Download PP 2 - Personal Web Pages

t tests
comparing two means
Overall Purpose

A t-test is used to compare two
average scores.
 Sample data are used to answer a
question about population means.
 The population means and
standard deviations are not known.
The Three Types

There are three ways to use a
t-test in a comparative
educational study:
 1. To compare two groups
measured at one time.

Independent t-test
The Three Types

2. To compare a sample to a
population, or one group
measured at one point in time.

One sample t-test
The Three Types


3. To compare one group to itself
over time, or one group measured at
two times.
Dependent t-test
Assumptions





Normality
Homogeneity of Variance
Independence of Observations
Random Sampling
Population Variance Not Known
Additional Considerations

Confidence intervals can be
used to test the same hypotheses.

There is a unique critical t value
for each degrees of freedom
condition.
Additional Considerations

Random assignment is needed
to make causal inferences when
using the independent t-test.

If intact groups are compared,
examine differences on potential
confounding variables.
Additional Considerations
 The z value of 1.96 serves as a rough
guideline for evaluating a
t value.
 It means that the amount of difference in
the means is approximately twice as
large as expected due to sampling error
alone.
 Interpret the p value.
Statistical Significance

How do you know when there is
a statistically significant
difference between the average
scores you are comparing?
Statistical Significance

When the p value is less than
alpha, usually set at .05.
 What does a small p value
mean?
Statistical Significance
 If
the two population means are equal,
your sample data can still show a
difference due to sampling error.
 The
p value indicates the probability
of results such as those obtained, or
larger, given that the null hypothesis
is true and only sampling error has
lead to the observed difference.
Statistical Significance

You have to decide which is a more
reasonable conclusion:
 There is a real difference between
the population means.
Or
 The observed difference is due to
sampling error.
Statistical Significance

We call these conclusions:
 Rejecting the null hypothesis.
 Failing to reject the null
hypothesis.
Statistical Significance

If the p value is small, less than
alpha (typically set at .05), then
we conclude that the observed
sample difference is unlikely to
be the result of sampling error.
Statistical Significance

If the p value is large, greater
than or equal to alpha (typically
set at .05), then we conclude
that the difference you observed
could have occurred by
sampling error, even when the
null hypothesis is true.
Hypotheses

Hypotheses for the Independent t-test
Null Hypothesis:
 m1 = m2 or m1 - m2 = 0
Directional Alternative Hypothesis:
 m1 > m2 or m1 - m2 > 0
Non-directional Alternative Hypothesis:
 m1 =/= m2 or m1 - m2 =/= 0
where:
 m1 = population mean for group one
 m2 = population mean for group two
Hypotheses

Hypotheses for the One Sample t-test
Null Hypothesis:
 m = m0 or m - m0 = 0
Directional Alternative Hypothesis:
 m > m0 or m - m0 > 0
Non-directional Alternative Hypothesis:
 m =/= m0 or m - m0 =/= 0
where:
 m = population mean for group of interest (local
population)
 m0 = population mean for comparison (national norm)
Hypotheses

Hypotheses for the Dependent t-test
Null Hypothesis:
 d = 0 or m1 - m2 = 0
Directional Alternative Hypothesis:
 d > 0 or m1 - m2 > 0
Non-directional Alternative Hypothesis:
 d =/= 0 or m1 - m2 =/= 0
where:
 m1 = population mean for time one
 m2 = population mean for time two
 d = the average difference between time on and time
two.
Example

Our research design:

We could compare “leavers” and
“stayers” on their reported Classroom
Demands, Resources, and Stress.