Hypothesis Testing
Hypothesis Testing (Ht):
Introduction
• After discussing procedures for data
preparation and preliminary analysis, the
next step for many studies is hypothesis
testing.
• In this context, induction and deduction is
fundamental to hypothesis testing.
HT Introduction Contd.
• Induction and deduction are used together
in research reasoning.
• Researchers describe this process as the
double movement of reflective thought.
• Induction occurs when we observe a fact
and ask, :Why is this?’
• In answer to this we advance a tentative
explanation (hypothesis).
HT Introduction Contd.
• The hypothesis is plausible if it explains
the event or condition (fact) that prompted
the question.
• Deduction is the process by which we test
whether the hypothesis is capable of
explaining the fact
HT Introduction Contd.
• Example:
– You promote a product but sales don’t
increase. (Fact1)
– You ask the question “Why didn’t sales
increase?. (Induction)
– You infer a conclusion (hypothesis) to answer
the question: The promotion was poorly
executed. (Hypothesis)
HT Introduction Contd.
– You use this hypothesis to conclude (deduce)
that sales will not increase during a poorly
executed promotion. You know from
experience that ineffective promotion will not
increase sales. (Deduction1)
– We deduce that a well-executed promotion
will result in increased sales. (Deduction2)
– We run an effective promotion, and sales
increase. (Fact2).
HT Introduction Contd.
• In most research, the process may be
more complicated than this specific
example suggests.
• For instance,
– we often develop multiple hypotheses by
which to explain the problem in question;
– Then we design a study to test all hypotheses
at once;
– This is not only more efficient but also is a
good way to reduce the attachment (potential
bias) of the researcher for any given
hypothesis.
HT Introduction Contd.
• Inductive reasoning moves from specific facts to
general, but tentative, conclusions.
• With the help of probability estimates,
conclusions can be refined and results
discussed with a degree of confidence.
• Statistical inference is an application of inductive
reasoning.
• It allows us to reason from evidence found in the
sample to conclusions we wish to make about
the population.
HT Introduction Contd.
• Two major categories of statistical
procedures are:
– Inferential statistics and
– Descriptive statistics (describing distributions)
• Under inferential statistics, two topics are
discussed:
– estimation of population values, and
– testing statistical hypothesis
HT Introduction Contd.
• We evaluate the accuracy of hypotheses
by determining the statistical likelihood
that the data reveal true differences – not
random sampling error;
• We evaluate the importance of a
statistically significant difference by
weighing the practical significance of any
change that we measure.
Hypothesis: Definition
• A hypothesis is a hunch, assumption,
suspicion, assertion, or an idea about a
phenomenon, relationship or situation, the
reality or truth of which we do not know;
• The above become the basis for an inquiry
for a researcher.
• In most studies, the hypothesis will be
based upon either previous studies, or on
your own or some one else’s observation.
Definition contd.
• A hypothesis has at least three
characteristics:
 It is a tentative proposition;
 Its validity is unknown; and
 In most cases, it specifies a relationship
between two or more variables.
Functions of a Hypothesis
• The formulation of a hypothesis provides a
study with focus. It tells you what specific
aspects of a research problem to
investigate;
• It tells you what data to collect and what
not to collect, thereby providing focus to
the study;
Functions contd.
• As it provides a focus, the construction of
a hypothesis enhances objectivity in a
study; and
• It may enable you to add to the formulation
of theory. It helps you to specifically
conclude what is true or what is false.
Characteristics of a Hypothesis
• A hypothesis should be simple, specific
and conceptually clear:
– To be able to develop a good hypothesis one
must be familiar with the subject area (the
literature review is of immense help).
– The more insight one has into a problem, the
easier it is to construct a hypothesis.
Characteristics contd.
• A hypothesis should be capable of
verification:
– Methods and techniques must be available for
data collection and analysis;
– The researcher might, while doing research,
develop new techniques to verify it.
• A hypothesis should be related to the
existing body of knowledge; and it must
add to it.
Characteristics contd.
• A hypothesis should be operationalisable.
– It should be expressed in terms that can be
measured. If it can not be measured, it cannot
be tested and, hence no conclusions can be
drawn.
Approaches to Hypothesis
Testing
• Classical or sampling theory approach
– This, the more established approach,
represents an objective view of probability in
which the decision making rests totally on an
analysis of available sampling data.
– A hypothesis is established; it is rejected or
fails to be rejected, based on the sample data
collected.
• Bayesian statistics approach
HT Approaches contd.
• Bayesian statistics are an extension of the
classical approach.
– They also use sampling data, but they go beyond to
consider all other available information which consists
of subjective probability estimates stated in terms of
degrees of belief.
– These subjective estimates are based on general
experience rather than on specific collected data.
– Various decision rules are established, cost and other
estimates can be introduced, and the expected
outcome of combinations of these elements are used
to judge decision alternatives.
Statistical Significance
• Following classical statistics approach, we
accept or reject a hypothesis on the basis of
sampling information alone.
• Since any sample will almost surely vary
somewhat from its population, we must judge
whether the differences are statistically
significant or insignificant.
• A difference has statistical significance if there is
good reason to believe the difference does not
represent random sampling fluctuations only.
Statistical Significance: Example
• Honda, Toyota, and other auto companies
produce hybrid vehicles using an
advanced technology that combines a
small gas engine with an electric motor.
• The vehicles run on an electric motor at
slow speeds but shifts to both the gasoline
motor and the electric motor at city and
higher freeway speeds.
• Their advertising strategies focus on fuel
economy.
SS Example contd.
• Let’s say that the hybrid Civic has maintained an
average of about 50 miles per gallon (mpg) with
a standard deviation o 10 mpg.
• Suppose researchers discover by analyzing all
production vehicles that the mpg is now 51.
• Is this difference statistically significant from 50?
– Yes it is, because the difference is based on a census
of the vehicles and there is no sampling involved.
– It has been demonstrated conclusively that the
population average has moved from 50 to 51 mpg.
SS Example contd.
• Since it would be too expensive to analyze
all of a manufacturer’s vehicles frequently,
we resort to sampling.
• Assume a sample of 25 cars is randomly
selected and the average mpg is 54.
• Is this statistically significant?
– The answer is not obvious.
– It is significant if there is good reason to
believe the average mpg of the total
population has moved up from 50.
SS Example contd.
• Since the evidence consists only of a
sample consider the second possibility:
that this is only a random sampling error
and thus is not significant.
• The task is to decide whether such a result
from this sample is or is not statistically
significant.
• To answer this question, one needs to
consider further the logic of hypothesis
testing.
The Logic of Hypothesis Testing
• In classical tests of significance, two kinds
of hypotheses are used which are
– The null hypothesis and
– the alternative hypothesis
• The null hypothesis is used for testing.
– It is a statement that no difference exists
between the parameter (a measure taken by
a census of the population), and
– the statistic being compared to it (a measure
from a recently drawn sample of the
population).
The Logic of HT contd.
• Analysts usually test to determine whether
there has been no change in the
population of interest or whether a real
difference exists.
• A null hypothesis is always stated in a
negative form.
– In the hybrid-car example, the null hypothesis
states that the population parameter of 50
mpg has not changed.
The Logic of HT contd.
• The alternative hypothesis holds that there
has been a change in average mpg (i.e.,
the sample statistic of 54 indicates the
population value probably is no longer 50).
• The alternative hypothesis is the logical
opposite of the null hypothesis.
– The hybrid-car example can be explored
further to show how these concepts are used
to test for significance.
The Logic of HT contd.
• The null hypothesis (Ho): There has been
no change from the 50 mpg average.
• The alternative (HA) may take several
forms, depending on the objective of the
researchers.
• The HA may be of the “not the same” or
the “greater than” or “less than” form:
– The average mpg has changed from 50.
– The average mpg has increased (decreased)
from 50.
The Logic of HT contd.
• These types of alternative hypotheses
correspond with two-tailed and one-tailed
tests.
• A two-tailed test, or nondirectional test,
considers two possibilities:
– the average could be more than 50 mpg, or
– It could be less than 50.
• To test this hypothesis, the region of
rejection are divided into two tails of the
distribution.
The Logic of HT contd.
• Such hypothesis can be expressed in the
following form:
– Null
Ho:μ = 50 mpg
– Alternative HA:μ ≠ 50 mpg (not the same
case)
(See figure on the next slide)
The Logic of HT contd.
• A one-tailed test, ore directional test, places
the entire probability of an unlikely outcome
into the tail specified by the alternative
hypothesis. Such hypotheses can be
expressed in the following form:
– Null
Ho:μ ≤ 50 mpg
– Alternative HA:μ > 50 mpg (greater than case)
Or
– Null
Ho:μ ≥ 50 mpg
– Alternative HA:μ < 50 mpg (less than case)
The Logic of HT contd.
• In testing these hypotheses, adopt this
decision rule:
• Take no corrective action if the analysis
shows that one cannot reject the null
hypothesis.
• Note the language “cannot reject” rather
than “accept” the null hypothesis.
• It is argued that a null hypothesis can
never be proved and therefore cannot be
“accepted”.
Type I and Type II Errors
• In the context of testing of Hypothesis,
there are basically two types of errors we
can make.
• We may reject Ho when Ho is true, and we
may accept Ho when in fact Ho is not true.
• The former is known as Type I error and
the latter as Type II error.
Errors contd.
• Type I error is denoted by α (alpha) known
as α error, also called the level of
significance of test;
• Type II error is denoted by β (beta) known
as β error.
Errors contd.
• If type I error is fixed at 5%, it means that
there are about 5 chances in 100 that we
will reject Ho, when Ho is true.
• We can control Type I error just by fixing it
at a lower level.
– For instance, if we fix it at 1 percent, we will
say that the maximum probability of
committing a type I error would only be 0.01.
Errors contd.
• With a fixed sample size n, when we try to
reduce Type I error, the probability of
committing Type II error increases. There
is a trade off between the two
(β error = 1 – α error).
Statistical Testing Procedures
Testing for statistical significance involves
the following six-stage sequence:
1. State the null hypothesis (Ho):
Make a formal clear statement of the null
hypothesis and also of the alternative
hypothesis (Ha);
Stat. Testing Proc. contd.
2. Choose an appropriate statistical test:
Two types of tests are parametric (t-test)
and nonparametric (Chi-square).
In choosing a test, one can consider how
the sample is drawn, the nature of the
population, and the type of measurement
scale used.
Stat. Testing Proc. contd.
3. Select the desired level of
significance:
The choice of the level of significance
should be made before we collect the
data. The most common level is .05,
although .01 is also widely used. Other α
levels such as .10, .25, or .001 are
sometimes chosen.
Stat. Testing Proc. contd.
The exact level to choose is largely
determined by how much α risk one is willing
to accept and the effect that this choice has on
β risk. The larger the α, the lower is the β.
4. Compute the calculated difference value:
After the data are collected, use the formula for
the appropriate significance test to obtain the
calculated value.
Stat. Testing Proc. contd.
5. Obtain the critical test value:
After we compute the calculated t, Chisquare, or other measure, we must look
up the critical value in the appropriate
table for that distribution. The critical
value is the criterion that defines the
region of rejection from the region of
acceptance of the null hypothesis.
Stat. Testing Proc. contd.
6. Interpret the test:
For most tests, if the calculated value is
larger than the critical value, we reject
the null hypothesis and conclude that the
alternative hypothesis is supported
(although it is by no means proved). If
the critical value is larger, we conclude
we have failed to reject the null
hypothesis.
Probability Values (p Values)
• According to the “interpret the test” step of
the statistical test procedure, the
conclusion is stated in terms of rejecting or
not rejecting the null hypothesis based on
a rejection region selected before the test
is conducted.
• A second method of presenting the results
of a statistical test reports the extent to
which the test statistic disagrees with the
null hypothesis.
p Values
• This second method has become popular
because analysts want to know what
percentage of the sampling distribution lies
beyond the sample statistic on the curve,
and most statistical computer programs
report the results of statistical tests as
probability values (p values).
• The p value is the probability of observing
a sample value as extreme as, or more
extreme than, the value actually observed,
given that the null hypothesis is true.
p Values contd.
• The p value is compared to the
significance level (α), and on this basis the
null hypothesis is either rejected or not
rejected.
– If p value is less than the significance level,
the null hypothesis is rejected.
– If p value is greater than or equal to the
significance level. The null hypothesis is not
rejected.