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

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National Taiwan University7

(4) 2 is unknown

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National Taiwan University8

Tests on Variance

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National Taiwan University9

(1) is known

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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

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National Taiwan University17

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National Taiwan University18

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National Taiwan University19

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National Taiwan University20

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National Taiwan University21

t-test in R

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National Taiwan University22

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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National Taiwan University23

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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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d

H

H

Two-sided test One-sided test