Inferential Statistics
 Hypothesis testing (relationship
between 2 or more variables)
 We want to make inferences from a
sample to a population.
 A random sample allows us to infer from
a sample to a population.
Inferential Statistics
Significance Tests
 Z scores (one sample case)
 Difference of means tests
Two sample case (t-test)
Three or more sample case (ANOVA)
 Chi-Square
 Bi-Variate Correlation (One IV & One DV)
 Bi-Variate Regression (One IV & One DV)
 Multi-Variate Regression (Two or more IVs & One DV)
Level of Measurement & Significance Tests
Chi-Square
IV & DV are nominal and/or ordinal
t-test
IV is nominal (group like men & women)
DV is Interval/Ratio (or a scale)
ANOVA
IV is nominal (group with 3 or more categories)
DV is I/R (or a scale)
Regression
IV(s) & DV are I/R (or scales)
IV(s) can be dummy variables
Which Test Would you Use?
Hr: There is a relationship between:

gender & income (measured in dollars)

race (measured as Black, Latino/a,
Caucasian) and income

religious preference (catholic, protestant)
and attitudes toward abortion (favor,
oppose)

education (measured in years) and income

degree completed (HS or Less & College)
and income
Chi-Square
Chi-Square: a test of significance used
with cross tabulations of nominal/ordinal
level data.
Example:
Research question: Does political orientation
influence parenting style?
Political orientation: Conservative & Liberal
Parenting style: Permissive & Not Permissive
Why not simply compare the mean difference
between liberals and conservatives on parenting
style?
We are really saying:
Hr: The frequency (proportion) of liberals
who are permissive is not the same as
the frequency of conservatives who are
permissive.
The null (a hypothesis of no difference)
says:
Ho: The frequency (proportion) of liberals
who are permissive is the same as the
frequency of conservatives who are
permissive.
Chi-Square compares the observed
frequencies (from the data in your
sample) to expected frequencies.
Expected frequencies: These are the
frequencies we would expect if the null
were true (if there is no difference
between political view and parenting
style)
Example:
We do a cross tab of political orientation by
parenting style and our observed frequencies
are:
Political Orientation
Liberals
Conservatives
Child-rearing
Permissive
Not permissive
5
15
___
20
10
10
___
20
Are these differences significant?
Chi-Square test of significance:
 Chi-Square = ∑(fo- fe)2 / fe
Steps
Step 1.
We have the observed frequencies
Political Orientation
Liberals
Conservatives
Child-rearing
Permissive
5
10
Not permissive
15
10
___
___
20
20
Steps
Step 2. Need to calculate the expected frequencies.
Formula:

fe = (row marginal total) (column marginal total)
___________________________________
N
Expected Frequencies
See board
Step 3. Calculate Chi-Square
See board
Calculated Chi-Square for Political Views by
Parenting Style
Chi Square = 2.66
Df = (r-1)(c-1)
Df = (2-1) (2-1) = 1
Must have a Chi Square of 3.84 at p.=.05
to reject the null hypothesis.
Decision?
Review Alpha Levels
Alpha level the probability of making a Type I
error
Type I error (reject the null when it is true)
Set alpha level small (.05 or smaller) to minimize
risk.
The larger the sample the smaller the alpha
level should be.
Chi square is sensitive to N (large
N’s can yield significant results)
So, we use a measure of
association with Chi-square
Measures of association tell us
about the
strength of the relationship
Measures of Association
The type of measure used is
determined by the level of measurement
and the number of categories.
See handout
Interpret GSS Output
Crosstab
FEELINGS ABOUT PORNOGRAPHY LAWS * HOW FUNDAMENTALIST IS R CURRENTLY Crosstabulation
FEELINGS ABOUT
PORNOGRAPHY
LAWS
ILLEGAL TO ALL
ILLEGAL UNDER 18
LEGAL
Total
Count
Expected Count
% within HOW
FUNDAMENTALIST
IS R CURRENTLY
Count
Expected Count
% within HOW
FUNDAMENTALIST
IS R CURRENTLY
Count
Expected Count
% within HOW
FUNDAMENTALIST
IS R CURRENTLY
Count
Expected Count
% within HOW
FUNDAMENTALIST
IS R CURRENTLY
HOW FUNDAMENTALIST IS R
CURRENTLY
FUNDAME
NTALIST
MODERATE
LIBERAL
249
227
150
193.2
242.1
190.7
Total
626
626.0
46.3%
33.7%
28.2%
35.9%
275
328.1
431
411.1
357
323.8
1063
1063.0
51.1%
63.9%
67.2%
61.0%
14
16.7
16
20.9
24
16.5
54
54.0
2.6%
2.4%
4.5%
3.1%
538
538.0
674
674.0
531
531.0
1743
1743.0
100.0%
100.0%
100.0%
100.0%
Chi-Square
Chi-Square Tests
Pears on Chi-Square
Likelihood Ratio
Linear-by-Linear
Ass ociation
N of Valid Cases
Value
43.721a
43.149
37.689
4
4
Asymp. Sig.
(2-s ided)
.000
.000
1
.000
df
1743
a. 0 cells (.0%) have expected count les s than 5. The
minimum expected count is 16.45.
Measure of Association
Which should we use?
Cramer’s V = .112