1 Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison...

1Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICS Section 10-3 Contingency Tables: Independence

and Homogeneity


Definition

Contingency Table (or two-way frequency table)

a table in which frequencies correspond to two variables.

(One variable is used to categorize rows, and a second variable is used to categorize columns.)


Definition Contingency Table (or two-way frequency table)

a table in which frequencies correspond to two variables.

(One variable is used to categorize rows, and a second variable is used to categorize columns.)

Contingency tables have at least two rows and at least two columns.


Test of Independence

tests the null hypothesis that the row variable and column variable in a contingency table are not related. (The null hypothesis is the

statement that the row and column variables are independent.)

Definition


Assumptions1. The sample data are randomly selected.

2.The null hypothesis H0 is the statement that the

row and column variables are independent; the alternative hypothesis H1 is the statement that the

row and column variables are dependent.

3. For every cell in the contingency table, the expected frequency E is at least 5. (There is no requirement that every observed frequency must be at least 5.)


Test of IndependenceTest Statistic

Critical Values

1. Found in Table A-4 using

degrees of freedom = (r - 1)(c - 1)

r is the number of rows and c is the number of columns

2. Tests of Independence are always right-tailed.

X2 = (O - E)2

E


(row total) (column total)

(grand total)E =

Total number of all observed frequencies

in the table


Tests of Independence

H0: The row variable is independent of the column variable

H1: The row variable is dependent (related to) the column variable

This procedure cannot be used to establish a direct cause-and-effect link between variables in question.

Dependence means only there is a relationship between the two variables.


Expected Frequency for Contingency Tables


E = • •row total column total

grand total


grand totalgrand total


n • p


grand total



(probability of a cell)


n • p


grand total





n • p


grand total




E = (row total) (column total)(grand total)


Is the type of crime independent of whether the criminal is a stranger?

Stranger

Acquaintanceor Relative

12

39

379

106

727

642

Homicide Robbery Assault


Row Total

Column Total

Stranger


1118

787

1905

12

39

51

379

106

485

727

642

1369




Row Total

Column Total


Stranger




1118

787

1905

12

39

51

379

106

485

727

642

1369


Row Total

(29.93)

Column Total


E = (1118)(51)

1905= 29.93

Stranger




1118

787

1905

12

39

51

379

106

485

727

642

1369


Row Total

(29.93)

(21.07)

(284.64)

(200.36)

(803.43)

(565.57)

Column Total


E = (1118)(51)

1905= 29.93 E =

(1118)(485)

1905= 284.64

etc.

Stranger




1118

787

1905

12

39

51

379

106

485

727

642

1369


12

39

379

106

727

642

Homicide Robbery Forgery

(29.93)

(21.07)

(284.64)

(200.36)

(803.43)

(565.57

[10.741]Stranger

Acquaintance

or Relative

X2 = (O - E )2

E

(O -E )2

EUpper left cell: = = 10.741

(12 -29.93)2

29.93

(E)

(O - E )2

E



12

39

379

106

727

642


(29.93)

(21.07)[15.258]

(284.64)[31.281]

(200.36)[44.439]

(803.43)[7.271]

(565.57)[10.329]

[10.741]Stranger

Acquaintance

or Relative

X2 = (O - E )2

E

(O -E )2

EUpper left cell: = = 10.741

(12 -29.93)2

29.93

(E)

(O - E )2

E



12

39

379

106

727

642


(29.93)

(21.07)[15.258]

(284.64)[31.281]

(200.36)[44.439]

(803.43)[7.271]

(565.57)[10.329]

[10.741]Stranger

Acquaintance

or Relative

X2 = (O - E )2

E

(E)

(O - E )2

E


Test Statistic X2 = 10.741 + 31.281 + ... + 10.329 =

119.319


Test Statistic X2 = 119.319

with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2 degrees of freedom

Critical Value X2 = 5.991 (from Table A-4)




0

= 0.05

X2 = 5.991

RejectIndependence


Sample data: X2 =119.319

Fail to RejectIndependence




0

= 0.05

X2 = 5.991

RejectIndependence


Reject independence






0

= 0.05

X2 = 5.991

RejectIndependence


Reject independence



Claim: The type of crime and knowledge of criminal are independentHo : The type of crime and knowledge of criminal are independent H1 : The type of crime and knowledge of criminal are dependent




It appears that the type of crime and knowledge of the criminal are related.

0

= 0.05

X2 = 5.991

RejectIndependence


Reject independence




Figure 10-8

Relationships Among Components in X2 Test of Independence


Definition

Test of Homogeneity test the claim that different

populations have the same proportions of some characteristics


How to distinguish between a test of homogeneity and a

test for independence:

Were predetermined sample sizes used for different populations (test of homogeneity), or was one big sample drawn so both row and column totals were determined randomly (test of independence)?

1 Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison...

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