A. Zhensykbaev
Categorical data analysis
Categorical data analysis.
The binomial probability distribution analyzes the variables in one of two
responses (success or failure). Quantitative variables often allow more than
two categories for a response (for example, levels of education). Quantitative
data that fall in more more than two categories often result from a multino
mial experiment. Below we characterize it.
Multinomial probability distribution.
Properties of the multinomial experiment.
1. The experiment consists of
n
identical trials.
2. There are
k
possible outcomes to each trial.
3. The probabilities of the
k
outcomes,
p
1
, ...,
p
k
, remain the same from
trial to trial, and
p
1
+
· · ·
+
p
k
= 1
.
4. The trials are independent.
The number of observations that fall in each of the
k
classes denote as
n
1
, ...,
n
k
and
n
1
+
· · ·
+
n
k
=:
n.
Oneway Table.
The multinomial probability distribution is
P
(
x
) =
n
!
n
1
!
...n
k
!
p
n
1
1
...p
n
k
k
.
(this is the probability to obtain in
n
trials
n
1
times event
x
1
= 1
,...,
n
k
times event
x
k
=
k
). In particular, if
k
= 2
we have a binomial distribution.
Expected values of
x
= (
x
1
, ..., x
k
)
are
E
(
x
1
) =
np
1
, . . . , E
(
x
k
) =
np
k
.
We want to make inferences about the true proportions that occur in the
k
categories based on the sample information in the oneway table.
Test of a hypothesis about multinomial probabilities.
1
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A. Zhensykbaev
Categorical data analysis
We assume that the sample size
n
is large (this is satisfied if for every
i
the expected value
E
(
x
i
)
will be not less than 5.
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 Spring '13
 ChristopherStocker
 Math, Binomial, Probability, Probability theory, Binomial distribution, A. Zhensykbaev

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