Two Independent Samples
Chapter 10 presents T-Tests for the situation where
there are two completely separate samples that are independently
taken from two different populations. Different subjects
are used in each sample, and they are in no way matched
with each other. There does not need to be the same
number of subjects in each sample (i.e., sample sizes
can be different).
This situation is probably the most common experimental
design in Psychology. These designs are sometimes
called between-subjects or between-groups
designs.
T-Test for Two Independent Samples
Once again, the generic formula for the T-Statistic
is:
For the Independent Samples T-Statistic:
- Sample statistic: For Independent samples
the sample statistic is the difference between the
two sample means.
- Population Parameter: The population parameter
is the hypothesized difference between the two population
means.
- Estimated Standard error: The estimated
standard error is defined as:
This formula for the estimated standard error uses
the "pooled" (combined) errors for the two sample
means. The formula for this is:
- Independent Samples T-Statistic: Finally
we can see how the T-Statistic for independent samples
is defined:
- Hypothesis Testing: Hypotheses are constructed
just as before, except that they are about the differences
(usually involving hypotheses about zero differences).
There can be one-tail tests, but usually, tests
are two-tailed.
Dependent Samples T-Test Example:
Reading Ability
We use data concerning reading ability. (These data
are from page 543 of Moore and McCabe.) The data come
from a study in which an educator tested whether a new
directed reading activity help elementary school pupils
improve their reading ability. The two groups are a
classroom of 21 students who got the activity (the "Treatment"
group), and another classroom of 23 students who didn't
(the "Control" group). All students were given the Degree
of Reading Power test.
- The data report for the ViSta
Data is:
- The ViSta Applet
for these data yields the following workmap:
- We analyze these data using a one-tailed
test based on a directional hypothesis that the
directed reading activity will improve reading ability
scores (that the "Treatment" group will have higher
scores than the "Control" group).
Report-Model: The analysis of these data
produces the following model report:
From this report we observe that p=.0129. Thus,
we conclude that we can reject the null hypothesis
that the reading activity did not improve reading
ability scores. We also conclude that the reading
activity had a "significant statistical effect"
on the reading ability scores.
- Visualize-Model: As pointed out
in the chapter, the significance test requires that
the data come from populations that are normally
distributed with equal variance. The visualization
helps us see whether these assumptions are met.
Normality: Interpreting features of these
plots discussed above, we conclude that the data
are reasonably normal.
Equal Variance: The box-plot, however,
reveals that there may be more variation in the
control group than in the treatment group (the
box for the control group is taller than for the
treatment group, and the observation dots cover
a wider range for the control group). This may
mean that the value of p (.0129) may be too optimistic.
We note that we have one outlying control group
value. Perhaps we should remove it and reanalyze
the data.
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