Download The t-test

Inferential Statistics I:
The t-test
Experimental Methods and
Statistics
Department of
Cognitive Science
Michael J. Kalsher
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Outline
Definitions
Descriptive vs. Inferential Statistics
The t-test
- One-group t-test
- Dependent-groups t-test
- Independent-groups t-test
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The t-test:
•
Basic Concepts
Types of t-tests
- Independent Groups vs. Dependent
Groups
• Rationale for the tests
- Assumptions
•
•
•
•
Interpretation
Reporting results
Calculating an Effect Size
t-tests as GLM
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Beer and Statistics:
A Winning Combination!
William Sealy Gosset (1876–1937)
Famous as a statistician, best known by his pen name
Student and for his work on Student's t-distribution.
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The One Group t test
The One-group t test is used to compare a sample mean
to a specific value (e.g., a population parameter; a neutral point on a
Likert-type scale).
Examples:
1. A study investigating whether stock brokers differ from the general population on
some rating scale where the mean for the general population is known.
2. An observational study to investigate whether scores differ from some neutral point
on a Likert-type scale.
Calculation of ty :
Note: The symbol ty indicates
this is a t test for a single group
mean.
ty =
Mean Difference
Standard Error
(of the mean difference)
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Assumptions
The one-group t test requires the following statistical
assumptions:
1. Random and Independent sampling.
2. Data are from normally distributed populations.
Note: The one-group t test is generally considered robust against violation of this
assumption once N > 30.
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Computing the one-group
t test by hand
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Critical Values:
One-Group t test
Note: Degrees of Freedom = N - 1
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Computing the one-group
t test using SPSS
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Move DV to box labeled
“Test variable(s):
Type in “3” as a proxy
for the population mean.
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SPSS Output
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Reporting the Results:
One Group t test
The results showed that the students’ rated level of
agreement with the statement “I feel good about myself”
(M=3.4) was not significantly different from the scale’s
neutral point (M=3.0), t(4)=.784. However, it is important
to note several important limitations with this result,
including the use of self-report measures and the small
sample size (five participants). Additional research is
needed to confirm, or refute, this initial finding.
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11. Select both Time 1 and
Time 2, then move to the box
labeled “Paired Variables.”
12. Next, “click”, “Paste”.
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The Independent Groups t test:
Between-subjects designs
Assumption:
Participants contributing to the two means come from
different groups; therefore, each person contributes only
one score to the data.
Calculation of t:
t=
Mean Difference
Standard Error
(of the mean difference)
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Standard Error:
How well does my sample
represent the population?
6
When someone takes
a sample from a
population, they are
taking one of many
possible samples-each of which has its
own mean (and s.d.).
Sampling
Distribution
5
Frequency
4
3
We can plot the
sample means as a
frequency distribution
or sampling
distribution.
2
1
0
Sample Mean
10
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Standard Error:
How well does my sample
represent the population?
The Standard Error, or Standard Error of the Mean, is an
estimate of the standard deviation of the sampling distribution
of means, based on the data from one or more random
samples.
•
•
•
Large values tell us that sample means can be quite different, and
therefore, a given sample may not be representative of the population.
Small values tell us that the sample is likely to be a reasonably accurate
reflection of the population.
An approximation of the standard error can be calculated by dividing the
sample standard deviation by the square root of the sample size
SE = 
N
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Standard Error:
Applied to Differences
We can extend the concept of standard error to situations in
which we’re examining differences between means.
The standard error of the differences estimates the extent to
which we’d expect sample means to differ by chance alone-it is a measure of the unsystematic variance, or variance not
caused by the experiment.
An estimate of the standard error can be calculated by
dividing the sample standard deviation by the square root of
the sample size.
SE = 
N
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Computing the independentgroups t test by hand
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Sample Problem
Anxiety Scores
Liberal
Arts
Behavioral
Science
45
58
63
59
62
63
51
68
54
74
63
68
52
52
54
66
64
69
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A college administrator reads an article
in USA Today suggesting that liberal
arts professors tend to be more
anxious than faculty members from
other disciplines within the humanities
and social sciences. To test whether
this is true at her university, she carries
out a study to determine whether
professors teaching liberal arts courses
are more anxious than professors
teaching behavioral science courses.
Sample data are gathered on two
variables: type of professor and level
of anxiety.
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Critical Values:
Independent Groups t test
Note: Degrees of Freedom = N1 + N2 - 2
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Reporting the Results:
Independent Groups t test
On average, the mean level of anxiety among a
sample of liberal arts professors (M = 55.7) was
significantly lower than the mean level of anxiety
among a sample of behavioral science
professors (M = 63.4), t(18) = -2.54, p < .05, r2 =
.26. The effect size estimate indicates that the
difference in anxiety level between the two
groups of professors represents a large effect.
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Computing the independentgroups t test using SPSS
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Sample Problem
A researcher is interested in
comparing the appetite suppression
effects of two drugs, fenfluramine
and amphetamine, in rat pups. Fiveday-old rat pups are randomly
assigned to be injected with one of
the two drugs. After injection, pups
are allowed to eat for two hours.
Percent weight gain is then
measured.
Percent Weight Gain
Fenfluramine Amphetamine
2
8
3
10
3
4
4
7
4
9
5
3
6
7
Compute the independent groups ttest using the data at right.
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Is this a true experiment, quasiexperiment, or observational study?
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8
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SPSS Output: Independent-Groups t test
Group Statistics
Weightgain
DrugType
1
2
N
Mean
4.60
7.40
10
10
Std. Deviation
1.647
2.675
Std. Error
Mean
.521
.846
Independent Samples Test
Levene's Test for
Equality of Variances
F
Weightgain
Equal variances
ass umed
Equal variances
not as sumed
1.110
Sig.
.306
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-2.819
18
.011
-2.800
.993
-4.887
-.713
-2.819
14.964
.013
-2.800
.993
-4.918
-.682
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Calculating Effect Size:
Independent Samples t test
r=
t2
(-2.819)2
t2 + df
(-2.819)2 + 18
7.95
r =.5534
7.95 + 18
r2 = .306
Note: Degrees of
freedom calculated by
adding the two sample
sizes and then
subtracting the number
of samples:
df = 10 + 10 – 2 = 18
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Reporting the Results:
Independent Groups t test
On average, the percent weight gain of five-dayold rat pups receiving amphetamine (M = 7.4, SE
= .85) was significantly higher than the percent
weight gain of rat pups receiving fenfluramine (M
= 4.6, SE = .52), t(18) = -2.82, p < .05, r2 = .31.
The effect size estimate indicates that the
difference in weight gain caused by the type of
drug given represents a large, and therefore
substantive, effect.
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