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The Statistical Imagination
Chapter 11:
Bivariate Relationships:
t-test for Comparing the
Means of Two Groups
© 2008 McGraw-Hill Higher Education
Bivariate Analysis
• Bivariate – or “two variable” – analysis
involves searching for statistical
relationships between two variables
• A statistical relationship between two
variables asserts that the measurements of
one variable tend to consistently change
with the measurements of the other,
making one variable a good predicator of
the other
© 2008 McGraw-Hill Higher Education
Independent and
Dependent Variables
• The predictor variable is the
independent variable
• The predicted variable is the
dependent variable
© 2008 McGraw-Hill Higher Education
Three Approaches to Measuring
Statistical Relationships
1. Difference of means testing
(Ch. 11 & 12)
2. Counting the frequencies of joint
occurrences of attributes of two
nominal/ordinal variables (Ch. 13)
3. Measuring the correlation between two
interval/ratio variables (Ch. 14 & 15)
© 2008 McGraw-Hill Higher Education
Difference of Means Testing
• Compares means of an interval/ratio
variable among the categories or groups of
a nominal/ordinal variable
• Chapter 11. The two-group difference of
means test – for a dependent interval/ratio
and an independent dichotomous
nominal/ordinal variable
• Chapter 12. Analysis of variance – to test
for a difference among three or more group
means
© 2008 McGraw-Hill Higher Education
Frequencies of Joint Occurrences
of Two Nominal Variables
• Chapter 13. Chi-square test – to
determine a relationship between two
nominal variables
• Web site Chapter Extensions to
Chapter 13: Gamma test – to
determine a relationship between two
ordinal variables
© 2008 McGraw-Hill Higher Education
Measuring Correlation
• Chapter 14-15. Correlation – to
determine a relationship between two
interval/ratio variables
• Web site Extensions to Chapter 15:
Rank-order correlation test – to
determine a relationship between two
numbered ordinal level variables
© 2008 McGraw-Hill Higher Education
2-Group Difference of Means Test:
Independent Samples (t-test)
• Useful for testing a hypothesis
that the means of a variable differ
between two populations
comprised of different groups of
individuals
© 2008 McGraw-Hill Higher Education
When to Use an
Independent Samples t-test
• Two variables from one population and
sample, one interval/ratio and one
dichotomous nominal/ordinal
• Or: There are two populations and samples
and one interval/ratio variable; the samples are
representative of their population
• The interval/ratio variable is typically the
dependent variable
• The groups do not consist of same subjects
• Population variances are assumed equal
© 2008 McGraw-Hill Higher Education
Features of an
Independent Samples t-test
• The t-test focuses on the computed
difference between two sample
means and addresses the question of
whether the observed difference
between the sample means reflects a
real difference in the population
means or is simply due to sampling
error
© 2008 McGraw-Hill Higher Education
Features of an Independent
Samples t-test (cont.)
• Step 1. Stating the H0:
The mean of population 1 equals the
mean of population 2
• That is, there is no difference in the
means of the interval/ratio variable, X,
for the two populations
© 2008 McGraw-Hill Higher Education
Features of an Independent
Samples t-test (cont.)
• Step 2. The sampling distribution is the
approximately normal t-distribution
• The pooled variance formula for the standard
error is used when we can assume that
population variances are equal
• The separate variance formula for the
standard error is used when we cannot
assume that population variances are equal
© 2008 McGraw-Hill Higher Education
Features of an Independent
Samples t-test (cont.)
• Step 4. The effect is the difference
between the sample means
• The test statistic is the effect divided
by the standard error
• The p-value is estimated using the
t-distribution table
© 2008 McGraw-Hill Higher Education
Assumption of Equality
of Population Variances
• When one sample variance is not larger than twice
the size of the other, this suggests that the two
population variances are equal and we assume
equality of variances
• We may use the pooled variance estimate of the
standard error
• Equality of variances is also termed homogeneity
of variances or homoscedasticity
© 2008 McGraw-Hill Higher Education
Assumption of Equality
of Population Variances (cont.)
• Heterogeneity of variances, or
heteroscedasticity, is when variances
of the two populations appear
unequal
• Here we use the separate variance
estimate of the standard error and
calculate degrees of freedom
differently
© 2008 McGraw-Hill Higher Education
Test for Nonindependent or
Matched-Pair Samples
• This is a test of the difference of
means between two sets of scores of
the same research subjects, such as
two questionnaire items or scores
measured at two points in time
• This test is especially useful for
before-after or test-retest
experimental designs
© 2008 McGraw-Hill Higher Education
When to Use a
Nonindependent Samples t-test
• There is one population with a
representative sample from it
• There are two interval/ratio variables with
the same score design
• Or: There is a single variable measured
twice for the same sample subjects
• There is a target value of the variable
(usually zero) to which we may compare
the mean of the differences between the
two sets of scores
© 2008 McGraw-Hill Higher Education
Features of a Nonindependent
Samples or Matched-Pair t-test
• Step 1. Stating the H0:
The mean of differences between
the scores in a population is
equal to zero
© 2008 McGraw-Hill Higher Education
Nonindependent Samples or
Matched-Pair t-test (cont.)
• Step 2. The sampling distribution is
the approximately normal
t-distribution
• The standard error is calculated as
the standard deviation of differences
between scores divided by the square
root of n - 1
© 2008 McGraw-Hill Higher Education
Nonindependent Samples or
Matched-Pair t-test (cont.)
• Step 4. The effect is the mean of
differences between scores
• The test statistic is the effect
divided by the standard error
• The p-value is estimated using
the t-distribution table
© 2008 McGraw-Hill Higher Education
Distinguishing Between Practical
and Statistical Significance
• A hypothesis test determines significance
in terms of likely sampling error – whether
a sample difference is so large that there
probably is a difference in the populations
• Practical significance is an issue of
substance. A statistically significant
difference may not be practically significant
© 2008 McGraw-Hill Higher Education
Practical and Statistical
Significance (cont.)
• E.g., a hypothesis test reveals a statistically
significant difference in the mean number of
personal holidays of men and women in a
corporation: women average 0.1 days per
year more. The test tells us with 95%
confidence that the 0.1 day difference in the
samples truly exists in the populations
• However, is one-tenth day per year
meaningful? Might such a small statistical
effect be accounted for by some other
variable?
© 2008 McGraw-Hill Higher Education
Four Aspects of
Statistical Relationships
• When examining a relationship between
two variables, we can address four things:
existence, direction, strength, and practical
applications
• These four aspects provide a checklist for
what to say in writing up the results of a
hypothesis test
© 2008 McGraw-Hill Higher Education
Existence of a Relationship
• Existence: On the basis of statistical
analysis of a sample, can we conclude that
a relationship exists between two variables
among all subjects in the population?
• Established by rejection of the H0
• Testing for the existence of a relationship is
the first step in any analysis. If a
relationship is found not to exist, the other
three aspects of a relationship are irrelevant
© 2008 McGraw-Hill Higher Education
Direction of a Relationship
• Direction: Can the dependent variable
be expected to increase or decrease
as the independent variable
increases?
• Direction is stated in the alternative
hypothesis (HA) of step 1 of the six
steps of statistical inference
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Strength of a Relationship
• Strength: To what extent are
errors reduced in predicting the
scores of a dependent variable
when an independent variable is
used as a predictor?
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Practical Applications
of a Relationship
• Practical Applications: In practical,
everyday terms, how does
knowledge of a relationship
between two variables help us
understand and predict outcomes
of the dependent variable?
© 2008 McGraw-Hill Higher Education
Existence of a Relationship for
2-Group Difference of Means Test
• Existence: Established by using
independent samples or
nonindependent samples t-test
• When the H0 is rejected, a
relationship exists
© 2008 McGraw-Hill Higher Education
Direction of a Relationship for
2-Group Difference of Means Test
• For the two group tests, direction
and strength are not relevant
• Direction: Not relevant
• Strength: Not relevant
© 2008 McGraw-Hill Higher Education
Practical Applications of Relationship
for a 2-Group Difference of Means Test
• Practical Applications: Describe
the effect of the test in everyday
terms, where the effect of the
independent variable on the
dependent variable is the
difference between sample
means
© 2008 McGraw-Hill Higher Education
Statistical Follies
• Avoid a common tendency:
Difference in means testing is so
widely used that researchers
often focus too heavily on mean
differences while ignoring the
differences in variances (or
standard deviations)
© 2008 McGraw-Hill Higher Education