TESTING OF HYPOTHESIS-1
Ph.D. Coursework Lecture
Prof.K.K.Achary
BASIC CONCEPTS
• Every research study is seeking answers to
questions of scientific interest.
• Ex: What is the optimum dose of a drug to be
administered in case patients showing some
symptoms of mental illness?
• Whether drug A is more effective than drug B
for treating disease X?
• Does training the healthcare workers improve
their commitment to service?
• In all these examples , the researcher wants to
study scientifically and establish the answers.
• The answer to these questions should be
sought based on evidence, i.e., using sample
data to support your research study.
• The research studies usually involve
estimation of a population parameter or
comparing two or more populations
responding to an intervention/treatment.
• The purpose of hypothesis testing is to aid the
researcher in reaching a conclusion
concerning a population by examining a
sample from that population.
• Definitions:
• A hypothesis may be defined as a statement
about one or more populations.
• It is concerned about the parameters of the
population.
• Researchers are concerned about two types of
hypotheses : research hypotheses and statistical
hypotheses.
• Research hypothesis is the conjecture or
supposition that motivates the research. This may
be based on researchers own experience or
motivated by work carried out by other
researchers.
• Research hypothesis leads to statistical
hypothesis.
• Statistical hypotheses are hypotheses that are
stated in such a way that they may be
evaluated by appropriate statistical
techniques.
• The process of testing a statistical hypothesis
can be seen as a logical sequence of actions
and decisions.
• The important steps in this sequence are the
following:
• DATA: It must be understood as it forms the basis
of testing procedure. The type of data –discrete
or continuous, nominal or ordinal/categorical are
to be decided as this will decide the test to be
used.
• ASSUMPTIONS: Assumptions are very important
as they provide the basis for using a particular
test procedure. Common assumptions are
normality, equality of variances, independence of
samples etc.
• HYPOTHESIS: There are two hypotheses
involved in hypothesis testing - null hypothesis
and alternative hypothesis.
• Null hypothesis is the hypothesis to be tested;
it is the hypothesis of no difference
• It is a statement of agreement with the
conditions assumed to be true in the
population. It is the complement of the
conclusion the researcher is seeking.
• Ex: Suppose the researcher wants to show
that drug A is more effective than drug B.
• Then the null hypothesis is a statement about
equal effect of the two drugs.
• Drug A is equal or less effective than drug B
• Null hypothesis is designated by the symbol
H0
• The other hypothesis is called the alternative
hypothesis.
• It is a statement about what the researcher
believes to be true if the test of null
hypothesis using the sample data rejects the
null hypothesis
• It is same as the research hypothesis and
shown by the symbol H1 (orH A )
• Examples: in determining the dose level,we
have to say about the the mean dose level for
the population. For this case, the null
hypothesis is :
H 0 : μ  250mg
•
The alternativ e hypothesis could be
stated in one of the following forms
H1 :   250(or   250 )
or H1 :   250
• Your research hypothesis should be placed as
the alternative hypothesis
• The null hypothesis should contain a
statement of equality/inequality, either
 or  or 
• The null hypothesis is the hypothesis to be
tested.
• The null and alternative hypotheses are
complementary
• Neither hypothesis testing nor statistical
inference,in general, leads to the proof of a
hypothesis; it merely indicates whether the
hypothesis is supported or not supported by
the available data.
• When we fail to reject a null hypothesis, we do
not say that it is true,but that it may be true.
•
Not rejecting H 0 neednot imply H 0isTRUE
TEST STATISTIC
• A test statistic is a function of the sample
data.Depending on the hypothesis to be
tested, the test statistic is defined/derived.
• It serves as a decision maker , since the
decision to reject or accept the null hypothesis
depends on the magnitude of the test
statistic.
• General rule for writing a test statistic:
relevant stat. - hypothesis ed parameter
test statistic 
standard error of relevant stat.
x  0
z
,  known, 0 is
/ n
hypothesis ed value of parameter 
Distribution of test statistic
• The key to statistical inference is the sampling
distribution – the distribution of the relevant
statistics used to formulate the test statistic.
• If the hypothesis is about the population
mean, then the relevant statistic is the sample
mean. The sampling distribution of the sample
mean plays the key role in constructing the
test statistic.
• The distribution of test statistic is derived
when the null hypothesis is true.
• The test statistic shown below has standard
normal distribution.
x  0
z
,  known, 0 is
/ n
hypothesis ed value of parameter 
Decision Rule
• Consider all values of the test statistic and plot
them on a line.We can form the set of values into
two groups.
• One set of values which are less likely to occur if
the null hypothesis is true. This group constitutes
what is known as the rejection region.
• The other set of values are those which are more
likely to occur if the null hypothesis is true.This
group constitutes what is known as the
nonrejection region.
• Decision rule:
• Reject the null hypothesis if the value of the test
statistic computed from the sample is a value in
the rejection region.
• Do not reject the null hypothesis if the value of
the test statistic is in the nonrejection region.
• Since the value of the test statistic depends on its
distribution , the decision rule is a probability
statement/probabilistic decision rule.
Level of Significance
• The level of significance is the probability of
rejecting a true null hypothesis.
• It specifies the area under the curve of the
probability distribution of the test statistic for its
values above the rejection region.
• Rejecting a true null hypothesis would constitute
an error. This is called the type -1 error. This is
denoted by α ( alpha ) . Type -1 error has to be a
small value. Usually α =0.01,0.05,0.10
• When we fail to reject a null hypothesis , the risk
of failing to reject a false null hypothesis is
present.
• This is an error again!. The error committed when
a false null hypothesis is not rejected is called
type-2 error. The probability of committing a
type-2 error is denoted by β (beta ).
• We fix type-1 error at a low level but we have no control
over type -2 error.
• The following table shows the
actions/decisions of a researcher based on
the situation reflected by the test hypothesis.
Ho
True
Reject
Fail to
Reject
Type I
Error
O.K.
False
O.K
Type II
Error
Calculation of test statistic
• Using the sample data we compute the test
statistic. This value is compared with the
values in the rejection and nonrejection
regions. These regions are specified when we
fix the type-1 error.
• The hypothesis is rejected if the computed
value of the test statistic falls in rejection
region and it is not rejected if the value falls in
the nonrejection region.
• If null hypothesis is not rejected it does not
mean the null hypothesis is accepted. We
should say that the null hypothesis is ” not
rejected”.
• We avoid using “ accept “ because we might
have committed “ type -2 error “. This error
could be quite high and hence we do not
commit “ acceptance “.
What is p-value?
• p-value is the probability that tells how unusual
our sample results are, given that the null
hypothesis is true.It indicates the probability that
the sample results are not likely to have occurred,
if the null hypothesis is true.
• Assuming H0 is true, the probability that the test
statistic will take on values as or more extreme
than the observed test statistic (computed from
the sample data) is called the p-value of the test.
• The smaller the P-value, the stronger the
evidence against H0.
• To take a decision about the null
hypothesis,we compare the p-value with the
level of significance α.
• If p-value is greater than or equal α, we say that the
test is “ not significant”. There is not enough
evidence to reject the null hypothesis.
• If p-value is less than α, we say that the test is
“significant”. There is enough evidence to reject the
null hypothesis.