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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
2Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.)
3Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
4Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
5Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.)
6Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
7Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
(row total) (column total)
(grand total)E =
Total number of all observed frequencies
in the table
8Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
9Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Expected Frequency for Contingency Tables
10Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
E = • •row total column total
grand total
Expected Frequency for Contingency Tables
grand totalgrand total
11Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n • p
E = • •row total column total
grand total
Expected Frequency for Contingency Tables
grand totalgrand total
(probability of a cell)
12Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n • p
E = • •row total column total
grand total
Expected Frequency for Contingency Tables
grand totalgrand total
(probability of a cell)
13Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n • p
E = • •row total column total
grand total
Expected Frequency for Contingency Tables
grand totalgrand total
(probability of a cell)
E = (row total) (column total)(grand total)
14Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
15Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Row Total
Column Total
Stranger
Acquaintanceor Relative
1118
787
1905
12
39
51
379
106
485
727
642
1369
Homicide Robbery Assault
Is the type of crime independent of whether the criminal is a stranger?
16Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Row Total
Column Total
E = (row total) (column total)(grand total)
Stranger
Acquaintanceor Relative
Homicide Robbery Assault
Is the type of crime independent of whether the criminal is a stranger?
1118
787
1905
12
39
51
379
106
485
727
642
1369
17Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Row Total
(29.93)
Column Total
E = (row total) (column total)(grand total)
E = (1118)(51)
1905= 29.93
Stranger
Acquaintanceor Relative
Homicide Robbery Assault
Is the type of crime independent of whether the criminal is a stranger?
1118
787
1905
12
39
51
379
106
485
727
642
1369
18Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Row Total
(29.93)
(21.07)
(284.64)
(200.36)
(803.43)
(565.57)
Column Total
E = (row total) (column total)(grand total)
E = (1118)(51)
1905= 29.93 E =
(1118)(485)
1905= 284.64
etc.
Stranger
Acquaintanceor Relative
Homicide Robbery Assault
Is the type of crime independent of whether the criminal is a stranger?
1118
787
1905
12
39
51
379
106
485
727
642
1369
19Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
Is the type of crime independent of whether the criminal is a stranger?
20Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
39
379
106
727
642
Homicide Robbery Forgery
(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
Is the type of crime independent of whether the criminal is a stranger?
21Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
39
379
106
727
642
Homicide Robbery Forgery
(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
Is the type of crime independent of whether the criminal is a stranger?
Test Statistic X2 = 10.741 + 31.281 + ... + 10.329 =
119.319
22Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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)
23Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X2 = 119.319
with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2 degrees of freedom
0
= 0.05
X2 = 5.991
RejectIndependence
Critical Value X2 = 5.991 (from Table A-4)
Sample data: X2 =119.319
Fail to RejectIndependence
24Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X2 = 119.319
with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2 degrees of freedom
0
= 0.05
X2 = 5.991
RejectIndependence
Critical Value X2 = 5.991 (from Table A-4)
Reject independence
Sample data: X2 =119.319
Fail to RejectIndependence
25Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X2 = 119.319
with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2 degrees of freedom
0
= 0.05
X2 = 5.991
RejectIndependence
Critical Value X2 = 5.991 (from Table A-4)
Reject independence
Sample data: X2 =119.319
Fail to RejectIndependence
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
26Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X2 = 119.319
with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2 degrees of freedom
It appears that the type of crime and knowledge of the criminal are related.
0
= 0.05
X2 = 5.991
RejectIndependence
Critical Value X2 = 5.991 (from Table A-4)
Reject independence
Sample data: X2 =119.319
Fail to RejectIndependence
27Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Figure 10-8
Relationships Among Components in X2 Test of Independence
28Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Definition
Test of Homogeneity test the claim that different
populations have the same proportions of some characteristics
29Chapter 10. Section 10-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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)?