STATISTICSHYPOTHESES TEST (I)

Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering

National Taiwan University

Examples of hypothesis tests

• Based on historical records, do female students really perform better in statistics class than male students?

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What is hypothesis test

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Null and alternative hypotheses

• Two hypotheses and are defined as: (Null hypothesis) (Alternative hypothesis)

• A procedure for deciding whether to accept (or more precisely, fail to reject) the hypothesis or to accept the hypothesis (or reject ) is called a “test procedure” or simply a “test”.

11 : H

0H

1H 0H

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00 : H

Simple and Composite Hypotheses

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The Critical Region and Test Statistics

• Consider the following hypotheses test:

• Suppose that we are given a random sample of size n, , from a distribution with parameter . Let S denote the sample space of the n-dimensional random vector .

00 : H 11 : H

nXX ,...,1

),...,( 1 nXXX

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• In order to carry out the test we can partition the sample space S into two disjoint subsets So

and S1. Subset So contains the values of X for

which we will accept , and subset S1

contains the values of X for which we will reject . The subset for which will be rejected is called the “critical region” of the test.

0H

0H0H

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Why is it fixed?

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Critical region – fixed due to specification of Ho , distribution of the test statistic, and the level of significance.

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The Power Function

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Types of error

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Making a test have a specific significance level

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Example

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Why?

The probability density function of the test statistic is known.

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Power function of the test

C=6

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Power functions

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

p

Pow

er f

un

ctio

n

C=6

C=7

C=8

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Now, let’s set the size of the random sample n = 20 and conduct the same test.

Let .

]3.0[

1.0|]3.0[3.0

pCXP

ppCTPp

T

20

11 ii

n

ii XXT

cc

ii pp

cpcTPcXT

20

20

1

)1(20

][

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Example

• Suppose that is a random sample of size n and we wish to test the hypotheses:

),...,( 1 nXXX

00 : H

01 : H

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328.119,1.0 t

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Noncentral t distribution in R

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Non-centrality parameter

ncp=0ncp=1,-1ncp=2,-2ncp=3,-3

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Guidelines for Hypothesis Testing

1. When testing a hypothesis concerning the value of some parameter , the statement of equality will always be included in H0. In this way H0 pinpoints a specific numerical value that could be the actual value of . This value is called the null value and is denoted by 0.

2. Whatever is to be detected or supported is the alternative hypothesis.

3. It is hoped that the evidence leads us to reject H0 and thereby to accept H1.

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• A confidence interval is just the flip side of a hypothesis test.

• If the hypothesis test fails to reject H0, then the parameter from H0 is definitely within the confidence interval.

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• Review of the confidence interval & acceptance interval

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nX

)2/)1((

2/10

n

z

0

nX of interval acceptance %100

of interval confidence %100

0

• Using the acceptance interval for hypothesis test

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nX

nX of interval acceptance %100 sided-One

0

0

1

1

2

2

n

zXPn

zXPzn

XP nn

n

111

/One-sided acceptance interval One-sided confidence interval

01

00

:

:

H

H

Critical region

The highest probability of committing a type-one error.

Ho is trueH1 is true

Test statistic

There exists a dual relationship between a hypothesis test and its corresponding confidence interval estimation.