Hypothesis Testing
A procedure for determining which of
two (or more) mutually exclusive
statements is more likely true
 We classify hypothesis tests in terms
of

– Parametric vs. Non-parametric
– Directional vs. Non-directional
Hypothesis Tests

Parametric Tests - tests about specific
population parameters (μ, σ2, etc.)
– Is μ1 different from a predetermined
value?
– Is μ1 different from μ2 ?

Non-Parametric Tests - tests about the
shape of the population (medians?)
– Is this population different from another
population?
Hypothesis Tests

Non-Directional Hypothesis
– Is μ1 different from μ2 ?
– Is the distribution of scores in group 1
different than those in group 2?

Directional Hypotheses
– Is μ1 greater than μ2 ?
– Is μ1 less than μ2 ?
– Is the distribution of scores in group 1 to
the right (greater than) of those in group
2?
The Six-Steps of
Hypothesis Testing
1. State and Check Assumptions
2. Generate Null and Alternative
Hypotheses
3. Chose the Sampling Distribution of the
Test Statistic
4. Set Significance Level
5. Compute the Test Statistic
6. Draw Conclusions
1. State and Check
Assumptions

There are three requirements for
hypothesis testing to work
– Assumptions about the population
– Assumptions about the sample
Assumptions about the
Population

“Assumption of Normality” - the
population is normally distributed or the
sample size is sufficiently large so that
the CLT comes into play and
Assumptions about Variance

Depending of the type of test, some
important features of variance may
come into play
– Is σ or σ2 known?
– “Homogeneity of Variance” - the variance
of the populations being compared have
equal variance
Assumptions about the
Sample

The sample has been obtained using
independent random sampling
What if the assumptions
can’t be met?

“Violation” - when assumptions are not
met
– Violation of Normality = “Non-normal”
– Violation of Homogeneity of Variance =
“Heterogeneous Variance”

“Robust” - a test’s ability to “deal
with” violations
– “a t-test is robust to violations of
normality”
BUT, ...
 No
tests are robust to violations of
the random sampling assumption.
 If you do not have a random
sample, probability theory will not
work and therefore inferential
statistical techniques will fail.
2. Generate Null and
Alternative Hypotheses

These two hypotheses, designated HO
and HA (or H1), are mutually exclusive
– mutually exclusive - the don’t overlap
Properties of HO
Specifies no difference or no change
from a standard or theoretical value
 Always specifies something about a
particular population parameter
 Used in constructing a sampling
distribution

– For the subsequent quantitative work, the
null hypothesis is assumed to be true
Properties of HA (or H1)
About the same aspect of the
population as HO
 Usually stated in general terms
 Mutually exclusive - no overlap with HO
 Used in making a decision
 Can be directional or non-directional

Directional vs.
Non-directional HAs
Non-directional HA - usually stated as
“does not equal” or “is different than”
 Directional HA - stated as “greater
than” or “less than”

– note that a non-directional hypothesis is
equal to the two directional hypotheses
“greater than” or “less than”
3. Chose the Sampling
Distribution
Depending on the type of data,
assumptions, and hypotheses certain
distributions of the test statistic require
selection
 The second half of this course will be
devoted to making the “best” decision
about which test statistic to choose (z,
t, F, etc.)

4. Set Significance Level
Hypothesis testing is sometimes called
“significance testing”
 The significance level is the basis for
making our decision

– “rejection region” - the value specified by
the significance level of the test
– “critical value” - the value of the test
statistic specified by the significance level
that begins the rejection region
Significance Level
The probability value associated with
the decision rule is called the
significance level of the test
 Significance level is represented by the
Greek letter alpha (α)
 The actual value of α is up to you

What is the significance
level?




Hypothesis testing entails determining which
of two hypotheses (HO and H1) is more likely
correct.
But “more likely” is a subjective evaluation
on your part.
If you were to obtain a statistic that was
unlikely if HO were assume to be true, would
you be willing to accept the H1?
How “unlikely” does it need to be for you to
be convinced?
Typical Significance level
A Typical significance level : α = .05
 We are convinced that results are
considered as significant different from
the HO when they are in the most
extreme 5% (a proportion of .05) of all
possible outcomes specified in HO

Using α

With our significance level, we
determine a decision rule
– Critical Value – the sufficiently extreme
value of the statistic such that if our
statistic is more extreme, we reject the HO
– p-value – if the probability of our statistic,
assuming HO is true, is less than α, we
reject the HO
5. Compute the Test Statistic

From the random sample obtained, the
test statistic is computed using various
formulae
– z-statistic
– binomial probability
– t-statistic
– F-statistic
– etc.
6. Draw Conclusions

Based on the test statistic just
computed, we do one of two things:
– Reject the null hypothesis and accept the
alternative hypothesis, or
– Do not reject the null hypothesis and do
not accept the alternative hypothesis

Note that we NEVER accept the null
hypothesis, we only fail to reject it!
More on Decisions

When completed, a significance test
tells us the probability of obtaining our
results when the null hypothesis is true
p(Results|Ho is True)

If that probability is small, smaller than
our significance level (α), it is likely that
Ho is not true and we reject it