Introduction to Hypothesis
Testing
CJ 526 Statistical Analysis in
Criminal Justice
Hypotheses
A hypothesis is a prediction about the
outcome of a research study
Hypothesis Testing
Hypothesis testing is an inferential
procedure that uses sample data to
evaluate the credibility of a
hypothesis about a population
Overview of Hypothesis Testing
1.
State a hypothesis about a
population
1.
Usually in terms of the value of a
population parameter
1.
Typically the mean or the difference between
means
Overview of Hypothesis Testing
-- Continued
If the data are consistent with the
hypothesis, conclude that the
hypothesis was reasonable, and fail
to reject it
Example



Babies birth weight will not differ between
smoking and non-smoking mothers (null)
Babies born to women who smoke during
pregnancy will be more likely to be of low
birth weight (alternative)
Independent Variable:
• Smoking during pregnancy

Dependent Variable:
• Birth weight
Example -- Continued
1.
2.
3.
4.
5.
Obtain a random sample of women who
are pregnant and smoke
Obtain a random sample of non-smoking
pregnant women, or compare to the
national average
Weigh the babies at birth
Compare sample data to hypothesis
Make decision:
1.
2.
Reject the null hypothesis
Fail to reject the hypothesis
Assumptions Behind Hypothesis
Testing
The effect of the Independent Variable
(treatment effect) is assumed to:
Add (or subtract) a constant from every
individual’s score
The Logic of Hypothesis Testing
1.
Can’t prove hypothesis
1.
Proof requires evidence for all cases
Steps in Hypothesis Testing
1.
Determine the number of
samples (groups, conditions)
1.
2.
3.
One
Two
k (three or more)
Steps in Hypothesis Testing -continued
2.
If there are two or more samples,
determine whether they are
independent or dependent
1.
2.
Same group (repeated-measures)
Match on some other variable(s) known to
influence DV (matched-subjects)
Steps in Hypothesis Testing -continued
3.
If there is one sample and the
Dependent Variable is at the
Interval or Ratio Level of
Measurement, is the standard
deviation of the population (,
sigma) known:
1.
2.
If  is known, use a One-Sample z-Test
If  is unknown, use a One-Sample t-Test
Steps in Hypothesis Testing -continued
4.
5.
6.
Identify the independent variable
Identify the dependent variable and
its level of measurement
Identify the population to which
inferences will be made
Steps in Hypothesis Testing -continued
7.
Determine the appropriate
inferential statistical test
1.
2.
3.
8.
9.
Number of samples
Nature of samples (if applicable)
Level of measurement of DV
State the null hypothesis
State the alternative hypothesis
Steps in Hypothesis Testing -continued
10.
State Decision Rule:
1.
11.
12.
13.
If the p-value of the obtained test statistic
is less than .05, reject the Null Hypothesis
Use SPSS to compute the obtained
test statistic
Make decision
Interpret results
Null Hypothesis
The null hypothesis predicts that the
Independent Variable (treatment)
will have no effect on the
Dependent Variable for the
population
Alternative Hypothesis
The alternative hypothesis predicts
that the Independent Variable
(treatment) will have an effect on
the Dependent Variable for the
population
Directional Alternative
Hypotheses
Researcher has reason to believe
before conducting the test that a
difference will lie in a specified
direction
1.
2.
Prior research
Theory
Non-directional Alternative
Hypotheses
Researcher has no reason to believe
that there will be a difference in a
specified direction
There is insufficient research or
information or theory to make a
specific prediction
Set the Criteria
Because of sampling error, there is
likely to be a discrepancy between
the sample mean and the
population mean
At what point does the difference
become meaningful and not just a
matter of chance?
3. Collect Sample Data
Use the data to calculate the obtained
test statistic, using the appropriate
statistical test, based on level of
measurement of the dependent
variable, number of samples,
whether the samples are
independent or related
4. Evaluate the Null Hypothesis
1. Reject the null hypothesis
1.
2.
3.
OR
If sample data is unlikely to have been
drawn from a population where the null
hypothesis is true
If the p-value of the obtained test statistic
is less than .05, meaning that the null
hypothesis is rejected and there is less
than a 5% probability that this decision is
incorrect
The alternative is accepted, that there is a
difference
Failure to Reject the Null
Hypothesis
1. Either:
1.
Treatment had an effect, could not
demonstrate it
• or
2.
Treatment had no effect
Errors in Hypothesis Testing
Actual State of Affairs
Belief
Decision
H0 is True
H0 is False
H0 is False
Reject H0
Type I Error
False Positive

Correct
Rejection
1-
Power
H0 is True
Fail to Reject
H0
Correct Failure Type II Error
to Reject
False Negative
1-

Type I Error
Committed when H0 is rejected as
false although it is true
Type II Error
Committed when H0 is not rejected
although it is false
Statistical Power
Probability that the test will correctly
reject a false null hypothesis
Power -- Continued
When a treatment effect exists
1.
2.
A study may fail to discover it (Type II
error, fail to reject a false null hypothesis)
A study may discover it (reject a false null
hypothesis)
Power -- Continued
Reducing alpha (.05 --> .01 --> .001)
1.
2.
Reduces power
Inverse relationship between Type I and
Type II errors
Power -- Continued
Some inferential statistical tests are
more powerful
Jury’s Decision
Did Not Commit Crime
Committed Crime
Guilty
Type I Error
Convict Innocent
Person
Correct Verdict
Convict Guilty
Person
Not Guilty
Correct Acquittal
Type II Error
Fail to Convict Innocent Fail to Convict
Person
Guilty Person
Level of Significance
Alpha: probability of committing a
Type I error
1.
2.
Reject H0 although it is true
Symbolized by 