Lecture Notes
Chapter Fourteen: Nonparametric Statistics
Randall Miller
1 |
P a g e
1.
Introduction: Distribution-Free Tests
Definition 14.1
Distribution-free tests
are statistical that do not rely on any underlying assumptions about the
probability distribution of the sampled population.
Definition 14.2
The branch of inferential statistics devoted to distribution-free test is call
nonparametrics
.
Definition 14.3
Nonparametric statistics (or test) based on the ranks of measurements are called
rank
statistics
(or
rank test
).
2.
Single-Population Inferences
Sign Test for a Population Median
η
One-Tailed Test
[
]
0
0
0
0
:
:
or
:
a
a
H
H
H
η
η
η
η
η
η
=
>
<
Test statistic
:
S
= Number of sample measurements
greater than
0
η
[or
S
= number of
measurements less than
0
η
]
Two-Tailed Test
0
0
0
:
:
a
H
H
η
η
η
η
=
≠
Test Statistic
:
S
= Larger of
1
2
and
S
S
where
1
S
is the
number of measurements less than
0
η
and
2
S
is the number of measurements
greater than
0
η
[
Note
: Eliminate observation from the analysis that are exactly equal to the hypothesized median,
0
η
.]
Observed significance level
:
p
-value =
(
)
P x
S
≥
Observed significance level
:
p
-value =
(
)
2
P x
S
≥
where
x
has a binomial distribution with parameter
n
and
p
= .5.
(Use Table II, Appendix A.)
Rejection region
:
z
z
α
>
Rejection region
:
/2
z
z
α
>
Where tabulated
z
0values can be found in Table IV, Appendix A.

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