Twosample
Difference of Means Test
 Independent
Samples design
 Classical design used
in psychology
 N subjects are
randomly assigned to two groups.
 Random
assignment insures that individuals have an
equal chance of being in either treatment
group
 After treatment,
the individuals are measured on the dependent
variable.
 A test of
differences in means between groups provides
evidence for the treatment's effect.
 Features of good
experimental design:
 Blind
assignment
 Blind
experimenter
 Doubleblind
study
 Approximation in
political and social research
 Little potential
for nor experience with experimental
design.
 Instead, social
researchers tend to select out two populations (or
samples) on the basis of some control variable
between the two groups.
 This is not as
neat as psychologists or statisticians like, but
it's what social researchers usually have to work
with.
 Related Samples
design
 The observations are
not made on independent samples but on matched
subjects or even the same subjects.
 Matchedpairs
design
 Cases in both
groups chosen to be similar on certain factors
thought to be causes of the dependent
variable.
 Examples:
 Individuals
matched on sex, height, weight,
age.
 Nations
matched on GNP, size, type of
government
 Because some
sources of variation are controlled, a
matchedpairs test is designed to detect smaller
variations in the dependent variable  such as
expenditures for defense, in the case of
nations.
 RepeatedMeasures
design (paired data in SPSS)
 Uses the same
cases but takes observations before and
after treatment.
 Reduces
possible variation even more.
 In political
and social research, may be used with repeated
measures on the same states over time.
 Summary:
 If random factors
only make the determination, the test for differences
of means should be conducted as if the data were from
independent samples.
 If the groups are
matched before the experiment or if the same subjects
are used, the test of difference of means should be
conducted as if the data were from related
samples.
 In political and
social research, most tests for the difference between
means employ independent samples.
The appropriate test for
difference of means in SPSS: TTest
 Available under the
Analyze Menu in SPSS 10
 Choose Compare
Means
 Select
IndependentSamples T Test  if your data are
divided into two unmatched groups, e.g.
 Democrats v.
Republicans
 Men v.
women
 Undergraduates v.
graduates
 Select
PairedSamples T Test  if your data consists
of repeated measures on the same units, e.g.
 comparing same
students' performance on 1/3 and 2/3
examinations
 comparing same
states' voting turnout in 1996 and 2000
presidential elections
The conceptual model underlying both types of TTest:
 compare the difference
of means for sample 1 with sample 2:
 Strictly speaking,
the test is for the difference computed between
the means of two samples, drawn and computed an
infinite number of times, compared with the
difference between the two means of their
parent populations:
 But since the two
population means, _{1
}and _{2},
are assumed to be equal, _{1}
 _{2}
= 0, so the second term drops out, and the test
becomes simply a test for the observed difference
between the sample means:
 As in the standard
"model" of hypothesis testing, this observed
difference in sample means can be evaluated by
dividing by some "standard error" allowable for sampling
differences.
 Assuming independent
samples, the STANDARD ERROR of the sampling
distribution of the DIFFERENCE IN MEANS is
 When _{1}
and _{2}
are known, the formula becomes
 When _{1}
and _{2}
are estimated by sample values and where: s = unbiased
estimate of ,
we can compute
 Thus, the formula
using
as an estimate of becomes
 Using estimated values
for the population variances produces the same formulate,
except that when SAMPLE SIZES are SMALL (30 cases or
less), the tdistribution must be used.
 For historical reasons,
the difference of means test is referred to as the
ttest, even thought the t distribution is really
necessary only for small samples.
