Hypothesis Testing
Module 3.3: Parametric Tests
c
University of New South Wales
School of Risk and Actuarial Studies
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Hypothesis Testing
Introduction
Fisher’s exact test
Fisher’s exact test
Contingency tables
One sample Multinomial
r
Sample Multinomial
Contingency tables
One sample Multinomial
Contingency tables
Examples
Contingency tables
Examples
Nonparametric tests
Two sample test
Nonparametric tests
Test on quantiles
Goodness of fit tests
An overview
Goodness of fit tests
Exercises
Summary
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Hypothesis Testing
Introduction
Introduction
I
Last two weeks we have seen hypothesis testing;
One part of it was the test statistic;
I
To do so, we assumed a distribution for
X
.
I
For example, normal sample and testing mean with unknown
variance,
¯
X

μ
S
/
√
n
∼
t
n

1
.
However, there are cases where we don’t know the distribution
of
X
.
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Hypothesis Testing
Introduction
Introduction
I
Use
nonparametric methods
:
I
The nonparametric methods do not rely on the estimation of
the parameters describing the distribution.
I
We introduce the following tests:
I
Fisher exact test;
I
Quantile test;
I
Signed rank test;
I
Wilcoxon paired sample test.
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Hypothesis Testing
Introduction
Introduction
I
We also introduce an important concept: Chisquared tests.
I
Contingency tables;
I
Test of independence;
I
Test of homogeneity.
I
Verify distributional assumption—Goodnessoffit tests:
I
Chisquared test;
I
AndersonDarling & Cramérvon Mises tests;
I
KolmogorovSmirnoff & Kuiper tests.
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Hypothesis Testing
Fisher’s exact test
Fisher’s exact test
Example: Fisher’s exact test
I
By random selection, 24 male and female employees are
selected. Their promotion records are tabulated below,
Number
Male
Female
Promote
x
=
21
14
M
=
35
Hold file
3
10
13
n
=
24
24
N
=
48
I
We want to test the following statement
Are promotion decisions made independent of gender?
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Hypothesis Testing
Fisher’s exact test
Fisher’s exact test
Hypergeometric distribution
I
Take
n
draws from a sample without replacement;
Let the total number of objects be
N
(
≥
n
)
;
Have two possible different outcomes (e.g. “
blue ball
” and
“
red ball
”), one of them (say
blue balls
) has
M
(
≤
N
)
objects.
I
Let
X
be the random variable of
blue
balls selected.
This provides us the
Hypergeometric distribution
:
X
∼
HYP
(
n
,
M
,
N
)
with
n
=
1
,
2
, . . . ,
N
,
M
=
0
,
1
, . . . ,
N
.
The probability mass function of a Hypergeometric distribution
is given by:
p
X
(
x
) =
(
M
x
)
·
(
N

M
n

x
)
(
N
n
)
.
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 Normal Distribution, Statistical tests, Chisquare distribution, Pearson's chisquare test, Nonparametric statistics