Download Paired t-test

STATISTICS
HYPOTHESES TEST (II)
One-sample tests
on the mean and variance
Professor Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University
Tests on Mean
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
2
(1) 2 is known
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
3
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
4
(2) 2 is unknown
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
5
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
6
(3) 2 is known
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
7
(4) 2 is unknown
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
8
Tests on Variance
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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(1)  is known
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
10
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
11
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
12
(2)  is unknown
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
13
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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(3)  is known
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
16
(4)  is unknown
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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t-test in R
For a two-tailed test, the p-value in R is
twice of the exceedance probability shown
in the above figure.
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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Paired t-test
• The paired t test provides a hypothesis
test of the difference between
population means for a pair of random
samples whose differences are
approximately normally distributed.
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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• If the population from which paired
differences to be analyzed by a paired t
test were sampled violate one or more of
the paired t test assumptions, the results
of the analysis may be incorrect or
misleading. For example, if the
assumption of independence for the
paired differences is violated, then the
paired t test is simply not appropriate.
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
26
• Note that the two values that make up
each paired difference need not be
independent, and in fact are expected to
be correlated, such as before and after
measurements. If you treat paired data
as coming from two independent
samples, such as doing an inappropriate
two-sample unpaired t test instead of a
paired t test, then you may sacrifice
power.
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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• If the assumption of normality is violated,
or outliers are present, then the paired t
test may not be the most powerful test
available, and this could mean the
difference between detecting a true
difference or not. A nonparametric test
or employing a transformation may
result in a more powerful test. For
example, if the distribution of the paired
differences is not symmetric, a
transformation may produce symmetry.
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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• The test statistic is calculated as:
T
d
Sd
n
~ tn1 , if d  0
H 0 : d  0
H 0 : d  0
H1 :  d  0
H1 :  d  0
Two-sided test
One-sided test
5/25/2017
Department of Bioenvironmental Systems Engineering
National Taiwan University
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