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QMDS 202 Data Analysis and Modeling

Chapter 15 Chi-Squared Tests

The Contingency Table Test

A Contingency table is one that shows all the classifications of the variables

 being studied, that is, it accounts for all contingencies in a particular situation.

For a contingency table that has r rows and c columns, the χ2  test can be

generalized as a test of:

a !ndependence or 

 b "omogeneity.

The test statistic #( )

∑  −

=i

ii

e

e f   2

2 χ 

v # (r – 1)(c – 1)

 g 

cr i

 f 

 f  f e   =

 f r  # row total

 f c # column total

 f  g  # grand total

For a test of independence we have one population and we are testing whether two characteristics in the population are independent. $ith a test of homogeneity we

have more than one population and we are testing to see if a characteristic is the same

across populations.

!ndependence Test

"%: A and B are independent

 A

a1   a2   a3

 B b1

a1∩b1 f 11

a2∩b1 f 12

a3∩b1 f 13   fr 1

b2

a1∩b2

 f 21

a2∩b2

 f 22

a3∩b2

 f 23   fr 2

 fc1   fc2   fc3   f  g 

!f A and B are independent, then

 g 

r 

 g 

c

  f  

  f  

  f  

  f  b P a P ba P    &&&&&&   '''   ×=×=∩

∴The e(pected fre)uency 'ei of 'a1∩b1

&

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  # g 

cr  g 

 g 

r 

 g 

c g 

  f  

  f    f    f  

  f  

  f  

  f  

  f    f  ba P    &&&&&&   '   =××=×∩

"omogeneity Test

"%:  P 'm1| A # P 'm1| B # P 'm1|C 

  and   P 'm2| A # P 'm2| B # P 'm2|C 

Samples

 A B C 

 M m1

'm1| A

 f 11

'm1| B

 f 12

'm1|C 

 f 13   fr 1

m2

'm2| A

 f 21

'm2| B

 f 22

'm2|C 

 f 23   fr 2

 fc1   fc2   fc3   f  g 

 fc1 # sample size of first sample

 fc2 # sample size of second sample

 fc3 # sample size of third sample

!f P 'm1| A# P 'm1| B# P 'm1|C , then all these probabilities e)ual to g 

r 

  f  

  f  m P    && '   =

.

*imilarly, P 'm2| A # P 'm2| B # P 'm2|C  # g 

r 

  f  

  f  m P    22 '   = .

∴ +(pected fre)uency 'ei of m& in sample A 

# ei of 'm1| A # P 'm1| A × sample size of A # &&

c

 g 

r    f    f  

  f  ×

  +(pected fre)uency 'ei of m2 in sample A 

# ei of 'm2| A # P 'm2| A × sample size of A # &2

c

 g 

r    f    f  

  f  ×

+(ample & A random sample of % men and -% women were ased whether or not

they lie a certain brand of product. Their responses are shown in the

following table:

/es 0o Total

1en & 2 %$omen 22 34 -%

Total 35 -3 &%%

6se the Chi7s)uare test to test that the liing of both se(es are not

related, using α # %.%.

*olution: "%: The liing of both se(es are not related 'independent

"&: The liing of both se(es are related 'dependentα # %.%

2

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χ27distribution will be used as the testing distribution.

r  # 2   c # 2

v # 'r  8 &'c 8 & # &

9eect "% if T* ; 3.4ei:

4.&,&%%

35,%=

×2.2

&%%

-3,%=

×

2.22&%%

35-%=

×4.35

&%%

-3-%=

×

( ) ( ) ( ) ( )%%52.%

4.35

4.3534

2.22

2.2222

2.2

2.22

4.&,

4.&,&  2222

=−

+−

+−

+−

=TS 

T* # %.%%52 is not greater than 3.4 ⇒ Cannot reect "%Conclusion: The liing of both se(es are not related.

+(ample 2 A business firm wants to determine whether the )uality of its product is

the same at all five of its production units. For this purpose it taes a

sample of &%% items at each of its production units and after testing the

items for )uality, obtains the following data. Test whether the

 probability of obtaining a satisfactory item is the same for all five

 production units, using α # %.%.

6nit & 6nit 2 6nit 3 6nit 6nit Total

*atisfactory items 42

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( ) ( ) ( )&5.&&

&2

&2&-...

44

44s product have a significant effect in

losing weight, using α # %.%&.

*olution: The re)uired contingency table:

"%:

"&:

α # %.%&

χ27distribution will be used as the testing distribution.

r  # c #

v # 'r  8 &'c 8 & #

9eect "% if T* ;

ei:

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T* #

*tatistical decision:

Conclusion:

9eview ?roblems: &.24, &.2