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 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 2

(1) 2 is known 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 3

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(2) 2 is unknown 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 5

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(3) 2 is known 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 7

(4) 2 is unknown 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 8

Tests on Variance 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 9

(1) is known 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 10

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(2) is unknown 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 13

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(3) is known 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 15

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(4) is unknown 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 17

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t-test in R 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 22 For a two-tailed test, the p-value in R is twice of the exceedance probability shown in the above figure.

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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. 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 25

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. 1/31/2014 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. 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 27

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. 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 28

The test statistic is calculated as: 1/31/2014 Department of Bioenvironmental Systems Engineering National Taiwan University 29 Two-sided test One-sided test