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A Simulated Chi Square GoodnessofFit test.
The model for the
N
plays of a game that has
k
disjoint out
comes, events that we shall denote by
E
1
,
,
E
k
, with corresponding prob
abilities
p
i
=
P
(
E
i
)
,
1
±
i
±
k
, where
P
k
i
=1
p
i
= 1
. In the
N
plays of this
game, we dneote
N
i
as the number of times that the event
E
i
occurs. Then
it is obvious that
X
k
i
=1
N
i
=
N
.
It should be noticed that each
N
i
has the
Bin
(
N;p
i
)
distribution. Thus,
E
(
N
i
N
) =
p
i
,
1
±
i
±
k
. For example, if the game consists in the toss of a
fair die once, then if we let
E
i
denote the event that the die comes up with
i
dots, then
P
(
E
i
) =
1
6
,
1
±
i
±
6
.
The problem that comes with this model is this. We know the value of
N
and we observe the values of each
N
i
. Given these numbers, we wish to
test whether the assumed values of
p
1
,
,
p
k
are indeed the correct values.
Here is a practical example of this problem. There are three cities, call
them
A
,
B
and
C
. Their corresponding populations are
100
;
000
for city
A
,
200
;
000
for city
B
and
300
;
000
for city
C
. Suppose that some rare disease
is striking the citizenry, and suppose in a given year
50
cases are diagnosed
in city
A
,
70
are diagnosed in city
B
, and
80
are diagnosed in city
C
. The
problem that might arise is whether the rate of occurrence of this disease
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This note was uploaded on 06/10/2008 for the course MATH 7 taught by Professor Tucker during the Spring '08 term at UC Irvine.
 Spring '08
 TUCKER
 Statistics

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