 i.e., it
may be more or less 100)
Ha: µ>µ H0 (The alternative hypothesis is that the population mean is greater than 100)
Ha: µ< µ H0 (The alternative hypothesis is that the population mean is less than 100)
The null hypothesis and the alternative hypothesis are chosen before the
sample is drawn (the researcher must avoid the error of deriving hypothesis
from the data he collects and testing the hypothesis from the same data). In
the choice of null hypothesis, the following considerations are usually kept in
view: Alternative hypothesis is usually the one which wishes to prove and the
null hypothesis are ones that wish to disprove. Thus a null hypothesis
represents the hypothesis we are trying to reject, the alternative
hypothesis represents all other possibilities. If the rejection of a certain hypothesis when it is actually true involves
great risk, it is taken as null hypothesis because then the probability of
rejecting it when it is true is α (the level of significance) which is chosen
very small. Null hypothesis should always be specific hypothesis i.e., it should not
state about or approximately a certain value. Generally, in hypothesis testing we proceed on the basis of null
hypothesis, keeping the alternative hypothesis in view. Why so? The
answer is that on assumption that null hypothesis is true, one can assign
the probabilities to different possible sample results, but this cannot be Sikkim Manipal University Page No. 32 Research Methodology Unit 3 done if we proceed with alternative hypothesis. Hence the use of null
hypothesis (at times also known as statistical hypothesis) is quite
frequent. 3.4 Concepts of Hypothesis Testing
Basic concepts in the context of testing of hypothesis need to be explained.
3.4.1 The Level of Significance
This is a very important concept in the context of hypothesis testing. It is
always some percentage (usually 5%) which should be chosen with great
care, thought and reason. In case we take the significance level at 5%, then
this implies that H0 will be rejected when the sampling result (i.e., observed
evidence) has a less than 0.05 probability of occurring if H0 is true. In other
words, the 5% level of significance means that researcher is willing to take
as much as 5% risk rejecting the null hypothesis when it (H0) happens to be
true. Thus the significance level is the maximum value of the probability of
rejecting H0 when it is true and is usually determined in advance before
testing the
Decision Rule of Test of Hypothesis:
Given a hypothesis H0 and an alternative hypothesis H0 we make rule which
is known as decision rule according to which we accept H0 (i.e., reject Ha) or
reject H0 (i.e., accept a). For instance, if (H0 is that a certain lot is good (there
are very few defective items in it) against Ha that the lot is not good (there
are many defective items in it), that we must decide the number of items to
be tested and the criterion for accepting or rejecting the hypothesis. We
might test 10 items in the lot and plan our decision saying that if there are
none or only 1 defective item among the 10, we will accept H0 otherwise we
will reject H0 (or accept Ha). This sort of basis is known as decision rule. Sikkim Manipal University Page No. 33 Research Methodology Unit 3 Type I & Type II Errors
In the context of testing of hypothesis there are basically two types of errors
that researchers make. We may reject H0 when H0 is true & we may accept
H0 when it is not true. The former is known as Type I & the later is known as
Type II. In other words, Type I error mean rejection of hypothesis which
should have been accepted & Type II error means accepting of hypothesis
which should have been rejected. Type I error is donated by α (alpha), also
called as level of significance of test; and Type II error is donated by β(beta).
Decision
Accept H0 Reject H0 H0 (true) Correct decision Type I error (α error) Ho (false) Type II error (β error) Correct decision The probability of Type I error is usually determined in advance and is
understood as the level of significance of testing the hypothesis. If type I
error is fixed at 5%, it means there are about chances in 100 that we will
reject H0 when H0 is true. We can control type I error just by fixing it at a
lower level. For instance, if we fix it at 1%, we will say that the maximum
probability of committing type I error would only be 0.01.
But with a fixed sample size, n when we try to reduce type I error, the
probability of committing type II error increases. Both types of errors can not
be reduced simultaneously. There is a tradeoff in business situations,
decisionmakers decide the appropriate level of type I error by examining
the costs of penalties attached to both types of errors. If type I error involves
time & trouble of reworking a batch of chemicals that should have been
accepted, where as type II error means taking a chance that an entire group
of users of this chemicals compound will be poisoned, then in such a
situation one should prefer a type I error to a type II error means taking a
Sikkim Manipal University Page No. 34 Research Methodology Unit 3 chance that an entire group of users of this chemicals compound will be
poisoned, then in such a situation one should prefer a type II error. As a
result one must set very high level for type I error in one’s testing techniques
of a given hypothesis. Hence, in testing of hypothesis, one must make all
possible effort to strike an adequate balance between Type I & Type II error.
3.4.2 Two Tailed Test & One Tailed Test
In the context of hypothesis testing these two terms are quite important and
must be clearly understood. A twotailed test rejects the null hypothesis if,
say, the sample mean is significantly higher or lower than the hypnotized
value of the mean of the population. Such a test inappropriate when we
haveH0: µ= µ H0 and Ha: µ≠µ H0 which may µ>µ H0 or µ<µ H0. If significance
level is % and the twotailed test to be applied, the probability of the
rejection area will be 0.05 (equally split on both tails of curve as 0.025) and
that of the acceptance region will be 0.95. If we take µ = 100 and if our
sample mean deviates significantly from µ, in that case we shall accept the
null hypothesis. But there are situations when only onetailed test is
considered appropriate. A onetailed test would be used when we are to
test, say, whether the population mean in either lower than or higher than
some hypothesized value. 3.5 Procedure for Testing Hypothesis
To test a hypothesis means to tell (on the basis of the data researcher has
collected) whether or not the hypothesis seems to be valid. In hypothesis
testing the main question is: whether the null hypothesis or not to accept the
null hypothesis? Procedure for hypothesis testing refers to all those steps
that we undertake for making a choice between the two actions i.e.,
rejection and acceptance of a null hypothesis. The various steps involved in
hypothesis testing are stated below: Sikkim Manipal University Page No. 35 Research Methodology Unit 3 3.5.1 Making a Formal Statement
The step consists in making a formal statement of the null hypothesis (Ho)
and also of the alternative hypothesis (Ha). This means that hypothesis
should clearly state, considering the nature of the research problem. For
instance, Mr. Mohan of the Civil Engineering Department wants to test the
load bearing capacity of an old bridge which must be more than 10 tons, in
that case he can state his hypothesis as under:
Null hypothesis HO: µ =10 tons
Alternative hypothesis Ha: µ >10 tons
Take another example. The average score in an aptitude test administered
at the national level is 80. To evaluate a state’s education system, the
average score of 100 of the state’s students selected on the random basis
was 75. The state wants to know if there is a significance difference
between the local scores and the national scores. In such a situation the
hypothesis may be state as under:
Null hypothesis HO: µ =80
Alternative hypothesis Ha: µ ≠ 80
The formulation of hypothesis is an important step which must be
accomplished with due care in accordance with the object and nature of the
problem under consideration. It also indicates whether we should use a
tailed test or a two tailed test. If Ha is of the type greater than, we use alone
tailed test, but when Ha is of the type “whether greater or smaller” then we
use a twotailed test.
3.5.2 Selecting a Significant Level
The hypothesis is tested on a predetermined level of significance and such
the same should have specified. Generally, in practice, either 5% level or
1% level is adopted for the purpose. The factors that affect the level of
significance are:
Sikkim Manipal University Page No. 36 Research Methodology Unit 3 The magnitude of the difference between sample ; The size of the sample; The variability of measurements within samples; Whether the hypothesis is directional or non – directional (A directional
hypothesis is one which predicts the direction of the difference between,
say, means). In brief, the level of significance must be adequate in the
context of the purpose and nature of enquiry. 3.5.3 Deciding the Distribution to Use
After deciding the level of significance, the next step in hypothesis testing is
to determine the appropriate sampling distribution. The choice generally
remains between distribution and the t distribution. The rules for selecting
the correct distribution are similar to those which we have stated earlier in
the context of estimation.
3.5.4 Selecting A Random Sample & Computing An Appropriate Value
Another step is to select a random sample(S) and compute an appropriate
value from the sample data concerning the test statistic utilizing the relevant
distribution. In other words, draw a sample to furnish empirical data.
3.5.5 Calculation of the Probability
One has then to calculate the probability that the sample result would
diverge as widely as it has from expectations, if the null hypothesis were in
fact true.
3.5.6 Comparing the Probability
Yet another step consists in comparing the probability thus calculated with
the specified value for α, the significance level. If the calculated probability is
equal to smaller than α value in case of one tailed test (and α/2 in case of
twotailed test), then reject the null hypothesis (i.e. accept the alternative
hypothesis), but if the probability is greater then accept the null hypothesis.
Sikkim Manipal University Page No. 37 Research Methodology Unit 3 In case we reject H0 we run a risk of (at most level of significance)
committing an error of type I, but if we accept H0, then we run some risk of
committing error type II.
Flow Diagram for Testing Hypothesis
State H0 as well as Ha Specify the level of significance Decide the correct sampling distribution Sample a random sample and workout an appropriate value Calculate the probability that sample result would diverge as
widely as it has form expectations, if H0 were true Is this probability equal to or smaller than α value in case of
onetailed test and α/2 in case of twotailed test Run the risk of Reject H0 Accept H0 Run some risk of
committing type II
error Sikkim Manipal University Page No. 38 Research Methodology Unit 3 3.6 Testing of Hypothesis
The hypothesis testing determines the validity of the assumption (technically
described as null hypothesis) with a view to choose between the conflicting
hypotheses about the value of the population hypothesis about the value of
the population of a population parameter. Hypothesis testing helps to
secede on the basis of a sample data, whether a hypothesis about the
population is likely to be true or false. Statisticians have developed several
tests of hypothesis (also known as tests of significance) for the purpose of
testing of hypothesis which can be classified as: Parametric tests or standard tests of hypothesis ; Non Parametric test or distribution – free test of the hypothesis. Parametric tests usually assume certain properties of the parent population
from which we draw samples. Assumption like observations come from a
normal population, sample size is large, assumptions about the population
parameters like mean, variants etc must hold good before parametric test
can be used. But there are situation when the researcher cannot or does not
want to make assumptions. In such situations we use statistical methods for
testing hypothesis which are called non parametric tests because such tests
do not depend on any assumption about the parameters of parent
population. Besides, most nonparametric test assumes only nominal or
original data, where as parametric test require measurement equivalent to at
least an interval scale. As a result nonparametric test needs more
observation than a parametric test to achieve the same size of Type I &
Type II error.
3.6.1 Important Parametric Tests
The important parametric tests are: ztest ttest Sikkim Manipal University Page No. 39 Research Methodology x2test Unit 3 ftest All these tests are based on the assumption of normality i.e., the source of
data is considered to be normally distributed. In some cases the population
may not be normally distributed, yet the test will be applicable on account of
the fact that we mostly deal with samples and the sampling distributions
closely approach normal distributions.
Ztest is based on the normal probability distribution and is used for judging
the significance of several statistical measures, particularly the mean. The
relevant test statistic is worked out and compared with its probable value (to
be read from the table showing area under normal curve) at a specified level
of significance for judging the significance of the measure concerned. This is
a most frequently used test in research studies. This test is used even when
binomial distribution or tdistribution is applicable on the presumption that
such a distribution tends to approximate normal distribution as ‘n’ becomes
larger. Ztest is generally used for comparing the mean of a sample to some
hypothesis mean for the population in case of large sample, or when
population variance is known as ztest is also used for judging the
significance of difference between means to of two independent samples in
case of large samples or when population variance is known ztest is
generally used for comparing the sample proportion to a theoretical value of
population proportion or for judging the difference in proportions of two
independent samples when happens to be large. Besides, this test may be
used for judging the significance of median, mode, coefficient of correlation
and several other measures
Ttest is based on tdistribution and is considered an appropriate test for
judging the significance of sample mean or for judging significance of
difference between the two means of the two samples in case of samples
Sikkim Manipal University Page No. 40 Research Methodology Unit 3 when population variance is not known (in which case we use variance of
the sample as an estimate the population variance). In case two samples
are related, we use paired ttest (difference test) for judging the significance
of their mean of difference between the two related samples. It can also be
used for judging the significance of coefficient of simple and partial
correlations. The relevant test statistic, t, is calculated from the sample data
and then compared with its probable value based on tdistribution at a
specified level of significance for concerning degrees of freedom for
accepting or rejecting the null hypothesis it may be noted that ttest applies
only in case of small sample when population variance is unknown.
X2test is based on chisquare distribution and as a parametric test is used
for comparing a sample variance to a theoretical population variance is
unknown.
Ftest is based on fdistribution and is used to compare the variance of the
twoindependent samples. This test is also used in the context of variance
(ANOVA) for judging the significance of more than two sample means at
one and the same time. It is also used for judging the significance of multiple
correlation coefficients. Test statistic, f, is calculated and compared with its
probable value for accepting or rejecting the H0.
Self Assessment Questions
Fill in the Blanks
1.  is a negative statement.
2. Type II error is .
3.  is tentative statement. 3.7 Summary
A hypothesis is an assumption about relations between variables. It is a
tentative explanation of the research problem or a guess about the research
Sikkim Manipal University Page No. 41 Research Methodology Unit 3 outcome. Before starting the research, the researcher has a rather general,
diffused, even confused notion of the problem. A hypothesis gives a definite
point to the investigation, and it guides the direction on the study. A
hypothesis specifies the sources of data, which shall be studied, and in what
context they shall be studied. In the context of hypothesis testing these two
terms are quite important and must be clearly understood. A twotailed test
rejects the null hypothesis if, say, the sample mean is significantly higher or
lower than the hypnotized value of the mean of the population.
The hypothesis is tested on a predetermined level of significance and such
the same should have specified. Generally, in practice, either 5% level or
1% level is adopted for the purpose. After deciding the level of significance,
the next step in hypothesis testing is to determine the appropriate sampling
distribution. The hypothesis testing determines the validity of the assumption
(technically described as null hypothesis) with a view to choose between the
conflicting hypotheses about the value of the population of a population
parameter. Ztest is based on the normal probability distribution and is used
for judging the significance of several statistical measures, particularly the
mean. The relevant test statistic is worked out and compared with its
probable value (to be read from the table showing area under normal curve)
at a specified level of significance for judging the significance of the
measure concerned. This is a most frequently used test in research studies.
Ttest is based on tdistribution and is considered an appropriate test for
judging the significance of sample mean or for judging significance of
difference between the two means of the two samples in case of samples
when population variance is not known (in which case we use variance of
the sample as an estimate of the population variance). X2test is based on
chisquare distribution and as a parametric test is used for comparing a
sample variance to a theoretical population variance is unknown. Ftest is Sikkim Manipal University Page No. 42 Research Methodology Unit 3 based on fdistribution and is used to compare the variance of the twoindependent samples. 3.8 Terminal Questions
1. What is the meaning of Hypothesis? 2 What are the criteria for Hypothesis Construction? 3. What is the need for having Working Hypothesis?
3. What are the characteristics of Good Hypothesis?
4. What are the types of Hypothesis?
5. What is Type I & Type II Errors?
6. What are Two Tailed Test & One Tailed Test?
7. What are the procedure and Flow Diagram for Testing Hypothesis?
8. Which are the important Parametric Tests? 3.9 Answers to SAQs and TQs
SAQs
1. Null hypothesis.
2. Accepting a statement that is false.
3. Hypothesis
TQs
1. Section 3.1
2. Section 3.2.1
3. Section 3.2.3
4. Section 3.2.4
5. Section 3.3
6. Section 3.4.3
7. Section 3.4.4
8. Section 3.5
9. Section 3.6
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This note was uploaded on 04/15/2010 for the course MBA mba taught by Professor Smu during the Spring '10 term at Manipal University.
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