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STAT 409
Examples for 11/09/2009
Fall 2009
Pearson’s
χ
2
Test
for
Goodness of Fit
(
Based on Large
n
)
A random sample of size
n
is classified into
k
categories or cells.
Let
Y
1
, Y
2
, … , Y
k
denote the respective cell frequencies.
n
k
i
i
=
∑
=
1
Y
.
Denote the cell probabilities by
p
1
,
p
2
, … ,
p
k
.
H
0
:
p
1
=
p
10
,
p
2
=
p
20
,
…
,
p
k
=
p
k
0
.
1
1
0
=
∑
=
k
i
i
p
.
1
2
…
k
Total
Observed frequency
O
Y
1
Y
2
…
Y
k
n
Probability under
H
0
p
10
p
20
…
p
k
0
1
Expected frequency
E
under
H
0
n p
10
n p
20
…
n p
k 0
n
Test Statistic:
( 29
( 29
( 29
∑
∑
∑

=

=

=
=
=

cells
k
i
i
i
i
k
i
i
i
i
k
p
n
p
n
E
E
O
E
E
O
2
1
2
1
0
2
0
1
Y
Q
Rejection Region:
Reject
H
0
if
Q
k
– 1
≥
2
α
χ
,
d.f. =
k
– 1 = (number of cells) – 1
Pearson’s
χ
2
test is an approximate test that is valid only for large samples.
As a rule of thumb,
n
should be large enough so that expected frequency of each cell
is at least 5.
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Alex buys a package of Sour Jelly Beans.
On the package, it says that 50% of all
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 Spring '11
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