Download Week 7

Two main tasks in inferential statistics:
(revisited)
1) Estimation:
•
Use data to infer population parameter  e.g.,
estimate victimization rate from NCVS
• 2 main forms:
1) Point Parameter estimation
2) Confidence Intervals
2) Hypothesis Testing:
•
Use data to check the reasonableness of some general
hypothesis or prediction about population events  e.g.,
test if civil orders of protection lower violent victimizations
2nd Inferential task: Hypothesis-testing
– Most common form =
the “Null Hypothesis Test” in which:
•
•
Our “Research Hypothesis” is tested against a
“Null Hypothesis” that says the observed sample
result is due only to random sampling error
The Null Hypothesis = a prediction of “null
effects” (e.g., of no difference or no relation)


•
Differences might be observed in our sample data, but
no true differences exist in the population
Any observed differences are due to sampling error
The Research Hypothesis = a prediction that
there are real differences in population
Hypothesis-testing (continued):
– Basic task in NHT is comparable to
establishing criminal guilt in a trial
 Basic presumption = null condition (not guilty)
 We must present evidence that suggests guilt
beyond our doubts that the incriminating pattern
observed could just be coincidence or chance
 When the evidence is so strong that it exceeds
“reasonable doubt”, we infer “guilty” conclusion
 If evidence is not strong enough to exceed
reasonable doubt, we retain the presumption of
“not guilty” (null hypothesis)
Hypothesis-testing (continued):
– Basic form of the test
 Formulate 2 hypotheses (both are necessary):
1)
2)
Null Hypothesis (of no difference/effect)
Research Hypothesis (of some difference/effect)
 Compute a test statistic value from the sample
data with a known Null sampling distribution.
 Compare the value with the probability table.
 If the probability exceeds “reasonable doubt”,
then reject the Null H & accept Research H.
 If evidence does not exceed reasonable doubt,
then “retain” Null hypothesis
Hypothesis-testing (continued):
Test
Statistic
(T or Z)
=
Sample
Null H
- Value
Statistic
Standard Error
of Sample Statistic
[compare computed value of Test Statistic to the null probability distribution]
What does this look like?
Test of Single Mean
z
X 
X
or
X 
t
 X
Test of Difference Between Two Means
z
X1  X 2
 X X
1
or
2
X1  X 2
t 
 X  X
1
2
Hypothesis-testing (continued):
– Basic Steps in the process
 Specify the Null and Research hypotheses
 Select & compute sample statistic (z or t)
 Compare sample statistic to probability table
1) See if computed value exceeds critical value
(for acceptable error level); or
2) See if P-level (associated with sample
statistic) is smaller than acceptable error level
 Decide whether to reject the Null hypothesis or
retain the Null hypothesis (notice the wording)
 Rejecting the Null = Accepting the Research H.
Hypothesis-testing (continued):
– Testing a single mean
 Test sample mean against hypothesized
population parameter
t
X
^
X
– Testing two means
 Test the difference between two sample means
against the hypothesis that in the population the
parameter difference is zero
X1  X 2
t
 X  X
1
2
Decision Table for Inferences
Inference
Null H = True
Null H = True
Null H = False
Correct
Decision
Type I error
( )
Type II error
( )
Correct
Decision
Actual
Null H = False
Important Distinctions to note:
– Z vs. t statistics:
•
•
What’s the difference?
When to use one versus the other?
– One-tail vs. Two-tail tests:
•
•
What’s the difference?
When to use one versus the other?
– P-level (Sig.) vs. alpha-level:
•
•
What do they reference?
How do they relate?
─ “significance”: Statistical vs. Substantive
Null Hypothesis Testing example:
–
o
o
o
o
o
Test the H that juvenile delinquents have
subnormal IQs by collecting IQ scores on 16
identified delinquents
 Population: mean = 100, St Dev = 10
 Sample: mean = 95, St Dev = 9.2, N = 16
Are the delinquents subnormal in IQ?
How do we test this? (By hand? By SPSS?)
What hypotheses are we testing?
What statistic should we use?
What kind of test (1- or 2-tailed) to use?
Final Distinction to note:
– Means of Independent Groups vs.
Correlated (Paired) Groups:
• What’s the difference?
• When to use or the other?
• How are the computations different?
– Means vs. Proportions:
• What’s the difference?
• How are the computations different?