STATISTICS HYPOTHESES TEST (II) One-sample tests on the mean and variance Professor Ke-Sheng Cheng...
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STATISTICS HYPOTHESES TEST (II) One-sample tests on the mean and variance Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University
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Transcript of STATISTICS HYPOTHESES TEST (II) One-sample tests on the mean and variance Professor Ke-Sheng Cheng...
STATISTICSHYPOTHESES TEST (II)
One-sample testson the mean and variance
Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering
National Taiwan University
Tests on Mean
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University2
(1) 2 is known
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University3
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University4
(2) 2 is unknown
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University5
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University6
(3) 2 is known
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University7
(4) 2 is unknown
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University8
Tests on Variance
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University9
(1) is known
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University10
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University11
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University12
(2) is unknown
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University13
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University14
(3) is known
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University15
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University16
(4) is unknown
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University17
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University18
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University19
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University20
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University21
t-test in R
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University22
For a two-tailed test, the p-value in R is twice of the exceedance probability shown in the above figure.
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University23
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University24
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.
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University25
• 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.
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University26
• 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.
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University27
• 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.
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University28
• The test statistic is calculated as:
04/10/23Department of Bioenvironmental Systems Engineering
National Taiwan University29
0 if,~ d1 n
d
tnS
dT
0:
0:
1
0
d
d
H
H
0:
0:
1
0
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Two-sided test One-sided test
